# Second Order Elliptic Equations and Hodge-Dirac Operators

Erik Duse

###### Abstract

In this paper we show how a second order scalar uniformly elliptic equation on
divergence form with measurable coefficients and Dirichlet boundary conditions
can be transformed into a first order elliptic system with half-Dirichlet
boundary condition. This first order system involves Hodge-Dirac operators and
can be seen as a natural generalization of the Beltrami equation in the plane
and we develop a theory for this equation, extending results from the plane to
higher dimension. The reduction to a first order system applies both to linear
as well as quasilinear second order equations and we believe this to be of
independent interest. Using the first order system, we give a new
representation formula of the solution of the Dirichlet problem both on simply
and finitely connected domains. This representation formula involves only
singular integral operators of convolution type and Neumann series there of,
for which classical Calderón-Zygmund theory is applicable. Moreover, no use is
made of any fundamental solution or Green’s function beside fundamental
solutions of constant coefficient operators. Remarkably, this representation
formula applies also for solutions of the fully non-linear first order system.
We hope that the representation formula could be used for numerically solving
the equations. Using these tools we give a new short proof of Meyers’ higher
integrability theorem. Furthermore, we show that the solutions of the first
order system are Hölder continuous with the same Hölder coefficient as the
solutions of the second order equations. Finally, factorization identities and
representation formulas for the higher dimensional Beurling-Ahlfors operator
are proven using Clifford algebras, and certain integral estimates for the
Cauchy transform is extended to higher dimensions.

###### Contents

  1. 1 Introduction
    1. 1.1 Introduction
    2. 1.2 Main Results
    3. 1.3 Relation to Similar Results in the Literature
  2. 2 Notational Conventions
  3. 3 Algebraic preliminaries
    1. 3.1 Exterior and Clifford Algebra
    2. 3.2 Relation between Bivectors and Complex Structures
    3. 3.3 Exterior Extensions of Linear Maps and Grassmann Duality
  4. 4 Analytical preliminaries
    1. 4.1 Tangential and Normal Trace Maps.
    2. 4.2 Hodge-Dirac Operators on Euclidean Domains
    3. 4.3 Cauchy-Pompeiu Formula for the Hodge-Dirac Operators
  5. 5 Topological preliminaries
    1. 5.1 Harmonic Fields and De Rham Cohomology
  6. 6 Reduction of Second Order Equation to Dirac-Beltrami Equation
    1. 6.1 The Dirichlet Problem
    2. 6.2 Hodge Theory and Gauge Conditions
    3. 6.3 Converse Direction
    4. 6.4 Comparison to the two dimensional case
    5. 6.5 Multiply connected domains and inhomogeneous equations
    6. 6.6 Nonlinear uniformly elliptic equations
    7. 6.7 Ellipticity of the Dirac–Beltrami equation and Null-Lagrangian
    8. 6.8 Hodge Duality
  7. 7 The Cauchy Transform on ${\mathbb{R}}^{n}$.
    1. 7.1 Dirac Operators and the Cauchy Transforms
    2. 7.2 Hölder estimates for the Cauchy Transforms
    3. 7.3 BMO estimates for the Cauchy Transform
    4. 7.4 Compactness Properties of the Cauchy Transforms
  8. 8 The Beurling-Ahlfors and Riesz Transforms on ${\mathbb{R}}^{n}$.
    1. 8.1 The Beurling-Ahlfors Transform on ${\mathbb{R}}^{n}$.
    2. 8.2 Factorisation formula
    3. 8.3 Connection to the Cauchy Transforms on ${\mathbb{R}}^{n}$.
  9. 9 Hodge-Dirac Equation with Normal and Tangential Boundary Conditions
    1. 9.1 Tangential to Normal Hilbert Transform.
    2. 9.2 Inhomogenous Hodge–Dirac Equation
    3. 9.3 Dirichlet Problem for Laplace Equation
  10. 10 Representation Formulas for the Solution to the Dirichlet Problem
    1. 10.1 Reduction of Homogeneous Dirac–Beltrami Equation to an an Inhomogeneous Equation
    2. 10.2 Tangential and Normal Beurling–Ahlfors Operators
    3. 10.3 Reduction of Inhomogeneous Dirac–Beltrami Equation to an Inhomogeneous Hodge–Dirac Equation
    4. 10.4 Tangential Cauchy Transform
    5. 10.5 Representation Formula for the Solution of the Dirichlet Problem
    6. 10.6 Representation formula for solutions in the non-linear uniformly elliptic case
  11. 11 Local Estimates
    1. 11.1 Higher Integrability and Caccioppoli Inequalities
    2. 11.2 Hölder Regularity
  12. A Auxiliary Algebraic Results
    1. A.1 Cayley Transform
    2. A.2 Ellipticity Bounds
    3. A.3 Nonlinear Cayley Transform
  13. B Iteration of Sobolev Conjugate Exponent
  14. C Auxiliary Analytic Results
    1. C.1 Integration by Parts Formula
    2. C.2 Poincaré Inequalities
    3. C.3 Isoperimetric Type Inequalities

##  1 Introduction

###  1.1 Introduction

The aim of this paper is to show that contrary to common belief, there exists
a natural extension of complex analytic and quasiconformal methods to
dimensions greater than 2 in the study linear elliptic partial differential
equations. Using these methods we will derive global representation formulas
for solutions of the Dirichlet problem for scalar linear uniformly elliptic
partial differential equations with measurable coefficients on arbitrary
simply connected Lipschitz domains. In particular, this will prove existence
and uniqueness for weak solutions and also derive local interior estimates of
higher integrability, Hölder continuity and differentiability almost
everywhere.

Now what do we mean by complex analytic and quasiconformal methods? To set the
stage we are interested in the Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{ll}\text{div}A(x)\nabla
u(x)=0,&x\in\Omega,\\\ u(x)=\phi(x),&x\in\partial\Omega.\end{array}\right.$
(1.3)

where $\Omega$ is a simply connected Lipschitz domain, $A(x)\in
L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$ and $\phi(x)\in
W^{1/2,2}(\partial\Omega)$. We say that $u$ is a weak solution of (1.3) if for
all $\varphi\in C^{\infty}_{0}(\Omega)$ we have

$\displaystyle\int_{\Omega}\langle A(x)\nabla u(x),\nabla\varphi(x)\rangle
dx=0.$ (1.4)

In the plane we can introduce a and $A$-harmonic conjugate $v$ of $u$ by first
defining

$\displaystyle B(z)=A(z)\nabla u(z)$

where $\star$ is the _Hodge star map_ given by the counterclockwise rotation

$\displaystyle\star=\begin{bmatrix}0&-1\\\
1&0\end{bmatrix}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$

and $z=x+iy$.

If we set $E(z)=\nabla u(z)$ where $u$ is a solution to (1.3) in the plane,
then $E$ is a curl free vector field whereas $B$ is divergence free. Since the
Hodge star map interchanges curl and divergence free vector fields, it follows
that $\star B(z)$ is curl free and since $\Omega$ is simply connected by
assumption, $\star B$ potential $v$ such that $\nabla v(z)=\star A(z)\nabla
u(z)$ unique up to a constant. Moreover, the conjugate $v$ solves the second
order linear elliptic equation

$\displaystyle\text{div}\,A^{\star}(z)\nabla v(z)=0$

in $\Omega$, where

$\displaystyle A^{\star}(z)=\star^{t}A^{-1}(z)\star.$

Using $v$ we define the complex valued function $f(z):=u(z)+iv(z)$. Now we
define the pair of fields

$\displaystyle\mathcal{F}^{+}(z)$ $\displaystyle=E(z)+B(z)=(I+A(z))E(z),$
$\displaystyle\mathcal{F}^{-}(z)$ $\displaystyle=E(z)-B(z)=(I-A(z))E(z).$

This gives the algebraic equation

$\displaystyle\mathcal{F}^{-}(z)=\mathcal{M}(z)\mathcal{F}^{+}(z)$ (1.5)

relating the fields $\mathcal{F}^{+}$ and $\mathcal{F}^{-}$, where
$\mathcal{M}(z)=(I-A(z))(I+A(z))^{-1}$ is the Cayley transform of $A(z)$. On
the other hand a direct computation gives

$\displaystyle 2f_{\overline{z}}$ $\displaystyle=u_{x}-v_{y}+i(v_{x}-u_{y}),$
$\displaystyle 2\overline{f_{z}(z)}$
$\displaystyle=u_{x}+v_{y}+i(u_{y}-v_{x})$

and

$\displaystyle 2\mathcal{F}^{+}(z)$ $\displaystyle=(u_{x}-v_{y},v_{x}-u_{y}),$
$\displaystyle 2\mathcal{F}^{-}(z)$ $\displaystyle=(u_{x}+v_{y},u_{y}-v_{x}),$

so we have $f_{\overline{z}}=\mathcal{F}^{-}(z)$ and
$\overline{f_{z}(z)}=\mathcal{F}^{-}(z)$. Combining this with the algebraic
equation (1.5), we get the Beltrami equation

$\displaystyle f_{\overline{z}}=\mathcal{M}(z)\overline{f_{z}(z)}.$ (1.6)

Since any linear map $L\in\mathcal{L}({\mathbb{C}})$ can be written on the
form

$\displaystyle Lz=\alpha z+\beta\overline{z}$

for two complex numbers $\alpha,\beta\in{\mathbb{C}}$, simple algebra shows
that (1.6) can be written as an ${\mathbb{R}}$-linear Beltrami equation on the
standard form

$\displaystyle f_{\overline{z}}=\mu(z)f_{z}(z)+\nu(z)\overline{f_{z}(z)}.$
(1.7)

The advantage of working with the first order system (1.7) over the second
order equation (1.3) is that one can use the full machinery of complex
potential operators and quasiconformal methods to study (1.7). It is therefore
natural to ask if something similar can be done in higher dimension. We will
show in this paper that this is indeed the case. However, it will also be
shown that the higher dimensional case is in many respects more subtle and
difficult.

A first key step towards a generalization of equation (1.5) to higher
dimension with the purpose of studying scalar elliptic linear as well as non-
linear partial differential equations was taken by Iwaniec and Sbordone in
their paper [29], where they introduced the notion of _quasiharmonic fields_.
To explain this concept we first observe that the vector field
$B(x)=A(x)\nabla u(x)=0$ is divergence free in the sense of distributions
whenever $u$ is a weak solution to $\text{div}A(x)\nabla u(x)=0$. Of course
the field $E(x)=\nabla u(x)$ is curl free in the sense of distributions, and
therefore the pair $(B,E)$ form a so called div-curl couple in the terminology
of compensated compactness, e.g. [57]. If a symmetric endomorphism field
$A(x)$ (i.e. $A^{\ast}(x)=A(x)$) satisfies the symmetric ellipticity bounds,

$\displaystyle K^{-1}|v|^{2}\leq\langle A(x)v,v\rangle\leq
K|v|^{2}\,\,\,\text{ for a.e. $x\in\Omega$}$

then one can show that the double sided condition is equivalent to the one-
sided inequality

$\displaystyle|v|^{2}+|A(x)v|^{2}\leq(K+K^{-1})\langle
A(x)v,v\rangle\,\,\,\text{ for a.e. $x\in\Omega$}.$

Letting $\mathcal{K}=K+K^{-1}$ this inequality implies that any weak solution
$u$ give rise to the inequality

$\displaystyle|E(x)|^{2}+|B(x)|^{2}\leq\mathcal{K}\langle E(x),B(x)\rangle$
(1.8)

for the div-curl couple $(B,E)$. Generalising this in [29], one say that any
div-curl couple $(B,E)\in
L^{2}(\Omega,{\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ is a
$\mathcal{K}$-quasiharmonic field if $\text{div }B(x)=0$ and $\text{curl
}E(x)=0$ in the sense of distribution and $(B,E)$ satisfies (1.8). Just as in
the plane one introduces the fields $\mathcal{F}^{\pm}=E\pm B$, which again
gives the algebraic relation

$\displaystyle\mathcal{F}^{-}(x)=(I-A(x))(I+A(x))^{-1}\mathcal{F}^{+}(x).$

The novelty of the present paper is that using Hodge decompositions one can
impose a gauge condition so that the there exists a multivector field $F$ with
the property that

$\displaystyle\mathcal{F}^{-}(x)=\mathcal{D}^{\tiny-}F(x),\,\,\,\,\mathcal{F}^{+}(x)=\mathcal{D}^{\tiny+}F(x),$

where $(\mathcal{D}^{\tiny-},\mathcal{D}^{\tiny+})$ is a pair of Hodge-Dirac
operators, one formally self-adjoint and the other formally skew-adjoint,
taking the role of the Wirtinger derivatives
$(\partial_{\overline{z}},\overline{\partial_{z}})$ in the plane. More
precisely, the Hodge-Dirac operators are related to the exterior derivative on
multivectors (i.e. differential forms) through

$\displaystyle\mathcal{D}^{\pm}=d\pm\delta,$

where $\delta=-d^{\ast}$ is the interior derivative. The benefit of
introducing the potential field $F$ is that this allows one to use powerful
integral operator techniques form Clifford and harmonic analysis to study the
original Dirichlet problem. As a by product, we will see that this also allows
one to generalize much of the quasiconformal methods in the plane used to to
study second order elliptic equations to higher dimensions.

It is interesting to note that Clifford analysis techniques have also been
used successfully in [61, 62] to prove global well-posedness of the water wave
problem in 3 dimensions.

###  1.2 Main Results

######  Theorem 1.1.

Let $\Omega\subset{\mathbb{R}}^{n}$ with $n>2$ be a bounded simply connected
$C^{2}$-domain. Furthermore let $A\in
L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n})$ be a normal matrix that
satisfies the ellipticity bounds

$\displaystyle\lambda|v|^{2}\leq\langle
A(x)v,v\rangle,\quad\Lambda|v|^{2}\leq\langle A^{-1}(x)v,v\rangle$ (1.9)

for all most every $x\in\Omega$ and every $v\in{\mathbb{R}}^{n}$. Let $u\in
W^{1,2}(\Omega)$ be any weak solution of

$\displaystyle\left\\{\begin{array}[]{ll}\text{div}\,A(x)\nabla
u(x)=0,&x\in\Omega,\\\ u(x)=\phi(x)\in
W^{1/2,2}(\partial\Omega),&x\in\partial\Omega.\end{array}\right.$ (1.12)

Define the bivector field $v\in W^{1,2}(\Omega,\Lambda^{2}{\mathbb{R}}^{n})$
to be the unique solution to the gauge condition equation

$\displaystyle\left\\{\begin{array}[]{ll}\delta v(x)=A(x)\nabla
u(x),&x\in\Omega,\\\ dv(x)=0,&x\in\Omega,\\\
v_{T}(x)=0,&x\in\partial\Omega,\end{array}\right.$ (1.16)

where $v_{T}$ denotes the tangential trace. Define the multivector field
$F=u+v\in W^{1,2}(\Omega,\Lambda^{(0,2)}{\mathbb{R}}^{n})$ and let

$\displaystyle\mathcal{M}(x)=(I-A(x))\circ(I+A(x))^{-1}$

denote the Cayley transform of $A$ and let $\widehat{\mathcal{M}}(x)$ denote
the exterior extension of $\mathcal{M}(x)$ defined in Definition 3.2. Then $F$
solves the half-Dirichlet problem for the Dirac-Beltrami equation

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny-}F(x)=\widehat{\mathcal{M}}(x)\mathcal{D}^{\tiny+}F(x)&x\in\Omega,\\\
F_{T}(x)=\phi(x)\in
W^{1/2,2}(\partial\Omega),&x\in\partial\Omega.\end{array}\right.$ (1.19)

Conversely, let $F\in W^{1,2}(\Omega,\Lambda^{(0,2)}{\mathbb{R}}^{n})$ be a
solution of (1.19) for some $\widehat{\mathcal{M}}(x)$ being the exterior
extension of a linear map $\mathcal{M}\in
L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$ satisfying the uniform
ellipticity condition

$\displaystyle\|\|\mathcal{M}(x)\|\|_{\infty}=M<1.$

Here $\|\mathcal{M}(x)\|$ denotes the operator norm of the linear map
$\mathcal{M}(x)$. Then the function $u=\langle F\rangle_{0}\in
W^{1,2}(\Omega,{\mathbb{R}})$ solves (1.12) with

$\displaystyle A(x)=(I-\mathcal{M}(x))\circ(I+\mathcal{M}(x))^{-1}.$

######  Remark 1.1.

The same results also applies to a Lipschitz domain $\Omega$, the only
difference is that the field $F$ need not belong to
$W^{1,2}(\Omega,\Lambda^{(0,2)})$. In this case $F$ belongs to the partial
Sobolev space

$\displaystyle F\in
W^{1,2}_{d,\delta}(\Omega,\Lambda^{(0,2)})=\\{F\in\mathscr{D}^{\prime}(\Omega,\Lambda^{(0,2)}):F,dF,\delta
F\in L^{2}(\Omega,\Lambda)\\}.$

Alternatively we may always change domain using a Lipschitz homeomorphism from
a Lipschitz domain to a $C^{2}$-domain. For a similar result for class of
uniformly non-linear elliptic second order equations see subsection 6.6.

For the corresponding statements in the case when $\Omega$ is finitely
connected or we have an inhomogeneous equation see Theorem 6.4 and Theorem 6.5
respectively. We now give solvability results for the general Dirac–Beltrami
equation for a general field $F\in W^{1,2}_{d,\delta}(\Omega,\Lambda)$ not
necessarily associated to a scalar second order equation where the solvability
cannot be deduced from the solvability of the scalar equation.

######  Theorem 1.2 (Solvability of the Dirac–Beltrami equation).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded simply connected Lipschitz
domain and let $\mathcal{M}\in
L^{\infty}(\Omega,\mathcal{L}(\Lambda{\mathbb{R}}^{n}))$ satisfying the
uniform ellipticity condition

$\displaystyle\|\|\mathcal{M}(x)\|\|_{\infty}=M<1.$

Then the equation

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)&x\in\Omega,\\\
F_{T}(x)=\phi(x)\in
W^{1,1/2}(\partial\Omega),&x\in\partial\Omega.\end{array}\right.$

has a solution $F\in W^{1,2}_{d,\delta}(\Omega,\Lambda^{(1,2)})$ for every
$\phi\in W^{1,1/2}(\Omega)$, unique up to elements in
$\mathcal{H}_{T}(\Omega)$.

######  Theorem 1.3.

The unique solution to the half-Dirichlet problem (1.19) is given by

$\displaystyle
F(x)=\mathcal{C}_{T}^{+}\circ\Pi_{\mathcal{M}}\mathcal{D}^{\tiny+}H(x)+H(x)$
(1.20)

where

$\displaystyle\Pi_{\mathcal{M}}$
$\displaystyle=(I-\mathcal{S}_{T}\circ\mathcal{M})^{-1}\circ\mathcal{S}_{T}\mathcal{M}$

and

$\displaystyle H(x)=\mathcal{E}^{-}\circ(I+\mathcal{H}_{TN}^{-})\phi.$

and $\mathcal{S}_{T}$ is the tangential Beurling-Ahlfors transform from
Definition 10.1, $\mathcal{C}^{+}_{T}$ is the tangential Cauchy transform from
Definition 10.2, $\mathcal{H}^{-}_{TN}$ is the tangential to normal Hilbert
transform from Definition 9.2 and $\mathcal{E}^{-}$ is the Cauchy integral
from Definition 4.5. In addition, we get a representation formula for the
unique solution to the Dirichlet problem 1.12 through

$\displaystyle u(x)=\langle F(x)\rangle_{0}.$ (1.21)

######  Remark 1.2.

For the solvability and representation formula for the boundary value problem
of the fully nonlinear Dirac–Beltrami equation see Theorem 10.2.

######  Theorem 1.4.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded domain and assume that $n\geq
3$ is odd. Let $\mathcal{M}\in
L^{\infty}(\mathcal{L}(\Lambda{\mathbb{R}}^{n}))$ and set
$M=\|\|\mathcal{M}(x)\|\|_{L^{\infty}}(\Omega)$. If

$\displaystyle M\|\mathcal{S}\|_{2n}<1,$

where $\|\mathcal{S}\|_{2n}$ denotes the norm of the Beurling-Ahlfors
transform on $L^{2n}(\Omega,\Lambda{\mathbb{R}}^{n})$, then any solution $F\in
W^{1,2}_{d,\delta}(\Omega,\Lambda{\mathbb{R}}^{n})$ of the Dirac–Beltrami
equation

$\displaystyle\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x),$

is locally Hölder continuous with Hölder coefficient $\alpha=1/2$.

######  Remark 1.3.

Note that in this generality the Hölder continuity of $F$ does not follow from
De Giorgi-Nash-Moser theory.

###  1.3 Relation to Similar Results in the Literature

The resolution of Kato’s square root conjecture in the seminal paper [3] have
given rise to a new developments in the solvability and well-posedness for
elliptic systems. In particular, this allowed for the bounded
$\mathcal{H}^{\infty}$-calculus of unbounded operators as developed by A.
McIntosh and collaborators to be used in [4, 5], to establish solvability and
well-posedness for very general boundary values of general second order
elliptic _systems_ of the form

$\displaystyle\mathcal{L}^{\alpha}u(t,x)=\sum_{i,j=0}^{n}\sum_{\beta=1}^{m}\partial_{i}\bigg{(}A_{i,j}^{\alpha,\beta}(t,x)\partial_{j}u^{\beta}(t,x)\bigg{)}=0,\quad\alpha=1,2,..,m,$
(1.22)

in the upper-half space
${\mathbb{R}}^{n+1}_{+}=\\{(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{n}:t>0\\}$,
where $\partial_{0}=\partial_{t}$, and $\partial_{j}=\frac{\partial}{\partial
x_{j}}$, $j=1,2,...,n$. It also allowed for the study regularity up to the
boundary for very general boundary conditions. Here $A\in
L^{\infty}({\mathbb{R}}^{n+1}_{+},\mathcal{L}({\mathbb{C}}^{(1+n)m}))$ and
satisfying the strict accretivity condition $(3)$ in [5]. Here complex valued
vector fields are allowed. At this level of generality solutions need not be
continuous and De Giorgi-Nash theory does not apply. Moreover, in [4, 5]
representation formulas for the Dirichlet problem for (1.22) are derived using
functional calculus for unbounded _non-selfadjoint_ bisectorial operators. The
method in [4, 5] is as in this work to reduce (1.22) to a first order system.
This equation is however very different from the Dirac–Beltrami equation
(1.19) in the present paper, and the techniques used in this paper are very
different. In particular we do not need to use any advanced functional
calculus for the representation formula (1.20) beyond Neumann series.
Moreover, our methods also apply to finitely connected domains and nonlinear
equations which are beyond the techniques in [5]. On the other hand, the
methods in [4, 5] answer different questions, in particular solvability and
boundary regularity in much more general spaces than $W^{1,2}$, and applies to
second order systems and boundary value problems to which the methods in this
paper cannot presently be used.

### Acknowledgements

Erik Duse was supported by the Knut and Alice Wallenberg Foundation KAW grant
2016.0416. The author thanks Björn Gustafsson for providing the proof of Lemma
7.3.

##  2 Notational Conventions

We will let $\mathscr{H}^{k}$ denote the $k$-dimensional Hausdorff measure
normalized so that $\mathscr{H}^{n}=\mathscr{L}^{n}$ where $\mathscr{L}^{n}$
is the Lebesgue measure of ${\mathbb{R}}^{n}$. Furthermore, for any Lipschitz
hypersurface $\Sigma\subset{\mathbb{R}}^{n}$ we will always let
$\sigma=\mathscr{H}^{n-1}\lfloor\Sigma$ denote the surface measure of $\Sigma$
without explicitly referring to $\Sigma$, i.e., we will omit any subscript
$\sigma_{\Sigma}$.

In what follows we will let $\sigma_{n-1}$ denote the $n-1$-dimensional volume
of the unit sphere $\mathbb{S}^{n-1}=\\{x\in{\mathbb{R}}^{n}:|x|=1\\}$ in
${\mathbb{R}}^{n}$ and $\omega_{n}$ the volume of the unit ball
$\mathbb{B}^{n}=\\{x\in{\mathbb{R}}^{n}:|x|<1\\}$ in ${\mathbb{R}}^{n}$.

If $V$ is a finite dimensional euclidean vector space we let $\mathcal{L}(V)$
denote the space of linear operators on $V$. If $A\in\mathcal{L}(V)$ we let
$\|A\|$ denote the operator norm of $A$ and $|A|$ the Hilbert-Schmidt norm of
$A$, i.e.,

$\displaystyle\|A\|$ $\displaystyle=\sup_{|x|=1}|Ax\|,$ $\displaystyle|A|$
$\displaystyle=\sqrt{\langle A,A\rangle}=\sqrt{\text{tr}(A^{\ast}A)}.$

Define the Hilbert space

$\displaystyle
L^{2}(\partial\Omega,\Lambda):=L^{2}(\partial\Omega,\sigma;\Lambda{\mathbb{R}}^{n})$

with the $L^{2}$-inner product

$\displaystyle\langle f,g\rangle=\int_{\partial\Omega}\langle f(x),g(x)\rangle
d\sigma(x),$

Similarly, we let

$\displaystyle
L^{2}(\partial\Omega,\Lambda^{k}):=L^{2}(\partial\Omega,\sigma;\Lambda^{k}{\mathbb{R}}^{n}),\quad
L^{2}(\partial\Omega,\Lambda^{(i_{1},i_{2},...,i_{p})}):=L^{2}(\partial\Omega,\sigma;\bigoplus_{l=1}^{p},\Lambda^{i_{l}}{\mathbb{R}}^{n}).$

We also write

$\displaystyle
W^{1,k}(\partial\Omega,\Lambda):=W^{1,k}(\partial\Omega,\Lambda{\mathbb{R}}^{n}),\quad
W^{1,k}(\partial\Omega,\Lambda^{p}):=W^{1,k}(\partial\Omega,\Lambda^{p}{\mathbb{R}}^{n}).$

for the classical Sobolev spaces. Since the letter $H$ will be used to denote
de Rham and Hardy spaces in this paper, in order to avoid confusion we will
not use the notation $H^{1}(\Omega,\Lambda)=W^{1,2}(\Omega,\Lambda)$ and
$H^{1/2}(\partial\Omega,\Lambda)=W^{1/2,2}(\partial\Omega,\Lambda)$ for the
trace space.

##  3 Algebraic preliminaries

###  3.1 Exterior and Clifford Algebra

In this section we give a brief introduction to the relevant algebraic
structures that will be used later on. For a more complete account we refer
the reader to [51].

Let $V$ be an $n$-dimensional real inner product vector space with inner
product denoted by $\langle\cdot,\cdot\rangle$. We denote by $\Lambda V$ the
exterior algebra of $V$ and by abuse of notation we still denote by
$\langle\cdot,\cdot\rangle$ the _induced_ euclidean inner product on $\Lambda
V$ given by the Gram determinant

$\displaystyle\langle u_{1}\wedge...\wedge u_{k},v_{1}\wedge...\wedge
v_{k}\rangle=\det[\langle u_{j},v_{j}\rangle]_{i,j=1}^{k}$

for $u_{1},...,u_{k},v_{1},...,v_{k}\in V$.

Let $\overline{n}:=\\{1,2,...,n\\}$. For an ordered subset
$s=\\{s_{1}<s_{2}<...<s_{k}\\}\subset\overline{n}$ we denote by $|s|=k$ its
cardinality. If $\\{e_{j}\\}_{j=1}^{n}$ is an ON-basis for $V$, we denote by
$\\{e_{s}\\}_{s\subset{\bar{n}}}$ the induced $ON$-basis for $\Lambda V$, with

$\displaystyle e_{s}:=e_{s_{1}}\wedge....\wedge e_{s_{k}}.$

Following [51], we define the left and right _adjoint product_ of $\wedge$,
called _left interior_ and _right interior multiplication_ , denoted
$\,\,\lrcorner\,\,$ and $\,\,\llcorner\,\,$ respectively, so that for all
$\Theta,w_{1},w_{2}\in\Lambda V$ we have

$\displaystyle\langle\Theta\,\,\lrcorner\,\,w_{1},w_{2}\rangle$
$\displaystyle=\langle w_{1},\Theta\wedge w_{2}\rangle,$ $\displaystyle\langle
w_{1}\,\,\llcorner\,\,\Theta,w_{2}\rangle$ $\displaystyle=\langle
w_{1},w_{2}\wedge\Theta\rangle$

Let $q$ be the associated quadratic form to the inner product
$\langle\cdot,\cdot\rangle$. We define the Clifford algebra $\Delta V^{q}$ to
be the associative algebra generated by the relations

$\displaystyle
e_{i}\text{\tiny{$\triangle$}}e_{j}+e_{j}\text{\tiny{$\triangle$}}e_{i}$
$\displaystyle=2\langle e_{i},e_{j}\rangle$

As vector spaces there exists a _canonical isomorphism_ (independent of basis)
such that $\Lambda V\cong\Delta V$, see Proposition 1.2 in [38, p. 10]. In
what follows we will therefore just write $\Lambda V$ for the underlying
vector space which we call the _exterior space_ , but equipped with different
products. Moreover, for any orthogonal multivectors
$e_{s},e_{t}\in\Lambda{\mathbb{R}}^{n}$, $e_{s}\wedge
e_{t}=e_{s}\text{\tiny{$\triangle$}}e_{t}$ and for any vector $v\in V$ and
multivector $w\in\Lambda V$ we have the Riesz identities

$\displaystyle v\text{\tiny{$\triangle$}}w$
$\displaystyle=v\,\,\lrcorner\,\,w+v\wedge w,$ (3.1) $\displaystyle
w\text{\tiny{$\triangle$}}v$ $\displaystyle=w\,\,\llcorner\,\,v+w\wedge v,$
(3.2)

connecting the Clifford product to the wedge products and the left and right
interior multiplication. Since $\Lambda V=\bigoplus_{k=0}^{n}\Lambda^{k}V$ is
a graded vector space, we denote by $\langle w\rangle_{k}$ the orthogonal
projection of $\Lambda V$ onto $\Lambda^{k}V$ so that any multivector
$w\in\Lambda V$ can be written $\langle w\rangle=\sum_{k=0}^{n}\langle
w\rangle_{k}$. Furthermore, we set

$\displaystyle\langle w\rangle_{ev}$
$\displaystyle=\sum_{\begin{subarray}{c}j=0\\\ 2j\leq n\end{subarray}}\langle
w\rangle_{2j},$ $\displaystyle\langle w\rangle_{odd}$
$\displaystyle=\sum_{\begin{subarray}{c}j=0\\\ 2j\leq n\end{subarray}}\langle
w\rangle_{2j+1}.$

We define the main involution and anti-involution by

$\displaystyle\widehat{w}$ $\displaystyle=\sum_{j=0}^{n}(-1)^{j}\langle
w\rangle_{j}$ $\displaystyle\overline{w}$
$\displaystyle=\sum_{j=0}^{n}(-1)^{j(j-1)/2}\langle w\rangle_{j},$

The main involution is both an exterior algebra as well as a Clifford algebra
automorphism, i.e. for all $w_{1},w_{2}\in\Lambda V$, $\widehat{w_{1}\wedge
w_{2}}=\widehat{w_{1}}\wedge\widehat{w_{2}}$ and
$\widehat{w_{1}\text{\tiny{$\triangle$}}w_{2}}=\widehat{w_{1}}\text{\tiny{$\triangle$}}\widehat{w_{2}}$.
The main anti-involution is both an exterior algebra as well as a Clifford
algebra anti-automorphism, i.e. for all $w_{1},w_{2}\in\Lambda V$,
$\widehat{w_{1}\wedge w_{2}}=\widehat{w_{2}}\wedge\widehat{w_{1}}$ and
$\widehat{w_{1}\text{\tiny{$\triangle$}}w_{2}}=\widehat{w_{2}}\text{\tiny{$\triangle$}}\widehat{w_{1}}$.
Using the involution we have the following useful commutation relations

$\displaystyle v\wedge w=\widehat{w}\wedge
v,\,\,v\,\,\lrcorner\,\,w=-\widehat{w}\,\,\llcorner\,\,v,\text{ for all $v\in
V$ and $w\in\Lambda W$}.$

Moreover, the anti-involution has the following useful properties with respect
to the inner product

$\displaystyle\langle w_{1}\text{\tiny{$\triangle$}}w_{2},w_{3}\rangle=\langle
w_{2},\overline{w_{1}}\text{\tiny{$\triangle$}}w_{3}\rangle=\langle
w_{1},w_{3}\text{\tiny{$\triangle$}}\overline{w_{2}}\rangle.$

Thus, the anti-involution expresses the adjoint operation of Clifford
multiplication. Furthermore, we let $\Lambda^{ev}V$ and $\Lambda^{od}V$
correspond to the orthogonal decomposition of $\Lambda V$ into the eigenspaces
corresponding to the eigenvalues $+1$ and $-1$ under the main involution.

######  Definition 3.1 (Hodge Duality).

Let $V$ be an $n$-dimensional Euclidean vector space and let $e_{V}$ denote
the volume form of $V$, which in an ON-basis $\\{e_{j}\\}_{j}$ of $V$ is given
by

$\displaystyle e_{V}=e_{1}\wedge...\wedge e_{n}.$

Then, for any $w\in\Lambda V$ we define the _left and right Hodge star map_ by

$\displaystyle\star w$
$\displaystyle:=e_{V}\,\,\llcorner\,\,w=e_{V}\text{\tiny{$\triangle$}}\overline{w},$
$\displaystyle w\star$
$\displaystyle:=w\,\,\lrcorner\,\,e_{V}=\overline{w}\text{\tiny{$\triangle$}}e_{V}.$

The Hodge star maps have the useful properties that $\star(w\star)=(\star
w)\star=w$, $\star\star w=\widehat{w}=w\star\star$ and $(w_{1}\wedge(\star
w_{2}))\star=\langle w_{1},w_{2}\rangle$ for any $w,w_{1},w_{2}\in\Lambda V$.

######  Lemma 3.1.

If $v\in V$ and $w\in\Lambda V$, then
$|v\text{\tiny{$\triangle$}}w|=|w\text{\tiny{$\triangle$}}v|=|v||w|$.

###### Proof.

$\displaystyle|v\text{\tiny{$\triangle$}}w|^{2}=\langle
v\text{\tiny{$\triangle$}}w,v\text{\tiny{$\triangle$}}w\rangle=\langle
w,v\text{\tiny{$\triangle$}}v\text{\tiny{$\triangle$}}w\rangle=\langle
w,|v|^{2}w\rangle=|v|^{2}|w|^{2}.$

∎

Note that in general it is not true that for any multivectors $w_{1},w_{2}$ we
have $|w_{1}\text{\tiny{$\triangle$}}w_{2}|\leq|w_{1}||w_{2}|$. What is true
is that we always have $|w_{1}\text{\tiny{$\triangle$}}w_{2}|\leq
2^{n}|w_{1}||w_{2}|$ where $n$ is the dimension of $V$.

We now assume that the inner product $\langle\cdot,\cdot\rangle_{+}$ on $V$ is
euclidean. We associate the anti-euclidean inner product
$\langle\cdot,\cdot\rangle_{-}=-\langle\cdot,\cdot\rangle_{+}$ on $V$. We
denote the two Clifford algebras with respect to the euclidean and anti-
euclidean inner product by $\Delta V^{+}$ and $\Delta V^{-}$ and denote the
associated Clifford products by $\triangle$ and $\triangledown$ respectively
and call them positive and negative Clifford multiplication. Note that for any
vectors $v_{1},...,v_{k},u_{1},...,u_{k}\in V$ we

$\displaystyle\langle v_{1}\wedge...\wedge v_{k},u_{1}\rangle_{-}$
$\displaystyle=\det[\langle v_{j},u_{k}\rangle_{-}]_{j,k}=\det[-\langle
v_{j},u_{k}\rangle_{+}]_{j,k}$
$\displaystyle=\det[\langle\widehat{v_{j}},u_{k}\rangle_{+}]_{j,k}=\langle\widehat{v_{1}\wedge...\wedge
v_{k}},u_{1}\rangle_{+}.$

Therefore, by multilinearity we have for all $w_{1},w_{2}\in\Lambda V$

$\displaystyle\langle
w_{1},w_{2}\rangle_{-}=\langle\widehat{w_{1}},w_{2}\rangle_{+}=\langle
w_{1},\widehat{w_{2}}\rangle_{+}.$

We now prove the following useful anti-commutation relations for positive and
negative Clifford multiplication with respect to vectors.

######  Proposition 3.1.

For all $u,v\in V$ and $w\in\Lambda V$ we have

$\displaystyle
u\text{\tiny{$\triangle$}}(v\text{\scriptsize{$\triangledown$}}w)=-v\text{\scriptsize{$\triangledown$}}(u\text{\tiny{$\triangle$}}w).$
(3.3)

and

$\displaystyle
v\text{\tiny{$\triangle$}}w=\widehat{w}\text{\scriptsize{$\triangledown$}}v.$
(3.4)

###### Proof.

We first prove (3.4). We use the Riesz identities for positive and negative
Clifford multiplication

$\displaystyle v\text{\tiny{$\triangle$}}w$
$\displaystyle=v\,\,\lrcorner\,\,w+v\wedge w,\quad
v\text{\scriptsize{$\triangledown$}}w=-v\,\,\lrcorner\,\,w+v\wedge w$
$\displaystyle w\text{\tiny{$\triangle$}}v$
$\displaystyle=w\,\,\llcorner\,\,v+w\wedge v,\quad
w\text{\scriptsize{$\triangledown$}}v=-w\,\,\llcorner\,\,v+w\wedge v.$

By Proposition 2.6.3 in [51],

$\displaystyle v\wedge w=\widehat{w}\wedge v,\quad
v\,\,\lrcorner\,\,w=-\widehat{w}\,\,\llcorner\,\,v$

Thus,

$\displaystyle v\text{\tiny{$\triangle$}}w$
$\displaystyle=v\,\,\lrcorner\,\,w+v\wedge
w=-\widehat{w}\,\,\llcorner\,\,v+\widehat{w}\wedge
v=\widehat{w}\text{\scriptsize{$\triangledown$}}v.$

We now prove (3.3). We use the anti-commutation relations from

$\displaystyle u\,\,\lrcorner\,\,(v\wedge
w)+v\wedge(u\,\,\lrcorner\,\,w)=\langle u,v\rangle w$

from Theorem 2.8.1 in [51] and expand using the Riesz identities. This gives
using Proposition 2.6.3 in [51],

$\displaystyle
u\text{\tiny{$\triangle$}}(v\text{\scriptsize{$\triangledown$}}w)$
$\displaystyle=-u\,\,\lrcorner\,\,(v\,\,\lrcorner\,\,w)+u\,\,\lrcorner\,\,(v\wedge
w)-u\wedge(v\,\,\lrcorner\,\,w)+u\wedge(v\wedge w)$ $\displaystyle=-(v\wedge
u)\,\,\lrcorner\,\,w-v\wedge(u\,\,\lrcorner\,\,w)+\langle u,v\rangle
w+v\,\,\lrcorner\,\,(u\wedge w)-\langle u,v\rangle w-v\wedge(u\wedge w)$
$\displaystyle=(u\wedge
v)\,\,\lrcorner\,\,w-v\wedge(u\,\,\lrcorner\,\,w)+v\,\,\lrcorner\,\,(u\wedge
w)-v\wedge(u\wedge w)$
$\displaystyle=v\,\,\lrcorner\,\,(u\,\,\lrcorner\,\,w)-v\wedge(u\,\,\lrcorner\,\,w)+v\,\,\lrcorner\,\,(u\wedge
w)-v\wedge(u\wedge w)$
$\displaystyle=v\,\,\lrcorner\,\,(u\text{\tiny{$\triangle$}}w)-v\wedge(u\text{\tiny{$\triangle$}}w)$
$\displaystyle=-v\text{\scriptsize{$\triangledown$}}(u\text{\tiny{$\triangle$}}w).$

∎

###  3.2 Relation between Bivectors and Complex Structures

For the space of bivectors $\Lambda^{2}V$ there is a direct connection to
geometry. In particular we have ;

######  Theorem 3.1 (Chap. 6.5 [40]).

Let $V$ be a euclidean vector space. Then there exists a Lie algebra
homomorphism

$\displaystyle\Lambda^{2}V\cong\mathfrak{so}(V),$

where the Lie bracket in $\Lambda^{2}V$ is given by the Clifford commutator,
i.e., for any $b_{1},b_{2}\in\Lambda^{2}V$

$\displaystyle[b_{1},b_{2}]_{\text{\tiny{$\triangle$}}}=b_{1}\text{\tiny{$\triangle$}}b_{2}-b_{2}\text{\tiny{$\triangle$}}b_{1}.$

In particular, for any skew symmetric linear map $A\in\mathcal{L}(V)$ there
exists a unique bivector $b$ such that for any $v\in V$

$\displaystyle Av=b\,\,\llcorner\,\,v.$

If we now assume that $\text{dim}V=2n$ and that we are given an ON-eigenbasis
$\\{e_{1}^{+},...,e_{n}^{+},e_{1}^{-},...,e_{n}^{-}\\}$ of $V$ we can form the
bivectors $J_{k}:=e_{k}^{+}\wedge e_{k}^{-}$ and the bivector

$\displaystyle J:=\sum_{k=1}^{n}e_{k}^{+}\wedge
e_{k}^{-}=\sum_{k=1}^{n}J_{k}.$

It is easy to any such $J$ becomes a complex structure on $V$. In particular,
we have $\overline{J}=-J$. Furthermore, in dimension two, $J=e_{1}\wedge
e_{2}$ coincides with the volume form and under the algebra isomorphism
$\Delta^{ev}V\cong{\mathbb{C}}$ we have the identification $J\cong i$.

###  3.3 Exterior Extensions of Linear Maps and Grassmann Duality

######  Definition 3.2.

For a linear operator $T\in\mathcal{L}(V)=\text{Hom}(V,V)$, we define the
_exterior extension_ of $T$, denoted $\widehat{T}$, to be unique linear
operator $\widehat{T}\in\mathcal{L}(\Lambda V)$ which is an exterior algebra
homomorphism such that

  * (i)

$\widehat{T}(1)=1$.

  * (ii)

$\widehat{T}|_{V}=T$.

  * (iii)

$\widehat{T}(\Lambda^{k}V)\subseteq\Lambda^{k}V$ for all $k=0,1,...,n$.

  * (iv)

For any $w_{1},w_{2}\in\Lambda V$, $\widehat{T}(w_{1}\wedge
w_{2})=\widehat{T}(w_{1})\wedge\widehat{T}(w_{2})$.

The restriction of $\widehat{T}$ to $\Lambda^{k}V$ will be denoted by
$\widehat{T}^{k}$.

Note that $\text{tr}(\widehat{T}^{n})=\det(T)$, and that the characteristic
polynomial of $T$ can be written according to

$\displaystyle\text{ch}_{T}(z)=\sum_{k=0}^{n}(-1)^{k}\text{tr}(\widehat{T}^{k})z^{n-k}.$

######  Lemma 3.2.

Let $V$ be a euclidean vectors space and let $\mathbf{R}(v)$ denote the
orthogonal reflection in the hyperplane orthogonal to the vector $v$. Then for
any $w\in\Lambda V$

$\displaystyle\widehat{\mathbf{R}(v)}w=v\text{\tiny{$\triangle$}}\widehat{w}\text{\tiny{$\triangle$}}v^{-1}$
(3.5)

Moreover, for any $T\in O(V)$, there exists a _rotor_
$q\in\text{Pin}(V)\subset\Delta V$ such that for any $w\in\Lambda V$,

$\displaystyle\widehat{\mathbf{T}}w=q\text{\tiny{$\triangle$}}\widehat{w}\text{\tiny{$\triangle$}}q^{-1},$
(3.6)

where $\text{Pin}(V)$ is the pin group, i.e., the double cover of $O(V)$. If
$T\in SO(V)$, then $q\in\text{Spin}(V)$, where $\text{Spin}(V)$ is the spin
group, i.e., the double cover of $SO(V)$.

###### Proof.

See Proposition 4.1.10 in [51]. ∎

Explicitly the Pin and the Spin groups can be represented within the Clifford
algebra in the following way. Again assume that $V$ is a euclidean vector
space and let $\Delta V$ denote the euclidean Clifford algebra with respect to
$V$. Define the _Clifford cone_ $\widehat{\Delta}V$ to consist of all
multivectors $q\in\Delta V$ such that there exists finitely many vectors
$v_{1},v_{2},...,v_{k}$ such that

$\displaystyle
q=v_{1}\text{\tiny{$\triangle$}}v_{2}\text{\tiny{$\triangle$}}...\text{\tiny{$\triangle$}}v_{k}.$

In particular, by Proposition 4.1.5 in [51], $q\in\widehat{\Delta}V$ if and
only if
$\widehat{q}\text{\tiny{$\triangle$}}v\text{\tiny{$\triangle$}}q^{-1}\in V$
for all $v\in V$. Then

$\displaystyle\text{Pin}(V)$ $\displaystyle=\\{q\in\widehat{\Delta}V:\langle
q,q\rangle=\pm 1\\},$ $\displaystyle\text{Spin}(V)$
$\displaystyle=\\{q\in\text{Pin}(V):\Delta^{ev}V\\},$

where $\Delta^{ev}V$ denotes the even subalgebra of $\Delta V$.

######  Definition 3.3.

Let $\nu\in V$ be a unit vector. Define the linear operators
$\mathbf{P}_{T}^{\nu}$ and $\mathbf{P}_{N}^{\nu}$ by

$\displaystyle\mathbf{P}_{T}^{\nu}w$
$\displaystyle:=\nu\,\,\lrcorner\,\,(\nu\wedge w),$
$\displaystyle\mathbf{P}_{N}^{\nu}w$
$\displaystyle:=\nu\wedge(\nu\,\,\lrcorner\,\,w),$

for all $w\in\Lambda V$.

In particular we note that $\mathbf{P}_{T}^{\nu}+\mathbf{P}_{N}^{\nu}=I$, and
that for any $u\in V$, $\mathbf{P}_{T}^{\nu}u$ is the orthogonal projection
onto the linear space spanned by $\nu$.

######  Definition 3.4 (Grassmann dual map).

Let $T\in\mathcal{L}(\Lambda V)$. The _Grassman dual linear map_
$T^{c}\in\mathcal{L}(\Lambda V)$ is defined by

$\displaystyle T^{c}(w):=T(\star w)\star$ (3.7)

for all $w\in\mathcal{L}(\Lambda V)$.

From the definition the following properties hold:

  * (i)

$(T^{c})^{c}=T$.

  * (ii)

$(T^{c})(e_{V})=e_{V}$.

  * (iii)

$(S\circ T)^{c}=S^{c}\circ T^{c}$

  * (iv)

$I^{c}=I$.

  * (v)

If $T(\Lambda^{p}V)\subset\Lambda^{q}V$ then
$T^{c}(\Lambda^{\text{dim}V-p}V)\subset\Lambda^{\text{dim}V-q}V$

  * (vi)

If $T(\Lambda^{p}V)\subset\Lambda^{p}V$ for all $p=0,1,...,\text{dim}V$, then
$(T^{\ast})^{c}=(T^{c})^{\ast}$.

Here $I$ denotes the identity map. Using the exterior extension and the
Grassmann dual the cofactor formula can be expressed as

$\displaystyle(\widehat{T})^{c}\circ\widehat{T^{\ast}}=\det(T)I$

for $T\in\mathcal{L}(V)$, see Proposition 3.7 [40].

######  Definition 3.5 (Self-dual linear maps).

A linear map $T\in\mathcal{L}(V)$ is said to be _self-dual_ if

$\displaystyle(\widehat{T})^{c}=\widehat{T}.$

######  Lemma 3.3.

If $T\in\mathcal{L}(V)$ is self-dual then $T$ is conformal.

###### Proof.

The cofactor formula and self-duality implies

$\displaystyle(\widehat{T})^{c}\circ\widehat{T^{\ast}}=\widehat{T}\circ\widehat{T^{\ast}}=\widehat{T\circ
T^{\ast}}=\det(T)I\Longrightarrow T\circ T^{\ast}=\det(T)I|_{V}.$

∎

For more background on the Grassmann dual we advice the reader to consult [40,
Chapter 3] and [10].

##  4 Analytical preliminaries

###  4.1 Tangential and Normal Trace Maps.

######  Definition 4.1.

A bounded domain $\Omega\subset{\mathbb{R}}^{n}$ will be called a _Lipschitz
domain_ or _strongly Lipschitz domain_ if for each point $p\in\partial\Omega$
there exists a Lipschitz function $\phi_{p}:{\mathbb{R}}^{n-1}\to{\mathbb{R}}$
and a neighbourhood $V_{p}$ of $p$ and a neighbourhood $U_{p}$ of $0$ in
${\mathbb{R}}^{n-1}$ and a rotation $R_{p}\in O({\mathbb{R}}^{n})$ such that

$\displaystyle\partial\Omega\cap U_{p}=R_{p}(\\{(y,\phi_{p}(y)):y\in
U_{p}\\}).$

These are the only Lipschitz domains that will be considered in this paper.

######  Definition 4.2.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a Lipschitz domain. The _tangential and
normal trace maps_ $\gamma_{T}:W^{1,2}(\Omega,\Lambda)\to
L^{2}(\partial\Omega,\Lambda)$ and $\gamma_{N}:W^{1,2}(\Omega,\Lambda)\to
L^{2}(\partial\Omega,\Lambda)$ respectively are defined by

$\displaystyle\gamma_{T}F(x)$
$\displaystyle:=\mathbf{P}_{T}^{\nu(x)}\circ\gamma F(x),$
$\displaystyle\gamma_{N}F(x)$
$\displaystyle:=\mathbf{P}_{N}^{\nu(x)}\circ\gamma F(x),$

for $\mathscr{H}^{n-1}$-a.e. $x\in\partial\Omega$, where $\nu$ is the outward
pointing unit normal on $\partial\Omega$ and $\gamma$ is the full Sobolev
trace map.

######  Remark 4.1.

Sometimes we will write $F_{T}$ and $F_{N}$ respectively instead of
$\gamma_{T}F$ and $\gamma_{N}F$.

There are other equivalent ways of defining the tangential and normal trace
maps. For a comparison between the different definitions we refer to the
discussion in [13, Chap. 3]

######  Definition 4.3.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded (strongly) Lipschitz domain.
The _nontangential maximal function_ $u^{\ast}:\partial\Omega\to{\mathbb{R}}$
of a function $u$, possibly vector valued, is defined by

$\displaystyle u^{\ast}(x):=\sup\\{|u(y)|:y\in\Omega,\,\text{ such that
}\,|x-y|\leq 2\,\text{dist}(y,\partial\Omega)\\}$

###  4.2 Hodge-Dirac Operators on Euclidean Domains

######  Definition 4.4 (Hodge-Dirac operators).

The Hodge-Dirac operators are defined by

$\displaystyle\mathcal{D}^{\tiny+}=d+\delta,\,\,\,,\mathcal{D}^{\tiny-}=d-\delta,$
(4.1)

where $d$ is the exterior derivative and $\delta$ is the interior derivative,
i.e., $\delta=-d^{\ast}$, where $d^{\ast}$ is the formal adjoint of $d$.

The symbols
$\sigma_{\mathcal{D}^{\tiny+}}(x,\xi),\sigma_{\mathcal{D}^{\tiny-}}(x,\xi)\in\mathcal{L}(V,\mathcal{L}(\Lambda
V))$ of $\mathcal{D}^{\tiny+}$ and $\mathcal{D}^{\tiny-}$ respectively is
given by the linear maps

$\displaystyle\sigma_{\mathcal{D}^{\tiny+}}(x,\xi)w=\xi\text{\tiny{$\triangle$}}w,\,\,\,\sigma_{\mathcal{D}^{\tiny-}}(x,\xi)w=\xi\text{\scriptsize{$\triangledown$}}w$

for all $w\in\Lambda V$ and $\xi\in V$. In particular,
$\sigma_{\mathcal{D}^{\tiny+}}(x,\xi)$ is self-adjoint and is skew-adjoint
$\sigma_{\mathcal{D}^{\tiny+}}(x,\xi)$.

We consider the Hodge-Dirac operators on $\Omega$ on a smooth bounded domain,
initially defined on $C_{0}^{\infty}(\Omega,\Lambda{\mathbb{R}}^{n})$ and
viewed as unbounded operators
$\mathcal{D}^{\pm}:L^{2}(\Omega,\Lambda{\mathbb{R}}^{n})\to
L^{2}(\Omega,\Lambda{\mathbb{R}}^{n})$. We recall that the maximal extension
$\mathcal{D}_{max}^{\pm}=((\mathcal{D}^{\pm})^{\ast})^{\ast}$, where
$\mathcal{D}^{\pm}$ acts on $C_{0}^{\infty}(\Omega,\Lambda{\mathbb{R}}^{n})$.
In particular the domain of $\mathcal{D}_{max}^{\pm}$ is given by

$\displaystyle\text{dom}(\mathcal{D}_{max}^{\pm})=\\{F\in\mathscr{D}^{\prime}(\Omega,\Lambda{\mathbb{R}}^{n}):F,\mathcal{D}_{max}^{\pm}F\in
L^{2}(\Omega,\Lambda{\mathbb{R}}^{n})\\}.$

It follows trivially that
$W^{1,2}(\Omega,\Lambda{\mathbb{R}}^{n})\subset\text{dom}(\mathcal{D}_{max}^{\pm})$.
On the other hand it is shown in Theorem [8, Theorem 3.2] that the trace map
$\gamma$ satisfies

$\displaystyle\gamma(\text{dom}(\mathcal{D}_{max}^{\pm}))\cap
W^{-1/2,2}(\partial\Omega,\Lambda)\neq\varnothing.$

Hence, the inclusion is strict since
$\gamma(W^{1,2}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n}))=W^{1/2,1}((\partial\Omega,\Lambda)$
by the trace theorem for Sobolev spaces, see [21].

Finally a direct computation shows that the fundamental solution of
$\mathcal{D}^{\tiny+}$ and $\mathcal{D}^{\tiny-}$ are given respectively by
$\pm E(x)$, where

$\displaystyle E(x)=\frac{1}{\sigma_{n-1}}\frac{x}{|x|^{n}}.$

When working with Dirac operators as well as the exterior and interior
derivative, it is many times useful for computations to introduce the formal
nabla symbol

$\displaystyle\nabla:=\sum_{j=1}^{n}e_{j}\partial_{j}.$

Using the formal nabla symbol we have for any $F\in
C^{\infty}(\Omega,\Lambda)$

$\displaystyle dF(x)=\nabla\wedge F(x),\,\,\,\delta
F(x)=\nabla\,\,\lrcorner\,\,F(x),\,\,\,\mathcal{D}^{\tiny+}F(x)=\nabla\text{\tiny{$\triangle$}}F(x),\,\,\,\mathcal{D}^{\tiny-}F(x)=\nabla\text{\scriptsize{$\triangledown$}}F(x).$

In the special case when $F=(f_{1},f_{2},...,f_{n})$ is a vector field, the
equation $\mathcal{D}^{\tiny+}F(x)=0$ is equivalent to the system

$\displaystyle\left\\{\begin{array}[]{ll}\sum_{j=1}^{n}\frac{\partial
f_{j}}{\partial x_{j}}=0,&\\\ \\\ \frac{\partial f_{i}}{\partial
x_{j}}=\frac{\partial f_{j}}{\partial x_{i}},&i\neq j,\end{array}\right.$

used in [56] to study boundary values of harmonic functions.

For any domain $\Omega\subset{\mathbb{R}}^{n}$ one can define an
$\Lambda{\mathbb{R}}^{n}$ valued dualities
$(\cdot,\cdot)_{\text{\tiny{$\triangle$}}}$ and
$(\cdot,\cdot)_{\text{\scriptsize{$\triangledown$}}}$, defined by

$\displaystyle(F,G)_{\text{\tiny{$\triangle$}}}:=\int_{\Omega}\overline{F(x)}\text{\tiny{$\triangle$}}G(x)dx,\,\,\,(F,G)_{\text{\scriptsize{$\triangledown$}}}:=\int_{\Omega}\overline{F(x)}\text{\scriptsize{$\triangledown$}}G(x)dx.$

In [20], it is shown the the Riesz representation theorem extends to such
generalized dualities. Moreover, the usual inner product is contained in the
generalized duality using the algebraic identity

$\displaystyle\langle
w_{1},w_{2}\rangle=\langle\overline{w_{1}}\text{\tiny{$\triangle$}}w_{2}\rangle_{0}$

for multivectors, which yield $\langle
F,G\rangle=\langle(F,G)_{\text{\tiny{$\triangle$}}}\rangle_{0}$. Using the
generalized duality we also have the vector valued integration by parts
formula

$\displaystyle(G,\mathcal{D}^{\tiny+}F)_{\text{\tiny{$\triangle$}}}=-(\mathcal{D}^{\tiny+}G,F)_{\text{\tiny{$\triangle$}}}+\int_{\partial\Omega}\overline{G(y)}\text{\tiny{$\triangle$}}\nu(y)\text{\tiny{$\triangle$}}F(y)d\sigma(y).$
(4.2)

on a smooth domain $C^{1}$-domain with outwards point unit normal $\nu$ and
any $F,G\in C^{1}(\overline{\Omega},\Lambda)$. In addition $\mathcal{D}^{\pm}$
satisfies the following versions of Stokes’ theorem

$\displaystyle\int_{\Omega}\mathcal{D}^{\tiny+}F(x)dx=\int_{\partial\Omega}\nu(y)\text{\tiny{$\triangle$}}F(y)d\sigma(y),\quad\int_{\Omega}\mathcal{D}^{\tiny-}F(x)dx=\int_{\partial\Omega}\nu(y)\text{\scriptsize{$\triangledown$}}F(y)d\sigma(y),$
(4.3)

valid for piecewise smooth bounded domains $\Omega$ and $F\in
W^{1,2}(\Omega,\Lambda)$. Finally, we may express the partial derivatives
$\partial_{j}F$ according

$\displaystyle\partial_{j}F(x)=\frac{1}{2}(e_{j}\text{\tiny{$\triangle$}}\mathcal{D}^{\tiny+}F(x)-e_{j}\text{\scriptsize{$\triangledown$}}\mathcal{D}^{\tiny-}F(x)).$

We now study how the Hodge-Dirac operators transform under the action of the
Hodge star maps.

######  Lemma 4.1.

If $F\in W^{1,2}(\Omega,\Lambda)$ then

$\displaystyle\mathcal{D}^{\tiny+}F(x)\star$
$\displaystyle=\mathcal{D}^{\tiny-}(\widehat{F(x)}\star),$
$\displaystyle\mathcal{D}^{\tiny-}F(x)\star$
$\displaystyle=\mathcal{D}^{\tiny+}(\widehat{F(x)}\star).$

###### Proof.

Using the algebraic identities from [51, Proposition 2.6.8]

$\displaystyle(\Theta_{1}\wedge\Theta_{2})\star$
$\displaystyle=\Theta_{2}\,\,\lrcorner\,\,(\Theta_{1}\star),$
$\displaystyle(\Theta_{1}\,\,\llcorner\,\,\Theta_{2})\star$
$\displaystyle=\Theta_{2}\wedge(\Theta_{1}\star).$

This gives using identities valid for $F\in W^{1,2}(\Omega,\Lambda)$ we get

$\displaystyle(\nabla\wedge F(x))\star$
$\displaystyle=(\widehat{F(x)}\wedge\nabla)\star=\nabla\,\,\lrcorner\,\,(\widehat{F(x)}\star),$
$\displaystyle(\nabla\,\,\lrcorner\,\,F(x))\star$
$\displaystyle=-(\widehat{F(x)}\,\,\llcorner\,\,\nabla)\star=-\nabla\wedge(\widehat{F(x)}\star).$

Hence,

$\displaystyle(\mathcal{D}^{\tiny-}F(x))\star$ $\displaystyle=(\nabla\wedge
F(x))\star-(\nabla\,\,\lrcorner\,\,F(x))\star=\nabla\,\,\lrcorner\,\,(\widehat{F(x)}\star)+\nabla\wedge(\widehat{F(x)}\star)=\mathcal{D}^{\tiny+}(\widehat{F(x)}\star)$
$\displaystyle(\mathcal{D}^{\tiny+}F(x))\star$ $\displaystyle=(\nabla\wedge
F(x))\star+(\nabla\,\,\lrcorner\,\,F(x))\star=\nabla\,\,\lrcorner\,\,(\widehat{F(x)}\star)-\nabla\wedge(\widehat{F(x)}\star)=\mathcal{D}^{\tiny-}(\widehat{F(x)}\star).$

∎

On the other hand in this paper we will mostly be concerned with multivector
fields $F$ of the form $F=u+v$, where $u\in W^{1,2}(\Omega,{\mathbb{R}})$ and
$v\in W^{1,2}(\Omega,\Lambda^{2}{\mathbb{R}}^{n})$ where in addition $dv=0$.
In this case $\overline{F}=u-v$, and

$\displaystyle\mathcal{D}^{\tiny+}\overline{F}=du-\delta
v=\mathcal{D}^{\tiny-}F,\,\,\,\mathcal{D}^{\tiny-}\overline{F}=du+\delta
v=\mathcal{D}^{\tiny+}F,$

which are analogous to the identities
$\partial_{\overline{z}}\overline{f}=\overline{\partial f}$ and
$\partial\overline{f}=\overline{\partial_{\overline{z}}f}$ for complex valued
fields in the plane.

######  Remark 4.2.

When acting on smooth fields $F\in
C^{\infty}(\Omega,\Lambda{\mathbb{R}}^{n})$, one easily checks that the Hodge-
Dirac operators $\mathcal{D}^{\pm}$ satisfies the (anti-)commutation relations

$\displaystyle\mathcal{D}^{\tiny+}\mathcal{D}^{\tiny-}+\mathcal{D}^{\tiny-}\mathcal{D}^{\tiny+}=0,\quad(\mathcal{D}^{\tiny+})^{2}=-(\mathcal{D}^{\tiny-})^{2}=\Delta,\quad\mathcal{D}^{\pm}\Delta=\Delta\mathcal{D}^{\pm}$

where $\Delta=d\delta+\delta d$ is the Hodge–Laplacian. A variation of the
Hodge–Dirac operators was used in E. Witten’s paper [60] on his supersymmetric
proof of the Morse inequalities. He used the “supersymmetry” operators

$\displaystyle Q_{1}=\mathcal{D}^{\tiny-},\quad Q_{2}=i\mathcal{D}^{\tiny+}$

###  4.3 Cauchy-Pompeiu Formula for the Hodge-Dirac Operators

######  Theorem 4.1.

If $F\in C^{1}(\Omega,\Lambda{\mathbb{R}}^{n})\cap
C^{0}(\overline{\Omega},\Lambda{\mathbb{R}}^{n})$ then the Cauchy-Pompieu
formula reads

$\displaystyle F(x)$
$\displaystyle=\frac{1}{\sigma_{n-1}}\int_{\Omega}\frac{y-x}{|y-x|^{n}}\text{\tiny{$\triangle$}}\mathcal{D}^{\tiny+}F(y)dy+\frac{1}{\sigma_{n-1}}\int_{\partial\Omega}\frac{y-x}{|y-x|^{n}}\text{\tiny{$\triangle$}}\nu(y)\text{\tiny{$\triangle$}}F(y)d\sigma(y),$
(4.4)
$\displaystyle=\frac{1}{\sigma_{n-1}}\int_{\Omega}\frac{x-y}{|x-y|^{n}}\text{\scriptsize{$\triangledown$}}\mathcal{D}^{\tiny-}F(y)dy+\frac{1}{\sigma_{n-1}}\int_{\partial\Omega}\frac{x-y}{|x-y|^{n}}\text{\scriptsize{$\triangledown$}}\nu(y)\text{\scriptsize{$\triangledown$}}F(y)d\sigma(y).$
(4.5)

. Moreover, if $\mathcal{D}^{\tiny+}F=0$ in $\Omega$ then the Cauchy integral
formula

$\displaystyle
F(x)=\int_{\partial\Omega}E(y-x)\text{\tiny{$\triangle$}}\nu(y)\text{\tiny{$\triangle$}}\gamma(F)(y)dy$
(4.6)

hold for _every_ $x\in\Omega$.

For a proof see Theorem 8.1.8 in [51].

######  Definition 4.5.

The Cauchy integrals $\mathcal{E}^{\pm}(f)$ of $f\in
L^{2}(\partial\Omega,\Lambda)$ is defined as

$\displaystyle\mathcal{E}^{+}(f)(x)$
$\displaystyle:=2\text{p.v.}\int_{\partial\Omega}E(y-x)\text{\tiny{$\triangle$}}\nu(y)\text{\tiny{$\triangle$}}f(y)d\sigma(y),$
$\displaystyle\mathcal{E}^{-}(f)(x)$
$\displaystyle:=2\text{p.v.}\int_{\partial\Omega}E(x-y)\text{\scriptsize{$\triangledown$}}\nu(y)\text{\scriptsize{$\triangledown$}}f(y)d\sigma(y)$

The boundedness of the operators $\mathcal{E}^{\pm}$ on
$L^{2}(\partial\Omega,\Lambda)$ on Lipschitz hypersurfaces $\partial\Omega$
follows from the seminal work in [16]. For more details see [32].

######  Theorem 4.2 (Sokhotski-Plemelj jump formulas).

Let $f\in L^{2}(\partial\Omega,\Lambda)$. Let $\Omega_{+}=\Omega$ and
$\Omega_{-}={\mathbb{R}}^{n}\setminus\overline{\Omega}$. Then

$\displaystyle\lim_{\begin{subarray}{c}y\to x\in\partial\Omega\\\
y\in\Omega^{\pm}\end{subarray}}\int_{\partial\Omega}E(y-x)\text{\tiny{$\triangle$}}\nu(y)\text{\tiny{$\triangle$}}f(y)d\sigma(y)=\frac{1}{2}(I\pm\mathcal{E}^{+})(f)(x).$

A proof of this fact follows from an easy modification to the vector valued
setting of the argument given in [59].

We have the Hardy space splitting of $L^{2}(\partial\Omega,\Lambda)$ according
to

$\displaystyle L^{2}(\partial\Omega,\Lambda)=H^{2}_{+}(\partial\Omega)\oplus
H^{2}_{-}(\partial\Omega).$

In particular $f\in H^{2}_{+}(\partial\Omega)$ if and only if

$\displaystyle\frac{1}{2}(I+\mathcal{E}^{+})(f)(x)=f(x)\text{ for a.e.
$x\in\partial\Omega$},$

that is if and only if $f$ solves the singular integral equation

$\displaystyle\frac{1}{\sigma_{n-1}}\text{p.v.}\int_{\partial\Omega}\frac{y-x}{|y-x|^{n}}\text{\tiny{$\triangle$}}\nu(y)\text{\tiny{$\triangle$}}f(y)d\sigma(y)=\frac{1}{2}f(x).$

##  5 Topological preliminaries

###  5.1 Harmonic Fields and De Rham Cohomology

Consider the Hilbert space $L^{2}(\partial\Omega,\Lambda)$ and the linear
subspaces $L^{2}(\partial\Omega,\Lambda^{k})$ consisting of homogeneous
multivector fields of degree $k$. At each point $x\in\partial\Omega$, the
tangent bundle $T_{x}\Omega$ has an orthogonal decomposition as
$T_{x}\Omega=T_{x}\partial\Omega\oplus N_{x}\partial\Omega$. This induces an
orthogonal splitting of the exterior tangent bundle at $x$ according to

$\displaystyle\Lambda T_{x}\Omega=\Lambda(T_{x}\partial\Omega\oplus
N_{x}\partial\Omega)\cong\Lambda T_{x}\partial\Omega\widehat{\otimes}\Lambda
N_{x}\partial\Omega.$

Since $\Lambda N_{x}\partial\Omega=\text{span}\\{1,\nu(x)\\}$, we see that

$\displaystyle\Lambda T_{x}\Omega\cong\Lambda
T_{x}\partial\Omega\widehat{\otimes}1\oplus\Lambda
T_{x}\partial\Omega\widehat{\otimes}\nu(x).$

Therefore, any $f\in L^{2}(\partial\Omega,\Lambda)$ can be written uniquely as

$\displaystyle f(x)=f_{T}(x)+f_{N}(x)=f_{T}(x)+\nu(x)\wedge g_{T}(x)$

with $f_{T},g_{T}\in L^{2}(\partial\Omega,\Lambda T\partial\Omega)$ and
$f_{N}\in\nu\wedge L^{2}(\partial\Omega,\Lambda T\partial\Omega)$ and where

$\displaystyle f_{T}(x)=\nu(x)\,\,\lrcorner\,\,(\nu(x)\wedge
f(x)),\,\,f_{N}(x)=\nu(x)\wedge(\nu(x)\,\,\lrcorner\,\,f(x)).$

Moreover $f_{T}$ and $f_{N}$ satisfies the following identities for
$\sigma$-a.e. $x\in\partial\Omega$

$\displaystyle\nu(x)\wedge f(x)=\nu(x)\wedge
f_{T}(x),\,\,\,\nu(x)\,\,\lrcorner\,\,f(x)=\nu(x)\,\,\lrcorner\,\,f_{N}(x).$

We define

$\displaystyle
L^{2}_{T}(\partial\Omega,\Lambda):=L^{2}(\partial\Omega,\sigma,\Lambda
T\partial\Omega),\,\,L^{2}_{N}(\partial\Omega,\Lambda):=\nu(x)\wedge
L^{2}(\partial\Omega,\sigma,\Lambda T\partial\Omega).$

In particular, $f\in L^{2}_{T}(\partial\Omega,\Lambda)$ if and only if for
$\sigma$-a.e. $x$ we have $\nu(x)\,\,\lrcorner\,\,f(x)=0$ and $f\in
L^{2}_{N}(\partial\Omega,\Lambda)$ if and only if for $\sigma$-a.e. $x$ we
have $\nu(x)\wedge f(x)=0$

######  Definition 5.1.

A $k$-multivector field $\omega\in
C^{\infty}(\Omega,\Lambda^{k}{\mathbb{R}}^{n})$ is called an _harmonic field_
if $d\omega=\delta\omega=0$ in $\Omega$. The space of harmonic $k$-multivector
fields in $\Omega$ is denoted by $\mathcal{H}^{k}(\Omega)$. A $k$-multivector
field $\omega\in C^{\infty}(\overline{\Omega},\Lambda^{k}{\mathbb{R}}^{n})$
such that $\omega_{T}=0$ on $\partial\Omega$ is called a _tangential harmonic
$k$-multivector_ and the space of tangential harmonic $k$-multivector fields
in $\Omega$ is denoted by $\mathcal{H}_{T}^{k}(\Omega)$. Similarly, A
$k$-multivector field $\omega\in
C^{\infty}(\overline{\Omega},\Lambda^{k}{\mathbb{R}}^{n})$ such that
$\omega_{N}=0$ on $\partial\Omega$ is called a _normal harmonic
$k$-multivector_ and the space of tangential harmonic $k$-multivector fields
in $\Omega$ is denoted by $\mathcal{H}_{N}^{k}(\Omega)$.

The Hodge-star map induces an isomorphism
$\star\mathcal{H}_{T}^{k}(\Omega)=\mathcal{H}_{N}^{n-k}(\Omega)$. Moreover we
have the de Rham–Hodge isomorphisms

$\displaystyle\mathbf{H}^{k}(\Omega)$
$\displaystyle\cong\mathbf{H}_{dR,abs}^{k}(\Omega)\cong\mathcal{H}_{N}^{k}(\Omega)$
$\displaystyle\mathbf{H}^{k}_{rel}(\Omega)$
$\displaystyle\cong\mathbf{H}_{dR,rel}^{k}(\Omega)\cong\mathcal{H}_{T}^{k}(\Omega)$

Here, $\mathbf{H}^{k}(\Omega)$ denotes the absolute cohomology groups and
$\mathbf{H}^{k}_{rel}(\Omega)$ denotes the relative cohomology groups. These
are also denoted by
$\mathbf{H}^{k}_{rel}(\Omega)=\mathbf{H}^{k}(\Omega,\partial\Omega)$ in the
literature. Therefore, normal and tangential boundary conditions are sometimes
called absolute and relative boundary conditions in the literature, e.g. [54,
15].

We will also set

$\displaystyle\mathcal{H}_{T}(\Omega)$
$\displaystyle=\bigoplus_{k=0}^{n}\mathcal{H}_{T}^{k}(\Omega)=\\{\omega\in
C^{\infty}(\overline{\Omega},\Lambda{\mathbb{R}}^{n}):d\omega=\delta\omega=0,\,\,\omega_{T}=0\\},$
$\displaystyle\mathcal{H}_{N}(\Omega)$
$\displaystyle=\bigoplus_{k=0}^{n}\mathcal{H}_{N}^{k}(\Omega)=\\{\omega\in
C^{\infty}(\overline{\Omega},\Lambda{\mathbb{R}}^{n}):d\omega=\delta\omega=0,\,\,\omega_{N}=0\\}.$

######  Lemma 5.1.

$\displaystyle\mathcal{H}_{T}(\Omega)$ $\displaystyle=\\{F\in
C^{\infty}(\overline{\Omega},\Lambda{\mathbb{R}}^{n}):,\,\,\mathcal{D}^{\tiny+}F=0,F_{T}=0\\}=\\{F\in
C^{\infty}(\overline{\Omega},\Lambda{\mathbb{R}}^{n}):,\,\,\mathcal{D}^{\tiny-}F=0,F_{T}=0\\},$
$\displaystyle\mathcal{H}_{N}(\Omega)$ $\displaystyle=\\{F\in
C^{\infty}(\overline{\Omega},\Lambda{\mathbb{R}}^{n}):,\,\,\mathcal{D}^{\tiny+}F=0,F_{N}=0\\}=\\{F\in
C^{\infty}(\overline{\Omega},\Lambda{\mathbb{R}}^{n}):,\,\,\mathcal{D}^{\tiny-}F=0,F_{N}=0\\},$

###### Proof.

If $\mathcal{D}^{\tiny+}F(x)=0$ in $\Omega$, then by integration by parts
formula (C.1)

$\displaystyle 0$
$\displaystyle=\int_{\Omega}\langle\mathcal{D}^{\tiny+}F(x),\mathcal{D}^{\tiny+}F(x)\rangle
dx=\int_{\Omega}\langle(d+\delta)F(x),(d+\delta)F(x)\rangle dx$
$\displaystyle=\int_{\Omega}(|dF(x)|^{2}+|\delta
F(x)|^{2})dx+2\int_{\Omega}\langle dF(x),\delta F(x)\rangle dx$
$\displaystyle=\int_{\Omega}(|dF(x)|^{2}+|\delta
F(x)|^{2})dx-2\int_{\Omega}\langle d^{2}F(x),F(x)\rangle
dx+2\int_{\partial\Omega}\langle\nu(x)\,\,\lrcorner\,\,F(x),dF(x)\rangle dx$
$\displaystyle=\int_{\Omega}(|dF(x)|^{2}+|\delta F(x)|^{2})dx.$

Consequently, $dF(x)=\delta F(x)=0$. ∎

In the case that $\Omega\subset{\mathbb{R}}^{n}$ is a simply connected domain,
so in particular $\Omega$ is diffeomorphic to the open unit ball
$\mathbb{B}_{1}(0)=\\{x\in{\mathbb{R}}^{n}:|x|<1\\}$, then by Poincaré lemma

$\displaystyle\mathbf{H}^{k}(\Omega)=\left\\{\begin{array}[]{ll}0&\text{if
$1\leq k\leq n$},\\\ {\mathbb{R}}&\text{if
$k=0$}\end{array}\right.,\,\,\mathbf{H}^{k}_{rel}(\Omega)=\left\\{\begin{array}[]{ll}0&\text{if
$0\leq k\leq n-1$},\\\ {\mathbb{R}}&\text{if $k=n$}\end{array}\right..$

Moreover one can show that $\mathcal{H}_{N}(\Omega)=\\{c:c\in{\mathbb{R}}\\}$
and $\mathcal{H}_{T}(\Omega)=\\{ce_{\mathbf{n}}:c\in{\mathbb{R}}\\}$, where
$e_{\mathbf{n}}=e_{1}\wedge e_{2}\wedge...\wedge e_{n}$ is the positively
oriented volume $n$-field for ${\mathbb{R}}^{n}$. Thus, the tangential
boundary value problem (9.4) always has solution whenever $\langle
F_{T}\rangle_{n}=0$. However, since $\langle F\rangle_{n}=\langle
F_{N}\rangle_{n}$, this is always true. Therefore, for a simply connected
domain, the tangential boundary value problem for the Hodge-Dirac operator is
always solvable. Moreover, since $\langle F\rangle_{0}=\langle
F_{T}\rangle_{0}$ it follows that $F_{N}\perp\mathcal{H}_{T}(\partial\Omega)$
automatically, and hence the solution is unique as well.

Moreover, by the Poincaré duality we have the isomorphisms between the
homology and cohomology groups

$\displaystyle\mathbf{H}_{k}(\Omega)\cong\mathbf{H}^{n-k}(\Omega,\partial\Omega),\,\,\,\mathbf{H}_{k}(\Omega,\partial\Omega)\cong\mathbf{H}^{n-k}(\Omega).$

In dimension 3 we have in particular (see [14, p. 423])

$\displaystyle\text{dim}\,\mathbf{H}_{0}(\Omega)$ $\displaystyle=\text{number
of components of $\Omega$}$ $\displaystyle\text{dim}\,\mathbf{H}_{1}(\Omega)$
$\displaystyle=\text{total genus of $\partial\Omega$}$
$\displaystyle\text{dim}\,\mathbf{H}_{2}(\Omega)$ $\displaystyle=\text{number
of components of $\partial\Omega$}-\text{number of components of $\Omega$}$
$\displaystyle\text{dim}\,\mathbf{H}_{3}(\Omega)$ $\displaystyle=0.$

##  6 Reduction of Second Order Equation to Dirac-Beltrami Equation

###  6.1 The Dirichlet Problem

We now consider the Dirichlet problem (1.3) and assume that the endomorphism
field $A\in L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$ is normal, i.e.,
$A^{\ast}(x)A(x)=A(x)A^{\ast}(x)$ for a.e. $x\in\Omega$ and satisfies the
uniform ellipticity bounds

$\displaystyle\lambda|v|^{2}\leq\langle
A(x)v,v\rangle,\quad\|A(x)\|\leq\Lambda\,\,\text{ for a.e. $x$},$ (6.1)

or in the case if $A$ is symmetric, i.e., $A^{\ast}(x)=A(x)$,

$\displaystyle\lambda|v|^{2}\leq\langle A(x)v,v\rangle\leq\Lambda|v|^{2}$

for all $v\in{\mathbb{R}}^{n}$ and a.e. $x\in\Omega$. Consider weak solution
$u\in W^{1,2}(\Omega)$ of the equation

$\displaystyle\text{div}A(x)\nabla u(x)=0$ (6.2)

in $\Omega$. Equation (6.2) is the Euler-Lagrange equation for the energy
functional

$\displaystyle\mathcal{E}_{A}[u]=\int_{\Omega}\langle A(x)\nabla u(x),\nabla
u(x)\rangle dx,$

and call

$\displaystyle\mathscr{E}_{A}[u]=\langle A(x)\nabla u(x),\nabla u(x)\rangle$

the energy density. The weak Euler-Lagrange equations for
$\mathcal{E}_{A}[\cdot]$ shows that $A(x)\nabla u(x)$ is a weakly divergence
free vector field for any weak solution $u$. The integration by parts formula
(C.1) shows that the energy density $\mathscr{E}_{A}[u]=\langle A(x)\nabla
u(x),\nabla u(x)\rangle$ is a _null-lagrangian_. Indeed,

$\displaystyle\mathcal{E}_{A}[u]$ $\displaystyle=\int_{\Omega}\langle
A(x)\nabla u(x),\nabla u(x)\rangle dx=\int_{\partial\Omega}\langle
u(y)\nu(y),A(y)\nabla u(y)\rangle d\sigma(y).$

We also notice that _conormal derivative_
$\partial_{\nu_{A}}u(x):=\langle\nu(x),A(y)\nabla u(x)\rangle$ is nothing but
the Neumann data for $\text{div}A\nabla u$, and so the energy of the solution
can be expressed a the pairing of the Dirichlet and Neumann data. If we let
$\Lambda_{DN}$ denote the _Dirichlet to Neumann map_ , the energy in terms of
the boundary data becomes

$\displaystyle\mathcal{E}_{A}[\phi]=\int_{\partial\Omega}\phi(x)\Lambda_{DN}(\phi)(x)d\sigma(x).$

###  6.2 Hodge Theory and Gauge Conditions

By assumption $\nabla u\in L^{2}(\Omega,{\mathbb{R}}^{n})$. Since $u$ is a
weak solution get the div-curl couple $(B,E)$ defined by

$\displaystyle E(x)=\nabla u(x),\,\,\,B(x)=A(x)\nabla u(x)$

where $dE(x)=0$ and $\delta B(x)=0$ in the sense of distributions. We have the
$L^{2}$-estimate

$\displaystyle\|B\|_{L^{2}}=\int_{\Omega}|B(x)|^{2}dx=\int_{\Omega}|A(x)\nabla
u(x)|^{2}dx\leq\|A\|_{\infty}\|\nabla
u\|_{L^{2}}\leq\Lambda_{E}\|E\|_{L^{2}}.$

Since $\delta B(x)=0$ we can always find a $v\in
W^{1,2}(U,\Lambda^{2}{\mathbb{R}}^{n})$ locally in a simply connected open set
$U\Subset\Omega$ such that $\delta v(x)=B(x)$. Of course such $v$ is not
unique and it is natural to try and impose the exact gauge condition $dv(x)=0$
so that $\mathcal{D}^{\tiny+}v=(d+\delta)v=B$ locally. If such condition can
be imposed, then this determines a unique $v$ up to harmonic bivector fields,
i.e. $h\in W^{1,2}(U,\Lambda^{2}{\mathbb{R}}^{n})$ such that $dh(x)=\delta
h(x)=0$. However, without boundary conditions this space is is still infinite
dimensional. A natural choice is to impose either tangential or normal
boundary conditions on $v$. If $U$ is simply connected this determines a
unique $v$ as Theorem 6.1 shows. If on the other hand we want to find a
_global_ bivector field $v$ there are topological restrictions for the
existence of such $v$.

We have the following theorem adapted to the Euclidean setting:

######  Theorem 6.1 (Theorem 3.2.5 in [54]).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a smooth ($C^{2}$) compact domain.
Given $\chi\in W^{s,p}(\Omega,\Lambda^{k+1}{\mathbb{R}}^{n})$, $\rho\in
W^{s,p}(\Omega,\Lambda^{k-1}{\mathbb{R}}^{n})$ and $\psi\in
W^{s+1-1/p,p}(\Omega,\Lambda^{k}{\mathbb{R}}^{n})|_{\partial\Omega}$, with
$s\geq 0$ and $p>1$ the boundary value problem

$\displaystyle\left\\{\begin{array}[]{ll}d\omega(x)=\chi(x),&x\in\Omega,\\\
\delta\omega(x)=\rho(x),&x\in\Omega,\\\
\omega_{T}(x)=\psi_{T},&x\in\partial\Omega.\end{array}\right.$

is solvable if and only if $\delta\rho=0$ in the sense of distributions and

$\displaystyle(\rho,\kappa)=\int_{\Omega}\langle\rho(x),\kappa(x)\rangle
dx=0,\,\,\forall\,\kappa\in\mathcal{H}_{T}^{k-1}(\Omega),$

and $d\chi=0$ in the sense of distributions, $\chi_{T}=d\psi_{T}$, and

$\displaystyle(\chi,\lambda)=\int_{\partial\Omega}\langle\psi_{T}(x),\lambda_{N}(x)\rangle
dx,\,\,\forall\,\lambda\in\mathcal{H}_{T}^{k+1}(\Omega),$

The solution is unique up to an arbitrary harmonic field
$h\in\mathcal{H}_{T}^{k}(\Omega)$. Moreover, we have the Sobolev norm estimate

$\displaystyle\|\omega\|_{W^{s+1,p}(\Omega)}\leq
C(\|\chi\|_{W^{s,p}(\Omega)}+\|\rho\|_{W^{s,p}(\Omega)}+\|\psi\|_{W^{s+1-1/p,p}(\partial\Omega)}).$
(6.3)

In the case of interest for us $k=2$, $s=0$ and $p=2$. We now _make the
choice_ to set $\chi=0$. Then of course $\chi=0\in
L^{2}(\Omega,\Lambda^{3}{\mathbb{R}}^{n})$, $\rho=B\in
L^{2}(\Omega,\Lambda^{1}{\mathbb{R}}^{n})$ and $\psi=0\in
W^{1/2,2}(\Omega,\Lambda^{2}{\mathbb{R}}^{n})|_{\partial\Omega}$ and
$d\psi=0\in W^{1,-1/2}(\Omega,\Lambda^{2}{\mathbb{R}}^{n})|_{\partial\Omega}$.
This gives us the boundary value problem for $v$

$\displaystyle\left\\{\begin{array}[]{ll}\delta v(x)=B(x),&x\in\Omega,\\\
dv(x)=0,&x\in\Omega,\\\ v_{T}(x)=0,&x\in\partial\Omega.\end{array}\right.$
(6.7)

According to Theorem 3.2.5 this problem has a solution if and only if

$\displaystyle
0=(\chi,\lambda)=\int_{\partial\Omega}\langle\psi_{T}(x),\lambda_{N}(x)\rangle
dx\,\,\forall\,\lambda\in\mathcal{H}_{T}^{2}(\Omega),$

which is automatically true since $\psi_{T}=0$ by choice and

$\displaystyle(\rho,\kappa)=(B,\kappa)=0,\,\,\forall\,\kappa\in\mathcal{H}_{T}^{1}(\Omega).$

This is not necessarily true unless
$\mathbf{H}_{rel}^{1}(\Omega)\cong\mathcal{H}_{T}^{1}(\Omega)=\\{0\\}$. We
therefore assume that $\mathbf{H}_{rel}^{1}(\Omega)=\\{0\\}$. This is
equivalent to assuming that $\Omega$ is simply connected. From now on we
therefore assume this. In particular (6.7) is equivalent to the inhomogenous
problem for the Hodge–Dirac operator

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny+}v(x)=B(x),&x\in\Omega,\\\
v_{T}(x)=0,&x\in\partial\Omega.\end{array}\right.$ (6.10)

This equation in turn can be reduced to the corresponding homogenous equation
by the Cauchy transform (see section 7). Applying Theorem 1.7 in [44] for the
homogenous Hodge–Dirac equation on Lipschitz domains one can extend the
Theorem 3.2.5 in [54] to Lipschitz domains. Alternatively one can directly
prove Theorem 3.2.5 for Lipschitz domains using Hodge decompositions for
multivector fields with zero tangential trace as in Chapter 10.3 Theorem
10.3.1 in [51]. We omit the details. Note however that in the case of
Lipschitz domain the estimate (6.3) need not hold, see section 6.7. Instead,
we have the estimate

$\displaystyle\|v\|_{W^{1,2}_{d,\delta}(\Omega)}\leq C\|B\|_{L^{2}(\Omega)}.$

Hence we have a unique $v$ up to an arbitrary harmonic field in
$h\in\mathcal{H}_{T}^{2}(\Omega)$. However, since
$\mathcal{H}_{T}^{2}(\Omega)=\\{0\\}$, this $v$ is unique. We call $v$ the
_tangential $A$-harmonic conjugate_ to $u$. The additional differential
condition $dv(x)=0$ in (6.7) can be viewed as imposing a gauge condition on
the auxiliary field $v$ much in the same way as one can impose gauge
conditions on the electromagnetic potential in electrodynamics. Moreover, by
Theorem 6.1 $v\in W^{1,2}(\Omega)$ in the case when $\Omega$ is a
$C^{2}$-domain. We now define the multivector field

$\displaystyle F(x)=u(x)+v(x)$

which in view of Theorem 6.1 lies in
$W^{1,2}_{d,\delta}(\Omega,\Lambda^{0}{\mathbb{R}}^{n}\oplus\Lambda^{2}{\mathbb{R}}^{n})$.
We note that

$\displaystyle\overline{F(x)}=u(x)-v(x),$

so that $u(x)=\frac{1}{2}(F(x)+\overline{F(x)})$ and
$v(x)=\frac{1}{2}(F(x)-\overline{F(x)})$. Since the anti-involution can be
thought of as generalization of complex conjugation we see that we can think
of $u$ and $v$ as the“’real” and “imaginary” part of $F$ in analogy with the
complex case. Furthermore, $F_{T}(x)=u(x)=\phi(x)$, $x\in\partial\Omega$. We
now relate the div-curl couple $(B,E)$ to $F$. We have

$\displaystyle E(x)+B(x)$ $\displaystyle=du(x)+\delta
v(x)=(d+\delta)u(x)+(d+\delta)v(x)=\mathcal{D}^{\tiny+}F(x),$ $\displaystyle
E(x)-B(x)$ $\displaystyle=du(x)-\delta
v(x)=(d-\delta)u(x)+(d-\delta)v(x)=\mathcal{D}^{\tiny-}F(x).$

And the other hand, the algebraic relation between $E$ and $B$ gives

$\displaystyle\mathcal{D}^{\tiny+}F(x)=E(x)+B(x)$
$\displaystyle=(I+A(x))E(x)=(I+A(x)^{-1})B(x),$
$\displaystyle\mathcal{D}^{\tiny-}F(x)=E(x)-B(x)$
$\displaystyle=(I-A(x))E(x)=-(I-A(x)^{-1})B(x).$

Hence,

$\displaystyle\mathcal{D}^{\tiny-}F(x)$
$\displaystyle=(I-A(x))\circ(I+A(x))^{-1}\mathcal{D}^{\tiny+}F(x)$

We set $\mathcal{M}(x):=(I-A(x))\circ(I+A(x))^{-1}$, the _Cayley transform_ of
$A$. Then we get the _Beltrami equation_

$\displaystyle\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x).$

Define

$\displaystyle\mathscr{M}(x):=\|\mathcal{M}(x)\|.$ (6.11)

Then ellipticity bound (1.9) together with Lemma A.4 implies that

$\displaystyle M:=\|\mathscr{M}(x)\|_{L^{\infty}(\Omega)}<1.$ (6.12)

If in addition $\Lambda=1$ and we set $L:=\lambda^{-1}$, then $M=(L-1)/(L+1)$.
If we furthermore set $\mathcal{K}:=2(1+M^{2})/(1-M^{2})$, then
$\mathcal{K}=(L^{2}+1)/(4L)$ and any solution $F$ satisfies the distortion
inequality

$\displaystyle|dF(x)|^{2}+|\delta F|^{2}\leq\mathcal{K}\langle dF(x),\delta
F(x)\rangle.$

Thus $F\in
W^{1,2}_{d,\delta}(\Omega,\Lambda^{0}{\mathbb{R}}^{n}\oplus\Lambda^{2}{\mathbb{R}}^{n})$
solves the half-Dirichlet boundary value problem

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)&x\in\Omega,\\\
F_{T}(x)=\phi(x),&x\in\partial\Omega.\end{array}\right.$

Here we see the analogy with classical linear Beltrami equation in the plane
$\overline{\partial}f(z)=\mathcal{M}(z)\overline{\partial f(z)}$, and we can
think of the pair of Hodge-Dirac operators
$(\mathcal{D}^{\tiny-},\mathcal{D}^{\tiny+})$ as a generalization of the pair
of complex derivatives $(\overline{\partial},\partial)$.

###  6.3 Converse Direction

Assume now that we have a multivector field $F\in
W^{1,2}_{d,\delta}(\Omega,\Lambda^{0}\oplus\Lambda^{2})$ with tangential
boundary value $F_{T}=\phi$. It then follows that
$\mathcal{D}^{\tiny+}F,\mathcal{D}^{\tiny-}F\in
L^{2}(\Omega,\Lambda^{1}\oplus\Lambda^{3})$. Thus in particular
$\langle\mathcal{D}^{\pm}\rangle_{3}$ need not be zero. However, in the
Beltrami equation
$\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)$, the
measurable field $\mathcal{M}\in
L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$, so $\mathcal{M}(x)$ is a
linear operator on vectors, and so the action on $\mathcal{M}$ on tri-vectors
are not a priori defined and the equation
$\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)$ does not
make sense unless $\mathcal{M}$ is extended as a measurable field
$\widetilde{\mathcal{M}}\in
L^{\infty}(\Omega,\mathcal{L}(\Lambda{\mathbb{R}}^{n}))$. The most natural
extension is to take $\widetilde{\mathcal{M}}=\widehat{\mathcal{M}}$, the
exterior extension of $\mathcal{M}$. In particular, the exterior extension is
grade preserving, so
$\widehat{\mathcal{M}}(\Lambda^{p}{\mathbb{R}}^{n})\subset\Lambda^{p}{\mathbb{R}}^{n}$,
and if $\mathcal{M}(x)$ is invertible, so is $\widehat{\mathcal{M}}(x)$ with
inverse $(\widehat{\mathcal{M}}(x))^{-1}=\widehat{\mathcal{M}^{-1}}(x)$.
Moreover, by Proposition A.1 in the Appendix, if $\|\mathcal{M}\|<1$ then
$\|\widehat{\mathcal{M}}\|\leq\|\mathcal{M}\|<1$.

In order not to overburden the notation, we will keep writing $\mathcal{M}(x)$
instead of $\widehat{\mathcal{M}}(x)$. With this in mind the Beltrami equation
$\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)$ is well
defined. Splitting $F=\langle F(x)\rangle_{0}+\langle
F(x)\rangle_{2}=u(x)+v(x)$, we find that

$\displaystyle\mathcal{D}^{\tiny+}F(x)=du(x)+\delta
v(x)+dv(x),\,\,\mathcal{D}^{\tiny-}F(x)=du(x)-\delta v(x)+dv(x).$

Since $\mathcal{M}$ is grade preserving by construction, the Beltrami equation
$\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)$ reduces to
the system

$\displaystyle du(x)-\delta v(x)$ $\displaystyle=\mathcal{M}(x)(du(x)+\delta
v(x)),$ (6.13) $\displaystyle dv(x)$ $\displaystyle=\mathcal{M}(x)dv(x).$
(6.14)

Assuming that $M<1$ so that the Dirac-Beltrami operator is elliptic, this
implies

$\displaystyle|dv(x)|\leq M|dv(x)|<|dv(x)|$

which implies that $dv(x)=0$. Hence $\mathcal{D}^{\pm}F(x)$ are vector fields
$v$ is a closed bivector field. Solving for $\delta v$ in equation (6.13)
gives

$\displaystyle\delta
v(x)=(I+\mathcal{M}(x))^{-1}\circ(I-\mathcal{M}(x))(du(x))=A(x)du(x).$

Since $\delta^{2}v(x)=0$, it follows that $u$ solves the scalar second order
(elliptic) equation

$\displaystyle\delta A(x)du(x)=0.$

Thus, the scalar part of $F$ with $F$ as above solves a linear $A$-harmonic
equation. On the other hand we may from (6.13) solve for $du(x)$ instead
giving

$\displaystyle du(x)$
$\displaystyle=(I-\mathcal{M}(x))^{-1}\circ(I+\mathcal{M}(x))(\delta v(x))$
$\displaystyle=A(x)^{-1}\delta v(x).$

Since $d^{2}u(x)=0$ in the sense of distributions it follows that $v$ solves
the equation

$\displaystyle dA(x)^{-1}\delta v(x)=0.$ (6.15)

We call (6.15) the _A-harmonic conjugate equation_ of $\text{div}A(x)\nabla
u(x)=0$.

To conclude, we have proven the following theorem:

######  Theorem 6.2.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a simply connected Lipschitz domain.
Let $F\in W^{1,2}_{d,\delta}(\Omega,\Lambda^{(0,2)})$ be the unique solution
to the half-Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny-}F(x)=\widehat{\mathcal{M}}(x)\mathcal{D}^{\tiny+}F(x)&x\in\Omega,\\\
F_{T}(x)=\phi(x),&x\in\partial\Omega.\end{array}\right.$

where $\widehat{\mathcal{M}}(x)$ is the exterior extension of a some
$\mathcal{M}\in L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$ satisfying
$\|\|\mathcal{M}(x)\|\|_{L^{\infty}}=m<1$. Then $u=\langle F\rangle_{0}\in
W^{1,2}(\Omega)$ is the unique weak solution of the Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{ll}\text{div}\,A(x)\nabla
u(x)=0&x\in\Omega,\\\ u(x)=\phi(x),&x\in\partial\Omega.\end{array}\right.,$

where $A(x)=(I-\mathcal{M}(x))\circ(I+\mathcal{M}(x))^{-1}$.

######  Remark 6.1.

In the special case when $\mathcal{M}(x)=\mu(x)I_{{\mathbb{R}}^{n}}$, for some
scalar function $\mu$, we may well take the extension
$\widetilde{\mathcal{M}}(x)=\mu(x)I_{\Lambda{\mathbb{R}}^{n}}$ instead of
$\widehat{\mathcal{M}}(x)$.

Finally, we recall Theorem 1 in [29]. Define the scalar function

$\displaystyle\lambda(x):=\left\\{\begin{array}[]{ll}\displaystyle\frac{|\mathcal{D}^{\tiny-}F(x)|}{|\mathcal{D}^{\tiny+}F(x)|}&\text{if
$\mathcal{D}^{\tiny+}F(x)\neq 0$}\\\ 0&\text{if
$\mathcal{D}^{\tiny+}F(x)=0$}\end{array}\right.,$

and the multivector field (being either a unit vector or 1)

$\displaystyle
a(x):=\left\\{\begin{array}[]{ll}\displaystyle\frac{\mathcal{D}^{\tiny-}F(x)-\lambda(x)\mathcal{D}^{\tiny+}F(x)}{|\mathcal{D}^{\tiny-}F(x)-\lambda(x)\mathcal{D}^{\tiny+}F(x)|}&\text{if
$\mathcal{D}^{\tiny-}F(x)-\lambda(x)\mathcal{D}^{\tiny+}F(x)\neq 0$}\\\
1&\text{if
$\mathcal{D}^{\tiny-}F(x)-\lambda(x)\mathcal{D}^{\tiny+}F(x)=0$}\end{array}\right..$

Then as $\mathcal{D}^{\tiny+}F(x)$ and $\mathcal{D}^{\tiny-}F(x)$ are vectors
and by Theorem 1 in [29] and Lemma 3.2 this can be written as

$\displaystyle\mathcal{D}^{\tiny-}F(x)=-\lambda(x)a(x)\text{\tiny{$\triangle$}}\mathcal{D}^{\tiny+}F(x)\text{\tiny{$\triangle$}}a(x)^{-1}.$

In particular, in view of Theorem 1 in [29] we may always assume that
$\mathcal{M}(x)$ is a conformal linear map. Moreover, if we define the
conformal map $\mathcal{C}(x)\in\mathcal{L}({\mathbb{R}}^{n})$ through

$\displaystyle\mathcal{C}(x)v=-\lambda(x)a(x)\text{\tiny{$\triangle$}}v\text{\tiny{$\triangle$}}a(x)^{-1}$

for all $v\in{\mathbb{R}}^{n}$, then

$\displaystyle\|\mathcal{C}(x)\|=\lambda(x)=\frac{|\mathcal{D}^{\tiny-}F(x)|}{|\mathcal{D}^{\tiny+}F(x)|}\leq\frac{\|\mathcal{M}(x)\||\mathcal{D}^{\tiny+}F(x)|}{|\mathcal{D}^{\tiny+}F(x)|}=\|\mathcal{M}(x)\|.$

and we see that the ellipticity bounds are preserved.

###  6.4 Comparison to the two dimensional case

In dimension two we recall the algebraic isomorphism
$\Delta^{ev}{\mathbb{R}}^{2}\cong\Lambda^{0}{\mathbb{R}}^{2}\oplus\Lambda^{2}{\mathbb{R}}^{2}\cong{\mathbb{C}}$
of the even subalgebra of the euclidean Clifford algebra
$\Delta{\mathbb{R}}^{2}$ with the complex numbers. In particular we can
identify the imaginary unit $i$ with the euclidean volume form $e_{12}$.
Moreover, the gauge condition $dv(x)=0$ for the $A$-harmonic conjugate of $u$
is _automatically_ satisfied and therefore vacuous in dimension two, contrary
to higher dimensions. Moreover, the Hodge star map can be expressed using
Clifford algebra by

$\displaystyle\star w=e_{12}\text{\tiny{$\triangle$}}w$

for any multivector $w\in\Lambda V$. Hence, using $v=\star(v\star)$ and
setting $w=v\star$, we have

$\displaystyle F=u+v=u+\star(v\star)=u+e_{12}w\cong u+iw,$

where $w$ is a scalar valued field. Hence, the usual stream function $v$ in
complex notation can be identified with $v\star$ in the framework of this
paper.

###  6.5 Multiply connected domains and inhomogeneous equations

Consider the Dirichlet problem for the Laplace equation

$\displaystyle\left\\{\begin{array}[]{ll}\Delta u(z)=0&z\in\Omega\\\
u(z)=\phi(z)&z\in\partial\Omega\end{array}\right.$ (6.18)

on a bounded smooth domain $\Omega\subset{\mathbb{C}}$ in the plane. If
$\Omega$ is simply connected it is well-known that $u$ is the real of a
holomorphic function $f$, i.e., $f=u+iv$, where $v$ is the harmonic conjugate
of $v$. In this case the Dirichlet problem for Laplace equation becomes
equivalent to the boundary value problem

$\displaystyle\left\\{\begin{array}[]{ll}\overline{\partial}f(z)=0&z\in\Omega\\\
\mathfrak{R}e[f(z)]=\phi(z)&z\in\partial\Omega\end{array}\right.$ (6.21)

for holomorphic functions. If on the other hand $\Omega$ is not simply
connected there need not exist a global harmonic conjugate $v$ of $u$, the two
boundary value problems are not equivalent. If one would like to avoid the use
of multivalued holomorphic functions in multiply connected domains there exist
a nice substitute called the logarithmic conjugation theorem.

######  Theorem 6.3.

Let $\Omega$ be a finitely connected domain with $K_{1},K_{2},...,K_{n}$
denoting the bounded components of the complement of $\Omega$. For each $j$,
let $a_{j}$ be in $K_{j}$. If $u$ is a real valued harmonic function on
$\Omega$, then there exists a holomorphic function $f$ on $\Omega$ and real
numbers $c_{1},c_{2},...,c_{n}$ such that

$\displaystyle
u(z)=\mathfrak{R}e[f(z)]+c_{1}\log|z-a_{1}|+c_{2}\log|z-a_{2}|+...+c_{n}\log|z-a_{n}|$

for every $z\in\Omega$.

For a proof see [6]. Using this theorem we see that the Dirichlet problem for
Laplace’s equation in a multiply connected domain is equivalent to boundary
value problem

$\displaystyle\left\\{\begin{array}[]{ll}\overline{\partial}f(z)=0&z\in\Omega\\\
\mathfrak{R}e[f(z)]=\phi(z)-\sum_{j=1}^{n}c_{j}\log|z-a_{j}|,&z\in\partial\Omega.\end{array}\right.$
(6.24)

Here the constants $c_{j}$ are determined by the period conditions. The
usefulness of Theorem 6.3 is that it reduces the boundary value problem for
Laplace equation on a multiply connected domain to a boundary value problem
for the Cauchy-Riemann operator. Unfortunately, the proof of Theorem 6.3 does
not generalize to higher dimension. Instead we will proceed by a different
route, again using Hodge theory.

Thus let $\Omega\subset{\mathbb{R}}^{n}$ be a finitely connected Lipschitz
domain and we consider the Dirichlet problem (1.3) on $\Omega$. Then
$\mathcal{H}_{T}^{1}(\Omega)$ is finite dimensional. Moreover we have the
unique orthogonal splitting

$\displaystyle A\nabla u=B=B_{T}^{\perp}+\omega_{T},$

where $B_{T}^{\perp}\in\mathcal{H}_{T}^{1}(\Omega)^{\perp}$ and
$\omega_{T}\in\mathcal{H}_{T}^{1}(\Omega)$. We can now apply Theorem 6.1, to
get a bivector field $v$ such that

$\displaystyle\left\\{\begin{array}[]{ll}dv(x)=0,&x\in\Omega,\\\ \delta
v(x)=B_{T}^{\perp}(x),&x\in\Omega,\\\
v_{T}(x)=0&x\in\partial\Omega.\end{array}\right.$

This $v$ is unique up to bivectors in $\mathcal{H}_{T}^{2}(\Omega)$. We define

$\displaystyle F(x)=u(x)+v(x).$

Then

$\displaystyle\mathcal{D}^{\tiny+}F(x)$ $\displaystyle=du(x)+\delta
v(x),\quad\mathcal{D}^{\tiny-}F(x)=du(x)-\delta v(x).$

On the other hand we consider the div-curl pair $(B,E)=(A(x)\nabla u(x),\nabla
u(x))$. We have

$\displaystyle E(x)\pm B(x)=\nabla u(x)\pm A(x)\nabla u(x)=du(x)\pm\delta
v(x)\pm\omega_{T}(x)=\mathcal{D}^{\pm}F(x)\pm\omega_{T}(x).$

Thus,

$\displaystyle\mathcal{D}^{+}F(x)+\omega_{T}(x)=(I+A(x))E(x),\quad\mathcal{D}^{-}F(x)-\omega_{T}(x)=(I-A(x))E(x).$

This finally gives

$\displaystyle\mathcal{D}^{-}F(x)-\omega_{T}(x)$
$\displaystyle=(I-A(x))\circ(I+A(x))^{-1}(\mathcal{D}^{+}F(x)+\omega_{T}(x))$
$\displaystyle\Longleftrightarrow$
$\displaystyle\mathcal{D}^{-}F(x)-M(x)\mathcal{D}^{+}F(x)$
$\displaystyle=(I+M(x))\omega_{T}(x).$

This leads to the boundary value problem

$\displaystyle\left\\{\begin{array}[]{cl}\mathcal{D}^{-}F(x)-M(x)\mathcal{D}^{+}F(x)=(I+M(x))\omega_{T}(x),&x\in\Omega,\\\
F_{T}=\phi(x),&x\in\partial\Omega.\end{array}\right.$ (6.27)

Now $\omega_{T}$ can be determined form the orthogonality criterion, namely
that $\delta v\perp\mathcal{H}_{T}^{1}(\Omega)$. Since $\delta v=\delta F$
this gives us the finite linear equation system

$\displaystyle\int_{\Omega}\langle\delta F(x),\kappa(x)\rangle
dx=0,\,\,\forall\,\kappa\in\mathcal{H}_{T}^{1}(\Omega)$

which uniquely determines $\omega_{T}$. Solutions of (6.27) are unique up to
elements in $\mathcal{H}_{T}^{2}(\Omega)$. By imposing the orthogonality
condition that $F\perp\mathcal{H}_{T}^{2}(\Omega)$ gives us a unique solution.
To conclude we have proven

######  Theorem 6.4.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded finitely connected domain. To
every solution of (1.3) on $\Omega$ there exists a unique solution $F\in
W^{1,2}_{d,\delta}(\Omega,\Lambda^{(0,2)})$ of the boundary value problem

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)+(I+\mathcal{M}(x))\omega_{T}&x\in\Omega,\\\
F_{T}=\phi&x\in\partial\Omega\\\ F\perp\mathcal{H}_{T}^{2}(\Omega)\\\
\omega_{T}\in\mathcal{H}_{T}^{1}(\Omega)\\\ \int_{\Omega}\langle\delta
F(x),\kappa(x)\rangle
dx=0,\,\,\forall\,\kappa\in\mathcal{H}_{T}^{1}(\Omega)\\\ \end{array}\right..$
(6.33)

We now consider the inhomogeneous Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{ll}\text{div}A(x)\nabla
u(x)=\text{div}\,G(x),&x\in\Omega,\\\
u(x)=\phi(x),&x\in\partial\Omega.\end{array}\right.$ (6.36)

on a simply connected Lipschitz domain $\Omega$ and vector field $G\in
L^{2}(\Omega,{\mathbb{R}}^{n})$.

We now use Theorem 6.1 to define the bivector field $v$ as the unique solution
of

$\displaystyle\left\\{\begin{array}[]{ll}dv(x)=0,&x\in\Omega,\\\ \delta
v(x)=\mathcal{A}(x)\nabla u(x)-G(x)&,x\in\Omega\\\
v_{T}(x)=0,&x\in\partial\Omega\end{array}\right..$

We now define

$\displaystyle F(x)=u(x)+v(x)$

Then

$\displaystyle\mathcal{D}^{\tiny+}F(x)$ $\displaystyle=du(x)+\delta
v(x)=du(x)+\mathcal{A}(x)du(x)-G(x),$ $\displaystyle\mathcal{D}^{\tiny-}F(x)$
$\displaystyle=du(x)-\delta v(x)=du(x)-\mathcal{A}(x)du(x)+G(x).$

Hence,

$\displaystyle du(x)=(I+\mathcal{A}(x))(\mathcal{D}^{\tiny+}F(x)+G(x))$

which gives

$\displaystyle\mathcal{D}^{\tiny-}F(x)$
$\displaystyle=(I-\mathcal{A}(x))du(x)+G(x)=(I-\mathcal{A}(x))(I-\mathcal{A}(x))(\mathcal{D}^{\tiny+}F(x)+G(x))+G(x)$
$\displaystyle=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)+(I+\mathcal{M}(x))G(x)$

In addition $F_{T}=\phi$. Thus we have proven the following theorem.

######  Theorem 6.5.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded simply connected Lipschitz
domain. To each solution of (6.36) there exists a unique solution $F\in
W^{1,2}_{d,\delta}(\Omega,\Lambda^{(0,2)})$ of the boundary value problem

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)+(I+\mathcal{M}(x))G(x)&x\in\Omega,\\\
F_{T}(x)=\phi,&x\in\partial\Omega\end{array}\right..$ (6.39)

for every $G\in L^{2}(\Omega,{\mathbb{R}}^{n})$.

###  6.6 Nonlinear uniformly elliptic equations

######  Definition 6.1.

Let $\mathcal{A}:\Omega\times{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be a mapping
that satisfies the conditions:

  * (i)

$\mathcal{A}:x\mapsto\mathcal{A}(x,\xi)$ is measurable for every
$\xi\in{\mathbb{R}}^{n}$.

  * (ii)

$\mathcal{A}:\xi\mapsto\mathcal{A}(x,\xi)$ is continuous for almost every
$x\in\Omega$.

  * (iii)

$\mathcal{A}(x,0)=0$ for all $x\in\Omega$.

  * (iv)

$\mathcal{A}$ satisfies the uniform ellipticity bound

$\displaystyle|\xi-\zeta|^{2}+|\mathcal{A}(x,\xi)-\mathcal{A}(x,\zeta)|^{2}\leq
K\langle\mathcal{A}(x,\xi)-\mathcal{A}(x,\zeta),\xi-\zeta\rangle\text{ for
a.e. $x\in\Omega$ and all $\xi,\zeta\in{\mathbb{R}}^{n}$}$

for some $K\geq 2$.

We want to solve Dirichlet problem for the nonlinear uniformly elliptic
equation

$\displaystyle\left\\{\begin{array}[]{ll}\text{div}\,\mathcal{A}(x,\nabla
u(x))=0&x\in\Omega,\\\ u(x)=\phi(x)&x\in\partial\Omega\end{array}\right..$
(6.42)

with structure field $\mathcal{A}$ as above for $u\in W^{1,2}(\Omega)$ and
$\phi\in W^{1/2,2}(\partial\Omega)$.

######  Definition 6.2.

Define the Cayley transform

$\displaystyle\mathcal{M}(x,\xi):=(I-\mathcal{A}(x,\xi))\circ(I+\mathcal{A}(x,\xi))^{-1}$

for each fixed $x$.

By Lemma A.5 $\mathcal{M}(x,\cdot):{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is
well defined for a.e. $x$. Moreover, $\mathcal{M}$ is measurable in $x$ and by
Lemma A.6 $\mathcal{M}$ is $\sqrt{\frac{K-2}{K+2}}$-Lipschitz continuous in
$\xi$ for a.e. $x$. We can now argue precisely as in the linear case. If
$\Omega$ is simply connected we define the bivector field $v$ to be a solution
of

$\displaystyle dv(x)=0,\,\,\,\delta v(x)=\mathcal{A}(x,\nabla
u(x)),\,\,\,v_{T}=0,$

and if $\Omega$ is multiply connected we do suitable modification as in
subsection 6.5. Then an identical argument proves that the multivector field
$F=u+v$ solves the boundary value problem

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x,\mathcal{D}^{\tiny+}F(x))&x\in\Omega,\\\
F_{T}(x)=\phi(x)&x\in\partial\Omega\end{array}\right..$ (6.45)

For the converse direction we note that
$\mathcal{D}^{\tiny+}F:\Omega\to\Lambda^{1}{\mathbb{R}}^{n}\oplus\Lambda^{3}{\mathbb{R}}^{n}$.
Thus $\mathcal{M}(x,\mathcal{D}^{\tiny+}F(x))$ is not a priori well-defined.
We therefore extend
$\mathcal{M}:\Omega\times{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ to a mapping
$\overline{\mathcal{M}}:\Omega\times\Lambda{\mathbb{R}}^{n}\to\Lambda{\mathbb{R}}^{n}$
by defining

$\displaystyle\overline{\mathcal{M}}(x,w)=\mathcal{M}(x,\langle
w\rangle_{1}).$

It the follows that the extended map is still
$\sqrt{\frac{K-2}{K+2}}$-Lipschitz. Indeed,

$\displaystyle|\overline{\mathcal{M}}(x,w_{1})-\overline{\mathcal{M}}(x,w_{2})|$
$\displaystyle=|\mathcal{M}(x,\langle w_{1}\rangle_{1})-\mathcal{M}(x,\langle
w_{2}\rangle_{1})|$ $\displaystyle\leq\sqrt{\frac{K-2}{K+2}}|\langle
w_{1}\rangle_{1}-\langle w_{2}\rangle_{1}|=\sqrt{\frac{K-2}{K+2}}|\langle
w_{1}-w_{2}\rangle_{1}|$
$\displaystyle\leq\sqrt{\frac{K-2}{K+2}}|w_{1}-w_{2}|.$

By abuse of notation we will write $\mathcal{M}$ instead of
$\overline{\mathcal{M}}$. An identical argument as in section 6.3, shows
$u=\frac{1}{2}(F+\overline{F})$ solves (6.42). We conclude this section by
mentioning that by rewriting the second order equation (6.42) as a nonlinear
Beltrami equation as in (6.45) allowed the authors of [1] to prove an improved
Hölder regularity for solutions of (6.42).

###  6.7 Ellipticity of the Dirac–Beltrami equation and Null-Lagrangian

We now consider the ellipticity of Dirac–Beltrami operator
$\mathcal{P}_{\mathcal{M}}=\mathcal{D}^{\tiny-}-\mathcal{M}(x)\mathcal{D}^{\tiny+}$.
We will consider the more general case when $\mathcal{P}_{\mathcal{M}}$ acts
on general multivector fields $F:\Omega\to\Lambda{\mathbb{R}}^{n}$ rather than
just special case when
$F:\Omega\to{\mathbb{R}}\oplus\Lambda^{2}{\mathbb{R}}^{n}$ considered in the
previous section. Following [8], the symbol
$\sigma_{\mathcal{M}}(x,\xi):=\sigma_{\mathcal{P}_{\mathcal{M}}}(x,\xi)$ acts
on multivectors $w\in\Lambda{\mathbb{R}}^{n}$ according to

$\displaystyle\sigma_{\mathcal{M}}(x,\xi)w=\xi\text{\scriptsize{$\triangledown$}}w-\mathcal{M}(x)\xi\text{\tiny{$\triangle$}}w,$

where $x,\xi\in{\mathbb{R}}^{n}$ and where $\mathcal{M}\in
L^{\infty}(\Omega,\mathcal{L}(\Lambda{\mathbb{R}}^{n}))$ such that
$\|\|\mathcal{M}\|\|_{L^{\infty}(\Omega)}<1$. Assume that for some $\xi\neq 0$
there exists an $w\neq 0$ such that $\sigma_{\mathcal{M}}(x,\xi)w=0$. Then

$\displaystyle\xi\text{\scriptsize{$\triangledown$}}w=\mathcal{M}(x)\xi\text{\tiny{$\triangle$}}w.$

Hence,

$\displaystyle|\xi\text{\scriptsize{$\triangledown$}}w|=|\xi||w|\leq\|\mathcal{M}(x)\|\xi||w|\Longrightarrow\|\mathcal{M}(x)\|\geq
1.$

Thus, $\mathcal{P}_{\mathcal{M}}$ is elliptic on a domain
$U\subset{\mathbb{R}}^{n}$ whenever $\sup_{x\in U}\|\mathcal{M}(x)\|<1$. By
Lemma A.2 in the appendix, this is equivalent to $A(x)$ being strictly
positive, i.e.,

$\displaystyle\langle A(x)\xi,\xi\rangle>0$

for all $\xi\in{\mathbb{R}}^{n}$ and all $x\in U$.

The Beltrami operator $\mathcal{P}_{\mathcal{M}}$ can be naturally defined on
$F\in\text{dom}_{max}(\mathcal{D}^{\tiny+})\cap\text{dom}_{max}(\mathcal{D}^{\tiny-})$,
where $\text{dom}_{max}$ is the maximal domain of an unbounded operator, (e.g.
[8]). In particular,

$\displaystyle W_{d,\delta}^{1,2}(\Omega,\Lambda)$
$\displaystyle=\text{dom}_{max}(\mathcal{D}^{\tiny+})\cap\text{dom}_{max}(\mathcal{D}^{\tiny-})$
$\displaystyle=\\{F\in\mathscr{D}^{\prime}(\Omega,\Lambda{\mathbb{R}}^{n}):F,\mathcal{D}^{\tiny+}F,\mathcal{D}^{\tiny-}F\in
L^{2}(\Omega,\Lambda{\mathbb{R}}^{n})\\}$
$\displaystyle=\\{F\in\mathscr{D}^{\prime}(\Omega,\Lambda{\mathbb{R}}^{n}):F,dF,\delta
F\in L^{2}(\Omega,\Lambda{\mathbb{R}}^{n})\\}.$

Without any assumptions on traces of $F$, one has for Lipschitz domains in
general the strict inclusion $W^{1,2}(\Omega,\Lambda)\subset
W_{d,\delta}^{1,2}(\Omega,\Lambda)$. This should be contrasted with the two
dimensional case. Indeed, by the identity

$\displaystyle\|Df(z)\|=|f_{\bar{z}}(z)|+|f_{z}(z)|$

for a mapping $f:\Omega\to{\mathbb{C}}$ it follows that

$\displaystyle
W^{1,2}(\Omega,{\mathbb{C}})=\\{f\in\mathscr{D}^{\prime}(\Omega,{\mathbb{C}}):f,f_{\bar{z}},f_{z}\in
L^{2}(\Omega,{\mathbb{C}})\\}.$

On the other hand we have for any $F\in W^{1,2}(\Omega,\Lambda)$ with
$\gamma_{T}F=0$ and $\Omega$ a bounded $C^{2}$-domain then by Gaffney’s
inequality with traces (see Corollary 2.1.6 in [54]) there exists a constant
$C>0$ depending only $\Omega$ such that

$\displaystyle\|F\|_{W^{1,2}(\Omega,\Lambda)}^{2}\leq
C\int_{\Omega}(|dF(x)|^{2}+|\delta F(x)|^{2}+|F(x)|^{2})dx$

Thus, if $F\in W_{d,\delta}^{1,2}(\Omega,\Lambda)$ and $\gamma_{T}F\in
W^{1/2,2}(\partial\Omega,\Lambda)$, we can write $F=F^{T}_{0}+\Phi$, with
$\Phi\in W^{1,2}(\Omega,\Lambda)$ and $F^{T}_{0}\in
W_{d,\delta}^{1,2}(\Omega,\Lambda)$ with $\gamma_{T}F^{T}_{0}=0$. Hence, by
the Gaffney inequality

$\displaystyle W^{1,2}(\Omega,\Lambda)=\\{F\in
W_{d,\delta}^{1,2}(\Omega,\Lambda):\gamma_{T}F\in
W^{1/2,2}(\partial\Omega,\Lambda)\\}.$

If however $\Omega$ is only a Lipschitz domain we have in general only
inclusion, see example 10.3.2 in [51].

We recall that the ellipticity condition (1.9) by Lemma A.4 implies the
distortion inequality

$\displaystyle|E(x)|^{2}+|B(x)|^{2}\leq\mathcal{K}\langle E(x),B(x)\rangle.$

for the div-curl couple $(B,E)=(\delta F,dF)$ and some $\mathcal{K}\geq 2$. In
[29], $\langle dF(x),\delta F(x)\rangle$ is called the Jacobian of the div-
curl pair $(B,E)$ and denoted by $J(x,\mathcal{F})$. However, since
$\mathscr{E}_{A}[u](x)=\langle dF(x),\delta F(x)\rangle$ and to avoid
confusion, we will instead refer to the _null-Lagrangian_ , e.g., [7, 28], as
the the energy density of the field $F$, and write $\mathscr{E}(x,F)$. Here we
recall that $\langle dF(x),\delta F(x)\rangle$ is a null-Lagrangian as

$\displaystyle\int_{\Omega}\langle dF(x),\delta F(x)\rangle dx=0$

for all $F\in C^{\infty}_{0}(\Omega,\Lambda{\mathbb{R}}^{n})$. Using that
$2(|dF(x)|^{2}+|\delta
F(x)|^{2})=|\mathcal{D}^{\tiny+}F(x)|^{2}+|\mathcal{D}^{\tiny-}F(x)|^{2}$ and
$4\langle dF(x),\delta
F(x)\rangle=|\mathcal{D}^{\tiny+}F(x)|^{2}-|\mathcal{D}^{\tiny-}F(x)|^{2}$,
the distortion inequality (6.46) can equivalently be expressed as

$\displaystyle|\mathcal{D}^{\tiny+}F(x)|^{2}+|\mathcal{D}^{\tiny-}F(x)|^{2}\leq\frac{1}{2}\mathcal{K}(|\mathcal{D}^{\tiny+}F(x)|^{2}-|\mathcal{D}^{\tiny-}F(x)|^{2}).$
(6.46)

We note that in dimension 2, using that $\|DF(x)\|^{2}=|\partial
F(x)|^{2}+|\overline{\partial}F(x)|^{2}$ and $\det(DF(x))=J(x,F)=|\partial
F(x)|^{2}-|\overline{\partial}F(x)|^{2}$ the inequality (6.46) can be written
as

$\displaystyle\|DF(x)\|^{2}\leq\mathcal{K}J(F,x),$

which is the classical differential inequality characterizing
$\mathcal{K}$-quasiregular mappings in the plane. Finally we note that in the
case when $\mathcal{M}(x)=\mathcal{M}$ is constant in the domain $\Omega$,
then for any smooth $U\Subset\Omega$ it follows form (4.3) that the following
version of Stokes’ theorem holds.

$\displaystyle\int_{U}\mathcal{P}_{\mathcal{M}}F(x)dx=\int_{\partial
U}\sigma_{\mathcal{M}}(\nu(y))F(y)d\sigma(y)$ (6.47)

for any $F\in W^{1,2}(\Omega,\Lambda)$.

When considering the general Dirac-Beltrami operator one should also
distinguish the special case when the linear operator
$\mathcal{M}(x)\in\mathcal{L}(\Lambda{\mathbb{R}}^{n})$ is grade preserving
for almost all $x\in\Omega$, i.e., when

$\displaystyle\mathcal{M}(x)(\Lambda^{k}{\mathbb{R}}^{n})\subset\Lambda^{k}{\mathbb{R}}^{n}$

for all $k=0,1,2,...,n$ and almost all $x\in\Omega$. Set
$\mathcal{M}_{k}(x)=\mathcal{M}_{k}(x)|_{\Lambda^{k}{\mathbb{R}}^{n}}$ and let
$F_{k}(x)=\langle F(x)\rangle_{k}$ for all $k=0,1,2,...,n$ so that
$F=\sum_{k=0}^{n}F_{k}$. Then for every $k=0,1,2,...,n$ (where $F_{j}$=0 if
$j<0$ or $j>n$)

$\displaystyle\langle\mathcal{D}^{\pm}F\rangle_{k}=dF_{k-1}+\delta F_{k+1},$

which gives the coupled system of equations

$\displaystyle dF_{k-1}-\delta F_{k+1}=\mathcal{M}_{k}(dF_{k-1}+\delta
F_{k+1})$

for $k=0,1,2,...,n$, or equivalently

$\displaystyle\delta F_{k}$
$\displaystyle=(I+\mathcal{M}_{k+1})^{-1}(I-\mathcal{M}_{k+1})dF_{k-2}$
$\displaystyle dF_{k}$
$\displaystyle=(I-\mathcal{M}_{k+1})^{-1}(I+\mathcal{M}_{k+1})\delta F_{k+2}\
$

Defining
$\mathcal{A}_{k}(x)=(I+\mathcal{M}_{k}(x))^{-1}(I-\mathcal{M}_{k}(x))$ and
using that $d^{2}=\delta^{2}=0$, we find that each component $F_{k}$ satisfies
the equations

$\displaystyle\delta\mathcal{A}_{k+1}(x)dF_{k}(x)$ $\displaystyle=0$ (6.48)
$\displaystyle d\mathcal{A}_{k-1}(x)^{-1}\delta F_{k}(x)$ $\displaystyle=0$
(6.49)

which form an _overdetermined elliptic system_. In the special case when
$\mathcal{M}(x)\equiv 0$, the Dirac–Beltrami equation reduces to the
homogeneous Hodge–Dirac equation

$\displaystyle\mathcal{D}^{\tiny-}F(x)=0.$

Since $(\mathcal{D}^{\tiny-})^{2}=-\Delta$, it follows that $\Delta
F_{k}=(d\delta+\delta d)F_{k}=0$ for each $k=0,1,2,...,n$. However, in view of
(6.48), we in fact have that

$\displaystyle\delta dF_{k}(x)$ $\displaystyle=0$ $\displaystyle d\delta
F_{k}(x)$ $\displaystyle=0$

or each $k=0,1,2,...,n$.

###  6.8 Hodge Duality

Assume that we have a self-dual Beltrami operator $\mathcal{M}(x)$, i.e., that
$\mathcal{M}^{c}(x)=\mathcal{M}(x)$. Then by definition any solution

$\displaystyle\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)$

is also a solution of

$\displaystyle\mathcal{D}^{\tiny-}F(x)=\mathcal{M}^{c}(x)\mathcal{D}^{\tiny+}F(x)\Longleftrightarrow\mathcal{D}^{\tiny-}F(x)=\star\mathcal{M}(x)(\mathcal{D}^{\tiny+}F(x)\star)\Longleftrightarrow\mathcal{D}^{\tiny-}F(x)\star=\mathcal{M}(x)(\mathcal{D}^{\tiny+}F(x)\star)$

Thus, by Lemma 4.1, any self-dual equation also solves

$\displaystyle\mathcal{D}^{\tiny+}(\widehat{F(x)}\star)=\mathcal{M}(x)\mathcal{D}^{\tiny-}(\widehat{F(x)}\star).$

Note that if $F:\Omega\to\Lambda^{ev}V$, then $\widehat{F(x)}=F(x)$ and so the
self-dual equations become

$\displaystyle\mathcal{D}^{\tiny+}(F(x)\star)=\mathcal{M}(x)\mathcal{D}^{\tiny-}(F(x)\star).$

In dimension two the Hodge star map coincides with the standard complex
structure $J$ of ${\mathbb{R}}^{2}$. In this case, the self-duality of
$\mathcal{M}(x)$ is equivalent to the condition that

$\displaystyle
J^{-1}\mathcal{M}(x)J=\mathcal{M}\Longleftrightarrow\mathcal{M}(x)J-J\mathcal{M}(x)=0\text{
for a.e. $x\in\Omega$},$

i.e., $\mathcal{M}(x)$ is ${\mathbb{C}}$-linear. In this case, Beltrami
equation reduces to the _anti_ -${\mathbb{C}}$-linear Beltrami equation

$\displaystyle\partial_{\overline{z}}f(z)=\mu(z)\overline{\partial f(z)}$

In subsection 6.3 we saw that any solution of
$\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)$ with $F\in
W^{1,2}(\Omega,\Lambda^{(0,2)})$ also solved an equation of the form

$\displaystyle\mathcal{D}^{\tiny+}F(x)=\mathcal{C}(x)\mathcal{D}^{\tiny-}F(x)$

where $\mathcal{C}(x)$ is a conformal map for a.e. $x$. In view of the
discussion in this subsection it follows that $F$ also solves its dual
equation

$\displaystyle\mathcal{D}^{\tiny+}(F(x)\star)=\mathcal{C}(x)\mathcal{D}^{\tiny-}(F(x)\star).$

##  7 The Cauchy Transform on ${\mathbb{R}}^{n}$.

###  7.1 Dirac Operators and the Cauchy Transforms

######  Definition 7.1 (Cauchy transform).

The _Cauchy transforms_ $\mathcal{C}^{\pm}$ are acting
$C^{\infty}_{0}({\mathbb{R}}^{n},\Lambda)$ are defined by

$\displaystyle\mathcal{C}^{+}\Phi(x)=\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\frac{y-x}{|y-x|^{n}}\text{\tiny{$\triangle$}}\Phi(y)dy,$
(7.1)
$\displaystyle\mathcal{C}^{-}\Phi(x)=\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\frac{x-y}{|x-y|^{n}}\text{\scriptsize{$\triangledown$}}\Phi(y)dy,$
(7.2)

for $\Phi\in C^{\infty}_{0}({\mathbb{R}}^{n},\Lambda)$.

We also recall the Riesz potentials

$\displaystyle\mathcal{I}_{\alpha}\Phi(x)=\frac{1}{\omega_{n}}\int_{{\mathbb{R}}^{n}}\frac{\Phi(y)}{|y-x|^{n-\alpha}}dy.$

where $0<\alpha<n$. We then clearly have the bound

$\displaystyle|\mathcal{C}^{+}\Phi(x)|\leq|\mathcal{I}_{1}|\Phi|(x)|$

for all $\Phi\in C^{\infty}_{0}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ and
all $x\in{\mathbb{R}}^{n}$.

The Hardy–Littlewood–Sobolev inequality for a Riesz potentials [55, p. 119]
shows that for $1<p<n$ and $p^{\ast}=\frac{np}{n-p}<+\infty$ there exists a
constant $A_{p}$ such that

$\displaystyle\|\mathcal{I}_{1}f(x)\|_{p^{\ast}}\leq A_{p}\|f\|_{p},$

for every $f\in L^{p}({\mathbb{R}}^{n})$. This immediately implies the
following theorem:

######  Theorem 7.1.

For all $1<p<n$ such that $p^{\ast}=\frac{np}{n-p}<+\infty$, the Cauchy
transforms $\mathcal{C}^{\pm}$ extends to continuous maps
$\mathcal{C}^{\pm}:L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})\to
L^{p^{\star}}(\Omega,\Lambda{\mathbb{R}}^{n})$. In particular there exists an
absolute constant $A_{p}=A(p,n)$ such that for all $f\in
L^{p^{\star}}(\Omega,\Lambda{\mathbb{R}}^{n})$.

$\displaystyle\|\mathcal{C}^{\pm}F\|_{p^{\star}}\leq A_{p}\|F\|_{p}.$

######  Theorem 7.2.

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{n}$. Let $G\in
L^{2}({\mathbb{R}}^{n},\Lambda)$. Then the weak derivatives satisfy

$\displaystyle\mathcal{D}^{\tiny+}\mathcal{C}^{+}G(x)=G(x)\,\,\,\text{and}\,\,\,\mathcal{D}^{\tiny-}\mathcal{C}^{-}G(x)=G(x).$

###### Proof.

First assume that $F\in C^{\infty}_{0}(\Omega)$. Then by the Cauchy-Pompieu
formula (4.4) that the solution to $\mathcal{D}^{\tiny+}F(x)=G(x)$ is given by
$F(x)=\mathcal{C}^{+}G(x)$. Since $C^{\infty}_{0}(\Omega)$ is dense in
$L^{2}(\Omega,\Lambda)$, the result follows by a standard approximation
argument. ∎

###  7.2 Hölder estimates for the Cauchy Transforms

######  Lemma 7.1.

If $n=2k$ then

$\displaystyle\frac{\xi-x}{|\xi-x|^{n}}-\frac{\xi-y}{|\xi-y|^{n}}=\frac{(\xi-x)\big{[}(\xi-y)^{n-1}-(\xi-x)^{n-1}\big{]}(\xi-y)}{|\xi-x|^{n}|y-\xi|^{n}}$

If $n=2k+1$ then

$\displaystyle\frac{\xi-x}{|\xi-x|^{n}}-\frac{\xi-y}{|\xi-y|^{n}}=\frac{(\xi-x)\big{[}|\xi-y|(\xi-y)^{n-2}-|\xi-x|(\xi-x)^{n-2}\big{]}(\xi-y)}{|\xi-x|^{n}|y-\xi|^{n}}.$

###### Proof.

If $n=2k$, then $|\xi-x|^{n}=(|\xi-x|^{2})^{k}=(\xi-x)^{2k}$. Consequently,

$\displaystyle(\xi-x)|\xi-y|^{n}-|\xi-x|^{n}(\xi-y)=(\xi-x)(\xi-y)^{2k-1}(\xi-y)+(\xi-x)(\xi-x)^{2k-1}(\xi-y)$
$\displaystyle=(\xi-x)\big{[}(\xi-y)^{2k-1}-(\xi-x)^{2k-1}\big{]}(\xi-y).$

If $n=2k+1$, then $|\xi-x|^{n}=|\xi-x|(|\xi-x|^{2})^{k}=|\xi-x|(\xi-x)^{2k}$.
Consequently,

$\displaystyle(\xi-x)|\xi-y|^{n}-|\xi-x|^{n}(\xi-y)=(\xi-x)|\xi-y|(y-\xi)^{2k-1}(\xi-y)-(\xi-x)|\xi-x|(\xi-x)^{2k-1}(\xi-y)$
$\displaystyle=(\xi-x)\big{[}|\xi-y|(\xi-y)^{2k-1}-|\xi-x|(\xi-x)^{2k-1}\big{]}(\xi-y).$

∎

######  Lemma 7.2.

If $n=2k$, $n>2$, we have the inequality

$\displaystyle\bigg{|}\frac{\xi-x}{|\xi-x|^{n}}-\frac{\xi-y}{|\xi-y|^{n}}\bigg{|}\leq
c_{n}\sum_{j=1}^{n-1}\frac{|x-y|^{j}|\xi-x|^{n-1-j}}{|\xi-x|^{n-1}|y-\xi|^{n-1}}$

If $n=2k+1$, $n>2$, we have the inequality

$\displaystyle\bigg{|}\frac{\xi-x}{|\xi-x|^{n}}-\frac{\xi-y}{|\xi-y|^{n}}\bigg{|}\leq
c_{n}\sum_{j=1}^{n-1}\bigg{(}\delta_{|\xi-x|>|\xi-y|}\frac{|x-y|^{j}|\xi-x|^{n-1-j}}{|\xi-x|^{n-1}|y-\xi|^{n-1}}+\delta_{|\xi-x|<|\xi-y|}\frac{|x-y|^{j}|\xi-y|^{n-1-j}}{|\xi-x|^{n-1}|y-\xi|^{n-1}}\bigg{)}.$

###### Proof.

First assume that $n=2k$. Set $\eta=\xi-x$. Then

$\displaystyle(\xi-y)^{n-1}-(\xi-x)^{n-1}=(\eta+x-y)^{n-1}+\eta^{n-1}=\eta^{n-1}+R-\eta^{n-1},$

where $R$ is sum of products, each containing at least one factor $(x-y)$.
Moreover, we have the estimate

$\displaystyle|R|\leq
c_{n}\sum_{j=1}^{n-1}|x-y|^{j}|\eta|^{n-1-j}=c_{n}\sum_{j=1}^{n-1}|x-y|^{j}|\xi-x|^{n-1-j}$

for some constant $c_{n}$ depending on $n$. Thus,

$\displaystyle\big{|}(\xi-x)\big{[}(\xi-y)^{n-1}-(\xi-x)^{n-1}\big{]}(\xi-y)\big{|}\leq
c_{n}\sum_{j=1}^{n-1}|\xi-x||x-y|^{j}|\xi-x|^{n-1-j}|\xi-y|.$

Now assume that $n=2k+1$. If $|\xi-y|>|\xi-x|$ set $\eta=\xi-x$, and we get

$\displaystyle|\xi-y|^{n-1}-|\xi-x|^{n-1}=|\eta+x-y|^{n-1}-|\eta|^{n-1}\leq\big{(}|\eta|+|x-y|\big{)}^{n-1}-|\eta|^{n-1}$
$\displaystyle\leq
c_{n}\sum_{j=1}^{n-1}|x-y|^{j}|\eta|^{n-1-j}=c_{n}\sum_{j=1}^{n-1}|x-y|^{j}|\xi-x|^{n-1-j}.$

If $|\xi-y|<|\xi-x|$ set $\zeta=\xi-y$, and we get

$\displaystyle|\xi-y|^{n-1}-|\xi-x|^{n-1}=|\zeta|^{n-1}-|\zeta+y-x|^{n-1}\leq\big{(}|\zeta|+|x-y|\big{)}^{n-1}-|\zeta|^{n-1}$
$\displaystyle\leq
c^{\prime}_{n}\sum_{j=1}^{n-1}|x-y|^{j}|\zeta|^{n-1-j}=c^{\prime}_{n}\sum_{j=1}^{n-1}|x-y|^{j}|\xi-y|^{n-1-j}.$

∎

######  Proposition 7.1.

Let $n>2$ and assume that $p>n$. If $\Phi\in
L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ then $\mathcal{C}^{+}\Phi$ is
Hölder continuous with Hölder constant $\displaystyle\alpha=1-\frac{n}{p}$.

######  Remark 7.1.

This is an extension of Theorem 4.3.13 in [2] for the Cauchy transform in the
plane to higher dimension. The only subtlety in proof compared to the plane is
the more complicated estimate in Lemma 7.2.

###### Proof.

We may assume that $|x-y|<1$. By Lemma 7.2,

$\displaystyle|\mathcal{C}^{+}\Phi(x)-\mathcal{C}^{+}\Phi(y)|=\bigg{|}\frac{1}{\omega_{n}}\int_{{\mathbb{R}}^{n}}\bigg{(}\frac{\xi-x}{|\xi-x|^{n}}-\frac{\xi-y}{|\xi-y|^{n}}\bigg{)}\text{\tiny{$\triangle$}}\Phi(\xi)d\xi\bigg{|}$
$\displaystyle\leq\frac{c_{n}}{\omega_{n}}\sum_{j=1}^{n-1}\int_{{\mathbb{R}}^{n}}\bigg{(}\frac{|x-y|^{j}|\xi-x|^{n-1-j}}{|\xi-x|^{n-1}|y-\xi|^{n-1}}+\frac{|x-y|^{j}|\xi-y|^{n-1-j}}{|\xi-x|^{n-1}|y-\xi|^{n-1}}\bigg{)}|\Phi(\xi)|d\xi$
$\displaystyle\leq\frac{c_{n}}{\omega_{n}}\sum_{j=1}^{n-1}|x-y|^{j}\int_{{\mathbb{R}}^{n}}\bigg{(}\frac{1}{|\xi-x|^{j}|y-\xi|^{n-1}}+\frac{1}{|\xi-x|^{n-1}|y-\xi|^{j}}\bigg{)}|\Phi(\xi)|d\xi.$

We now apply Hölder’s inequality with Hölder conjugate pair chosen so that
$(n-1)q<n$. Then we get

$\displaystyle|\mathcal{C}^{+}\Phi(x)-\mathcal{C}^{+}\Phi(y)|$
$\displaystyle\leq\frac{c_{n}}{\omega_{n}}\|\Phi\|_{L^{p}}\sum_{j=1}^{n-1}|x-y|^{j}\bigg{\\{}\bigg{(}\int_{{\mathbb{R}}^{n}}\frac{1}{|\xi-x|^{qj}|\xi-y|^{q(n-1)}}d\xi\bigg{)}^{1/q}+\bigg{(}\int_{{\mathbb{R}}^{n}}\frac{1}{|\xi-x|^{q(n-1)}|y-\xi|^{qj}}d\xi\bigg{)}^{1/q}\bigg{\\}}.$

Since

$\displaystyle\int_{{\mathbb{R}}^{n}}\frac{1}{|\xi-x|^{qj}|\xi-y|^{q(n-1)}}d\xi=\int_{{\mathbb{R}}^{n}}\frac{1}{|\eta|^{qj}|\eta+x-y|^{q(n-1)}}d\eta$
$\displaystyle=\int_{{\mathbb{R}}^{n}}\frac{|x-y|^{n}}{|x-y|^{qj+q(n-1)}|\zeta|^{qj}|\zeta+\frac{x-y}{|x-y|}|^{q(n-1)}}d\zeta$
$\displaystyle=|x-y|^{n-qj-q(n-1)}\int_{{\mathbb{R}}^{n}}\frac{1}{|\zeta|^{qj}|\zeta+\frac{x-y}{|x-y|}|^{q(n-1)}}d\zeta$

and similarly for the other integral, and since

$\displaystyle I:=\sup_{x\neq
y}\int_{{\mathbb{R}}^{n}}\frac{1}{|\zeta|^{qj}|\zeta+\frac{x-y}{|x-y|}|^{q(n-1)}}d\zeta<+\infty$

we find that

$\displaystyle|\mathcal{C}^{+}\Phi(x)-\mathcal{C}^{+}\Phi(y)|$
$\displaystyle\leq\frac{2Ic_{n}}{\omega_{n}}\|\Phi\|_{L^{p}}\sum_{j=1}^{n-1}|x-y|^{j}|x-y|^{(n-qj-q(n-1))/q}$
$\displaystyle\leq\frac{2nIc_{n}}{\omega_{n}}\|\Phi\|_{L^{p}}|x-y|^{1-n/p}$

∎

###  7.3 BMO estimates for the Cauchy Transform

In what follows we set $B_{1}(0)=\\{x\in{\mathbb{R}}^{n}:|x|<1\\}$ and recall
that $n|B_{1}(0)|=\omega_{n-1}$

######  Lemma 7.3.

$\displaystyle\mathcal{C}^{+}(\chi_{B_{1}(0)})(x)=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{n}x&\text{if
}|x|<1,\\\ &\\\ \displaystyle\frac{1}{n}\frac{x}{|x|^{n}}&\text{if
}|x|>1.\end{array}\right.$

###### Proof.

This follows from the well-known formula for the Newton potential of a ball
and the fact that $(\mathcal{D}^{\tiny+})^{2}=\Delta$. ∎

######  Lemma 7.4.

Let $\phi\in L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ with
$1<p<\infty$ and set $\phi_{r}(x)=\phi(rx)$. Then

$\displaystyle\mathcal{C}^{+}(\phi_{r})(x)=r^{-1}\mathcal{C}^{+}(\phi)(rx)$

###### Proof.

$\displaystyle\mathcal{C}^{+}(\phi_{r})(x)$
$\displaystyle=\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\frac{y-x}{|y-x|^{n}}\text{\tiny{$\triangle$}}\phi(ry)dy=\\{ry=\xi\\}$
$\displaystyle=\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\frac{r^{-1}\xi-x}{|r^{-1}\xi-x|^{n}}\text{\tiny{$\triangle$}}\phi(\xi)r^{-n}d\xi$
$\displaystyle=\frac{1}{r\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\frac{\xi-
rx}{|\xi-rx|^{n}}\text{\tiny{$\triangle$}}\phi(\xi)d\xi.$

∎

######  Proposition 7.2.

Let $\phi\in L^{n}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$. Then
$\mathcal{C}^{\pm}\phi\in BMO({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ and

$\displaystyle\|\mathcal{C}^{\pm}\phi\|_{BMO}\leq
C_{n}^{\pm}\|\phi\|_{L^{n}}.$

for some constant $C_{n}^{\pm}$ depending on $n$.

###### Proof.

It is enough to consider $\mathcal{C}^{+}$ as the proof for $\mathcal{C}^{-}$
is analogous. Since functions in
$BMO({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ are only defined up to
constants we first need to give a meaning to $\mathcal{C}^{+}\phi$ in $BMO$ as
an equivalence class modulo constants. Therefore we are free to add a constant
to the original definition that may depend on the function $\phi$ itself since
this gives the same element in $BMO$. For $\phi\in
L^{n}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ we define

$\displaystyle\mathcal{C}^{+}\phi(x):=\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\bigg{[}\frac{y-x}{|y-x|^{n}}-\frac{y}{|y|^{n}}\chi_{B_{1}(0)}(y)\bigg{]}\text{\tiny{$\triangle$}}\phi(y)dy.$

Then the integral converges for almost every $x$. Since
$C^{\infty}_{0}(\Omega,\Lambda{\mathbb{R}}^{n})$ is dense in
$L^{n}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ it is sufficient to prove
the BMO-estimate for a smooth compactly supported function.

For any ball $B_{r}(x_{0})$ set

$\displaystyle(\phi)_{B_{r}(x_{0})}:=\frac{1}{|B_{r}(x_{0})|}\int_{B_{r}(x_{0})}\phi(x)dx.$

After a translation we may assume that $x_{0}=0$. Set $\phi_{r}(x)=\phi(rx)$.
By Lemma 7.4,

Since

$\displaystyle\frac{1}{|B_{r}(0)|}\int_{B_{r}(0)}|\mathcal{C}^{+}\phi(x)-(\mathcal{C}^{+}\phi)_{B_{r}(0)}|dx=\\{r\xi=x\\}$
$\displaystyle=\frac{1}{|B_{r}(0)|}\int_{B_{1}(0)}|\mathcal{C}^{+}\phi(r\xi)-(\mathcal{C}^{+}\phi)_{B_{r}(0)}|r^{n}d\xi$
$\displaystyle=\frac{r}{|B_{1}(0)|}\int_{B_{1}(0)}|\mathcal{C}^{+}\phi_{r}(x)-(\mathcal{C}^{+}\phi_{r})_{B_{1}(0)}|dx.$

We see it is enough to prove that for all $r>0$

$\displaystyle\frac{r}{|B_{1}(0)|}\int_{B_{1}(0)}|\mathcal{C}^{+}\phi_{r}(x)-(\mathcal{C}^{+}\phi_{r})_{B_{1}(0)}|dx$
$\displaystyle\leq C_{n}\|\phi(x)\|_{n}$ $\displaystyle\Longleftrightarrow$
$\displaystyle\frac{1}{|B_{1}(0)|}\int_{B_{1}(0)}|\mathcal{C}^{+}\phi_{r}(x)-(\mathcal{C}^{+}\phi_{r})_{B_{1}(0)}|dx$
$\displaystyle\leq C_{n}\|\phi_{r}(x)\|_{n}.$

for some constant $C_{n}$ depending only on $n$.

Now,

$\displaystyle(\mathcal{C}^{+}\phi)_{B_{1}(0)}$
$\displaystyle=\frac{1}{|B_{1}(0)|}\int_{B_{1}(0)}\bigg{(}\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\bigg{[}\frac{y-x}{|y-x|^{n}}-\frac{y}{|y|^{n}}\chi_{B_{1}(0)}(y)\bigg{]}\text{\tiny{$\triangle$}}\phi(y)dy\bigg{)}dx$
$\displaystyle=\frac{1}{|B_{1}(0)|}\int_{{\mathbb{R}}^{n}}\bigg{(}\frac{1}{\sigma_{n-1}}\int_{B_{1}(0)}\frac{y-x}{|y-x|^{n}}dx\bigg{)}\text{\tiny{$\triangle$}}\phi(y)dy-\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\frac{y}{|y|^{n}}\chi_{B_{1}(0)}(y)\text{\tiny{$\triangle$}}\phi(y)dy$
$\displaystyle=-\frac{1}{|B_{1}(0)|}\int_{{\mathbb{R}}^{n}}\mathcal{C}^{+}(\chi_{B_{1}(0)})(x)\text{\tiny{$\triangle$}}\phi(y)dy-\frac{1}{\sigma_{n-1}}\int_{B_{1}(0)}\frac{y}{|y|^{n}}\text{\tiny{$\triangle$}}\phi(y)dy.$

Using Lemma 7.3

$\displaystyle-\frac{1}{|B_{1}(0)|}\int_{{\mathbb{R}}^{n}}\mathcal{C}^{+}(\chi_{B_{1}(0)})(x)\text{\tiny{$\triangle$}}\phi(y)dy$
$\displaystyle=-\frac{1}{|B_{1}(0)|}\int_{B_{1}(0)}\mathcal{C}^{+}(\chi_{B_{1}(0)})(x)\text{\tiny{$\triangle$}}\phi(y)dy-\frac{1}{|B_{1}(0)|}\int_{{\mathbb{R}}^{n}\setminus
B_{1}(0)}\mathcal{C}^{+}(\chi_{B_{1}(0)})(x)\text{\tiny{$\triangle$}}\phi(y)dy$
$\displaystyle=-\frac{1}{n|B_{1}(0)|}\int_{B_{1}(0)}y\text{\tiny{$\triangle$}}\phi(y)dy-\frac{1}{n|B_{1}(0)|}\int_{{\mathbb{R}}^{n}\setminus
B_{1}(0)}\frac{y}{|y|^{n}}\text{\tiny{$\triangle$}}\phi(y)dy$
$\displaystyle=-\frac{1}{\sigma_{n-1}}\int_{B_{1}(0)n}y\text{\tiny{$\triangle$}}\phi(y)dy-\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}\setminus
B_{1}(0)}\frac{y}{|y|^{n}}\text{\tiny{$\triangle$}}\phi(y)dy.$

Hence

$\displaystyle(\mathcal{C}^{+}\phi)_{B_{1}(0)}$
$\displaystyle=-\frac{1}{|B_{1}(0)|}\int_{{\mathbb{R}}^{n}}\mathcal{C}^{+}(\chi_{B_{1}(0)})(x)\text{\tiny{$\triangle$}}\phi(y)dy-\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}}\frac{y}{|y|^{n}}\text{\tiny{$\triangle$}}\phi(y)dy.$

Thus,

$\displaystyle\mathcal{C}^{+}\phi(x)-(\mathcal{C}^{+}\phi)_{B_{1}(0)}$
$\displaystyle=\frac{1}{\sigma_{n-1}}\int_{B_{1}(0)}\bigg{\\{}\frac{y-x}{|y-x|^{n}}+y\bigg{\\}}\text{\tiny{$\triangle$}}\phi(y)dy+\frac{1}{\sigma_{n-1}}\int_{{\mathbb{R}}^{n}\setminus
B_{1}(0)}\bigg{\\{}\frac{y-x}{|y-x|^{n}}+\frac{y}{|y|^{n}}\bigg{\\}}\text{\tiny{$\triangle$}}\phi(y)dy.$

Define the function

$\displaystyle\psi(x,y):=\left\\{\begin{array}[]{ll}\displaystyle\frac{y-x}{|y-x|^{n}}+y&\text{if
}|y|<1,\\\ &\\\ \displaystyle\frac{y-x}{|y-x|^{n}}+\frac{y}{|y|^{n}}&\text{if
}|y|>1.\end{array}\right.$

Then

$\displaystyle\frac{1}{|B_{1}(0)|}\int_{B_{1}(0)}|\mathcal{C}^{+}\phi(x)-(\mathcal{C}^{+}\phi)_{B_{1}(0)}|dx\leq\frac{1}{\sigma_{n-1}|B_{1}(0)|}\int_{B_{1}(0)}\bigg{(}\int_{{\mathbb{R}}^{n}}|\psi(x,y)||\phi(y)|dy\bigg{)}dx$
$\displaystyle=\frac{1}{\sigma_{n-1}|B_{1}(0)|}\int_{{\mathbb{R}}^{n}}\bigg{(}\int_{B_{1}(0)}|\psi(x,y)|dx\bigg{)}|\phi(y)|dy.$

We now estimate $\int_{B_{1}(0)}|\psi(x,y)|dx$. When $|y|<1$ we get

$\displaystyle\int_{B_{1}(0)}|\psi(x,y)|dx\leq\int_{B_{1}(0)}\frac{1}{|y-x|^{n-1}}dx+\int_{B_{1}(0)}|y|dx\leq
c_{1},$

for some constant $c_{1}$ independent of $y$. When $|y|>1$ we get

$\displaystyle\int_{B_{1}(0)}|\psi(x,y)|dx$
$\displaystyle\leq\int_{B_{1}(0)}\frac{1}{|y-x|^{n-1}}dx+\int_{B_{1}(0)}\frac{1}{|y|^{n}}dx$
$\displaystyle\leq\frac{1}{|y|^{n-1}}\bigg{(}\int_{B_{1}(0)n}\frac{1}{|1-y^{-1}\text{\tiny{$\triangle$}}x|^{n-1}}dx+\int_{B_{1}(0)}dx\bigg{)}$
$\displaystyle\leq\frac{c_{2}}{|y|^{n-1}}$

for some constant $c_{2}$ independent of $y$. Let
$g(y)=\int_{B_{1}(0)}|\psi(x,y)|dx$. This gives

$\displaystyle\|g\|_{L^{n/(n-1)}({\mathbb{R}}^{n})}^{(n-1)/n}=\int_{B_{1}(0)}c_{1}^{n/(n-1)}dy+c_{2}^{n/(n-1)}\int_{{\mathbb{R}}^{n}\setminus
B_{1}(0)}\frac{1}{|y|^{n}}dy<+\infty$

By Hölder’s inequality

$\displaystyle\frac{1}{|B_{1}(0)|}\int_{B_{1}(0)}|\mathcal{C}^{+}\phi(x)-(\mathcal{C}^{+}\phi)_{B_{1}(0)}|dx\leq\frac{1}{\sigma_{n-1}|B_{1}(0)|}\int_{{\mathbb{R}}^{n}}g(y)|\phi(y)|dy$
$\displaystyle\leq C_{n}\|\phi\|_{L^{n}}$

for some constant $C_{n}$, and the proof is complete. ∎

###  7.4 Compactness Properties of the Cauchy Transforms

In many applications one is more interested in the restriction of the Cauchy
transforms to bounded domains $\Omega\subset{\mathbb{R}}^{n}$. We write the
restriction of $\mathcal{C}^{\pm}F|_{\Omega}$ as multiplication with the
characteristic function $\chi_{\Omega}$. On bounded domains the Cauchy
transforms have the following compactness properties:

######  Theorem 7.3.

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{n}$. Then the following
operators are compact.

  * (i)

For $n<p\leq\infty$,

$\displaystyle\chi_{\Omega}\circ\mathcal{C}^{\pm}:L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})\to
C^{\alpha}(\Omega,\Lambda{\mathbb{R}}^{n}),\,\,\,\,0\leq\alpha<1-n/p$

  * (ii)

For $1\leq p<n$,

$\displaystyle\chi_{\Omega}\circ\mathcal{C}^{\pm}:L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})\to
L^{s}(\Omega,\Lambda{\mathbb{R}}^{n}),\,\,\,\,1\leq
s<p^{\star}=\frac{np}{n-p}$

###### Proof.

The proofs are the same as the proof of Theorem 4.3.14 in [2] with suitable
modifications. The proof of claim $(ii)$ follows from Proposition 7.1 and an
application of Arzela-Ascoli’s theorem. The proof of claim $(ii)$ follows from
Theorem 1 in [55, page 120] and Rellich-Kondrachov compact embedding theorem.
∎

##  8 The Beurling-Ahlfors and Riesz Transforms on ${\mathbb{R}}^{n}$.

###  8.1 The Beurling-Ahlfors Transform on ${\mathbb{R}}^{n}$.

In the ground breaking papers [22] and [25] the authors introduced a higher
dimensional generalisation of the classical Beurling-Ahlfors transform to
study quasiconformal $4$-manifolds and removable singularities of quasiregular
maps in even dimensions respectively. Furthermore this operator was also used
to give a local formulae for characteristic classes on even dimensional
quasiconformal manifolds, see [18]. This higher dimensional operator, denoted
by $\mathcal{S}$ is intimately connected with Hodge-Morrey decompositions of
differential forms on ${\mathbb{R}}^{n}$. Indeed, let $\omega\in
L^{p}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})$ be a differential
$k$-form. Then the Hodge-Morrey decomposition of $\omega$ is given by

$\displaystyle\omega=d\alpha+\delta\beta,$

with $\alpha\in\text{ker}(\delta)\cap
W^{1,p}({\mathbb{R}}^{n},\Lambda^{k-1}{\mathbb{R}}^{n})$ and
$\beta\in\text{ker}(d)\cap
W^{1,p}({\mathbb{R}}^{n},\Lambda^{k+1}{\mathbb{R}}^{n})$ and where
$\delta=d^{\ast}$. The Beurling-Ahlfors transform on
$\mathcal{S}:L^{p}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})\to
L^{p}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})$ is defined by the rule

$\displaystyle\mathcal{S}(\omega)=d\alpha-\delta\beta,$

or alternatively $\mathcal{S}=(d\delta-\delta d)\circ\Delta^{-1}$, where
$\Delta$ is the Laplacian.

In [25] Theorem 8.1 it is shown that the Beurling-Ahlfors operator has the
following properties:

  * (i)

$\mathcal{S}(d\alpha)=d\alpha$ for all $\alpha$ with $d\alpha\in
L^{p}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})$.

  * (ii)

$\mathcal{S}(\delta\alpha)=-\delta\alpha$ for all $\alpha$ with
$\delta\alpha\in L^{p}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})$.

  * (iii)

$\mathcal{S}$ is self-adjoint and $\mathcal{S}\circ\mathcal{S}=\text{Id}$ and
$\mathcal{S}^{-1}=\mathcal{S}$.

  * (iv)

$\mathcal{S}$ anti-commutes with the Hodge star, i.e.,

$\displaystyle\mathcal{S}\star=-\star\mathcal{S}.$

  * (v)

$\mathcal{S}:L^{2}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})\to
L^{2}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ is an isometry and
$\mathcal{S}$ is bounded on $L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$.

For applications it is of great interest to know the $L^{p}$-norm of
$\mathcal{S}$. Indeed, in the paper [25] Theorem 8.4 the authors prove the
following bound

$\displaystyle\|\mathcal{S}\|_{L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})}\leq
C(n)\max\bigg{\\{}\frac{1}{p-1},p-1\bigg{\\}},\,\,1<p<+\infty,$ (8.1)

where $n$ is the dimension and $C(n)$ is a polynomial in $n$ not depending on
$p$. In fact the authors conjecture that $C(n)=1$ for all $n$. This conjecture
remains open so far and has been the focus of great attention over the years.
It is even open in dimension $n=2$. The bound 8.1 has subsequently been
improved over the years by several authors, e.g., [12],[49],[50] using in
particular Bellman function techniques and Burkholder’s sharp inequality for
martingale transforms on $L^{p}$. The best bound so far is proven in [37] and
given by

$\displaystyle\|\mathcal{S}\|_{L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})}\leq(1+n/2)(p-1),$
(8.2)

where $p^{\star}:=\max\\{p/(p-1),p\\}$. In fact when $\mathcal{S}$ is
restricted to the spaces
$L^{p}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})$, then in [37] it is
shown that the $L^{p}$-norms are have the estimates

$\displaystyle\|\mathcal{S}\|_{L^{p}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})}\leq\bigg{(}\frac{2k(n-k)}{n}+1\bigg{)}(p^{\star}-1)\leq(2k+1)(p^{\star}-1).$
(8.3)

###  8.2 Factorisation formula

Let $\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ denote the
Schwartz space of differential forms on ${\mathbb{R}}^{n}$, i.e.

$\displaystyle\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n}):=\\{\omega\in
C^{\infty}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n}):\sup_{x}|x^{\beta}\partial^{\alpha}\omega(x)\
|\text{ for all muti-indices $\alpha$ and $\beta$}\\}.$

One easily checks that $\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$
is a Fréchet algebra, closed under the products
$\wedge,\,\,\lrcorner\,\,,\,\,\llcorner\,\,,\text{\tiny{$\triangle$}},\text{\scriptsize{$\triangledown$}}$.
Moreover, $\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ is dense in
$L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ for $1<p<+\infty$.

The Fourier transform $\mathcal{F}$ of a multicovector field
$\omega\in\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ is defined as

$\displaystyle\mathcal{F}(\omega)(\xi)=\frac{1}{(\sqrt{2\pi})^{n}}\int_{{\mathbb{R}}^{n}}\omega(x)e^{-i\langle
x,\xi\rangle}dx.$

In Theorem 8.2 in [25] it is shown that $\mathcal{S}$ is an isometry on
$L^{2}({\mathbb{R}}^{n},\Lambda^{k}{\mathbb{R}}^{n})$ for every $k$. Let
$M(\xi):{\mathbb{R}}^{n}\to\mathcal{L}({\mathbb{R}}^{n})$ be the orthogonal
transformation

$\displaystyle M(\xi)=I-2|\xi|^{-1}\xi\otimes\xi,$

being the reflection in the hyperplane orthogonal to $\xi$. Let
$\widehat{M}_{\xi}:\mathcal{L}(\Lambda{\mathbb{R}}^{n})\to\mathcal{L}(\Lambda{\mathbb{R}}^{n})$
denote its exterior extension (in the notation in [25]
$\widehat{M}(\xi)=M_{\\#}(\xi)$). In Theorem 8.10 in [25] is shown that the
Fourier multiplier of
$\mathcal{S}:L^{2}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})\to
L^{2}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ is given by

$\displaystyle\mathcal{F}(\mathcal{S}\omega)(\xi)$
$\displaystyle=\widehat{M}(\xi)\mathcal{F}(\omega)(\xi).$

Here $\widehat{M}(\xi)$ is identified with its complexification acting on
$\Lambda{\mathbb{R}}^{n}\otimes{\mathbb{C}}$ so that for any
$w_{1},w_{2}\in\Lambda{\mathbb{R}}^{n}$ we have
$\widehat{M}(\xi)(w_{1}+iw_{2})=\widehat{M}(\xi)(w_{1})+i\widehat{M}(\xi)(w_{2})$.

######  Lemma 8.1.

The Fourier multiplier of the Beurling transform can be expressed as

$\displaystyle\mathcal{F}(\mathcal{S}\omega)(\xi)$
$\displaystyle=-\frac{i\xi}{|\xi|}\text{\tiny{$\triangle$}}\widehat{\mathcal{F}(\omega)(\xi)}\text{\tiny{$\triangle$}}\frac{i\xi}{|\xi|}.$
(8.4)

######  Remark 8.1.

Note that here $\widehat{\omega(\xi)}$ denotes the involution on $\Lambda V$
($\Lambda V\otimes{\mathbb{C}}$) and not the Fourier transform.

We will give to different proofs.

###### Proof 1.

Since $M(\xi)$ is a reflection in the hyperplane orthogonal to $\xi$, Theorem
4.1.3 in [51] gives

$\displaystyle
M(\xi)v=-\xi\text{\tiny{$\triangle$}}v\text{\tiny{$\triangle$}}\xi^{-1}$

for every $v\in V$. Lemma 3.2 equation (3.5) and the fact that the Clifford
multiplicative inverse of a vector is given by $\xi^{-1}=\xi/|\xi|^{2}$ give

$\displaystyle\mathcal{F}(\mathcal{S}\omega)(\xi)$
$\displaystyle=\xi\text{\tiny{$\triangle$}}\widehat{\mathcal{F}(\omega)(\xi)}\text{\tiny{$\triangle$}}\xi^{-1}=\frac{\xi}{|\xi|}\text{\tiny{$\triangle$}}\widehat{\mathcal{F}(\omega)}\text{\tiny{$\triangle$}}\frac{\xi}{|\xi|}$
$\displaystyle=-\frac{i\xi}{|\xi|}\text{\tiny{$\triangle$}}\widehat{\mathcal{F}(\omega)}\text{\tiny{$\triangle$}}\frac{i\xi}{|\xi|}.$

∎

###### Proof 2.

We use the fact that
$\mathcal{F}(d\omega)(\xi)=i\xi\wedge\mathcal{F}(\omega)(\xi)$ and
$\mathcal{F}(\delta\omega)(\xi)=-i\xi\,\,\lrcorner\,\,\mathcal{F}(\omega)(\xi)$.
Set $\eta(\xi)=-|\xi|^{-2}\mathcal{F}(\omega)(\xi)$. Thus,

$\displaystyle\mathcal{F}((d\delta-\delta
d)\circ\Delta^{-1}\omega)(\xi)=i\xi\wedge(-i\xi\,\,\lrcorner\,\,\eta(\xi))+i\xi\,\,\lrcorner\,\,(i\xi\wedge\eta(\xi))$
$\displaystyle=(\xi\,\,\lrcorner\,\,+\xi\wedge)(-\xi\,\,\lrcorner\,\,\eta(\xi))+(\xi\,\,\lrcorner\,\,+\xi\wedge)(\xi\wedge\eta(\xi))=\xi\text{\tiny{$\triangle$}}(-\xi\,\,\lrcorner\,\,\eta(\xi)+\xi\wedge\eta(\xi))$
$\displaystyle=\xi\text{\tiny{$\triangle$}}(\widehat{\eta(\xi)}\,\,\llcorner\,\,\xi+\widehat{\eta(\xi)}\wedge)=\xi(\text{\tiny{$\triangle$}}\widehat{\eta(\xi)}\text{\tiny{$\triangle$}})$
$\displaystyle=\xi(\text{\tiny{$\triangle$}}-|\xi|^{-2}\widehat{\mathcal{F}(\omega)(\xi)}\text{\tiny{$\triangle$}})$
$\displaystyle=-\frac{i\xi}{|\xi|}\widehat{\mathcal{F}(\omega)}\text{\tiny{$\triangle$}}\frac{i\xi}{|\xi|}.$

Here we have used the fact that $\xi\wedge(\xi\wedge w)=0$ and
$\xi\,\,\lrcorner\,\,(\xi\,\,\lrcorner\,\,w)=0$ for any multivector $w$. ∎

######  Definition 8.1.

The scalar Riesz transforms $\mathcal{R}_{\nu}$ acting on
$L^{p}({\mathbb{R}}^{n})$ for any $\nu\in\mathbb{S}^{n-1}$ are given by the
singular integrals

$\displaystyle\mathcal{R}_{\nu}f(x)=\frac{2}{\sigma_{n}}\text{p.v.}\int_{{\mathbb{R}}^{n}}\frac{(x_{\nu}-y_{\nu})f(y)}{|x-y|^{n+1}}dy$

where $x_{\nu}=\langle x,\nu\rangle$.

The $L^{p}$-norms of $\mathcal{R}_{\nu}$ are given by
$\|\mathcal{R}_{\nu}\|_{p}=H_{p}$, (Theorem 12.11 in [26, p. 304]) for any
$\nu\in\mathbb{S}^{n-1}$ and the Fourier transforms of
$\mathcal{R}_{\nu}\omega$ are given by

$\displaystyle\mathcal{F}[\mathcal{R}_{\nu}\omega](\xi)=\frac{i\xi_{\nu}}{|\xi|}\mathcal{F}(\omega)(\xi).$

######  Definition 8.2.

Define the Clifford-Riesz transforms $\mathcal{R}^{+}$ and $\mathcal{R}^{-}$
are defined by

$\displaystyle\mathcal{R}^{+}f(x)$
$\displaystyle:=\frac{2}{\sigma_{n}}\text{p.v.}\int_{{\mathbb{R}}^{n}}\frac{x-y}{|x-y|^{n+1}}\text{\tiny{$\triangle$}}f(y)dy,$
$\displaystyle\mathcal{R}^{-}f(x)$
$\displaystyle:=\frac{2}{\sigma_{n}}\text{p.v.}\int_{{\mathbb{R}}^{n}}\frac{x-y}{|x-y|^{n+1}}\text{\scriptsize{$\triangledown$}}f(y)dy,$

These operators arise naturally when considering Hardy spaces of multivector
fields with respect to the Hodge-Dirac operators. For more on these operators
we refer the reader to Chapter 4 in [32] and [39]

######  Lemma 8.2.

$\mathcal{R}^{+}$ and $\mathcal{R}^{-}$ are bounded operators on
$L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ with norm estimates

$\displaystyle\|\mathcal{R}^{\pm}\|_{p}\leq
2H_{p}\frac{\sigma_{n-1}}{\sigma_{n}}.$

###### Proof.

It is enough to consider $\mathcal{R}^{+}$. We apply the real method of
rotation, Theorem 12.5.1 in [26].
$\Gamma:{\mathbb{R}}^{n}\to\mathcal{L}(\Lambda V)$ is given by

$\displaystyle\Gamma(x)=\frac{2}{\sigma_{n}}\frac{x}{|x|^{n-1}}\text{\tiny{$\triangle$}},$

and so for all $x\in\mathbb{S}^{n-1}$ and all $t\in{\mathbb{R}}$, we have
$|t|^{n+1}\Gamma(tx)=t\Gamma(x)$. Hence by Theorem 12.5.1. in [26]
$\mathcal{R}^{\pm}$ are bounded in
$L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ for all $1<p<\infty$ and

$\displaystyle\|\mathcal{R}^{+}\|_{p}\leq\frac{\pi}{2}H_{p}\int_{\mathbb{S}^{n-1}}|\Gamma(x)|d\sigma(x),$

where

$\displaystyle|\Gamma(x)|=\max\\{|\Gamma(x)w|:w\in\Lambda V,|w|=1\\}.$

Since $x\in\mathbb{S}^{n-1}$,
$2\omega_{n}^{-1}|\Gamma(x)|=\max\\{|x\text{\tiny{$\triangle$}}w|:w\in\Lambda
V,|w|=1\\}=\max\\{|w|:w\in\Lambda V,|w|=1\\}=1$ by Lemma 3.1. Thus,

$\displaystyle\int_{\mathbb{S}^{n-1}}|\Gamma(x)|d\sigma(x)=\frac{2}{\omega_{n}}\int_{\mathbb{S}^{n-1}}\frac{|x|}{|x|^{n+1}}d\sigma(x)=\frac{2\omega_{n-1}}{\omega_{n}}.$

∎

It is sometimes useful to introduce the formal Riesz transform symbol

$\displaystyle\mathbf{R}:=\sum_{j=1}^{n}\mathcal{R}_{e_{j}}e_{j}$

with respect to an ON-basis $\\{e_{j}\\}_{j}$ of ${\mathbb{R}}^{n}$ much in
the same way as the formal nabla symbol $\nabla$. Using this symbol we can
write formally

$\displaystyle\mathcal{R}^{+}F(x)=\mathbf{R}\text{\tiny{$\triangle$}}F(x),\,\,\,\mathcal{R}^{-}F(x)=\mathbf{R}\text{\scriptsize{$\triangledown$}}F(x),$

In addition, because of the algebraic decomposition of the action of Clifford
multiplication into exterior an interior multiplication we have

$\displaystyle\mathbf{R}\text{\tiny{$\triangle$}}F(x)=\mathbf{R}\wedge
F(x)+\mathbf{R}\,\,\lrcorner\,\,F(x),\,\,\,\mathbf{R}\text{\scriptsize{$\triangledown$}}F(x)=\mathbf{R}\wedge
F(x)-\mathbf{R}\,\,\lrcorner\,\,F(x),$

where

$\displaystyle\mathbf{R}\wedge F(x)$
$\displaystyle=\frac{2}{\sigma_{n}}\text{p.v.}\int_{{\mathbb{R}}^{n}}\frac{x-y}{|x-y|^{n+1}}\wedge
F(y)dy,$ $\displaystyle\mathbf{R}\,\,\lrcorner\,\,F(x)$
$\displaystyle=\frac{2}{\sigma_{n}}\text{p.v.}\int_{{\mathbb{R}}^{n}}\frac{x-y}{|x-y|^{n+1}}\,\,\lrcorner\,\,F(y)dy.$

######  Proposition 8.1.

The Beurling-Ahlfors operator factorises according to

$\displaystyle\mathcal{S}(\omega)=-\mathbf{R}\text{\tiny{$\triangle$}}(\widehat{\omega}\text{\tiny{$\triangle$}}\mathbf{R})=-(\mathbf{R}\text{\tiny{$\triangle$}}\widehat{\omega})\text{\tiny{$\triangle$}}\mathbf{R}$

for all $\omega\in L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ and
$1<p<\infty$.

###### Proof.

It is enough to establish the factorisation for
$\omega\in\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$. Indeed,
since the Schwartz space of multivector fields is dense in
$L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ for $1<p<\infty$, and
$\mathcal{S},\mathbf{R}\text{\tiny{$\triangle$}},\text{\tiny{$\triangle$}}\mathbf{R}$
and $\mathcal{R}\text{\scriptsize{$\triangledown$}}$ are bounded on
$L^{p}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$, the result follows by
continuity. The functions $x_{\nu}/|x|^{n+1}$ define tempered homogeneous
distributions for each $\nu$. Hence, by acting component wise
$\mathcal{R}_{\nu}\omega\in\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$
for every $\omega\in\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$ and
consequently also
$\mathbf{R}\text{\tiny{$\triangle$}}\omega,\omega\text{\tiny{$\triangle$}}\mathbf{R},\mathbf{R}\text{\scriptsize{$\triangledown$}}\omega\in\mathscr{S}({\mathbb{R}}^{n},\Lambda{\mathbb{R}}^{n})$.
Setting
$\eta(\xi)=\widehat{\mathcal{F}(\omega)(\xi)}\text{\tiny{$\triangle$}}i\xi/|\xi|$,
we apply Theorem 5 in [55, Chap. 3 p.73] first to
$-\frac{i\xi}{|\xi|}\text{\tiny{$\triangle$}}\eta$ and then to $\eta(\xi)$ and
the result follows. The other formula follows similarly. ∎

######  Remark 8.2.

In the complex case we set
$\mathcal{R}_{\mathbb{C}}=\mathcal{R}_{1}+i\mathcal{R}_{2}$. Then we have
$\mathcal{S}_{{\mathbb{C}}}=\mathcal{R}_{\mathbb{C}}^{2}$ where

$\displaystyle\mathcal{S}_{{\mathbb{C}}}f(z)=-\frac{1}{\pi}\text{p.v.}\int_{{\mathbb{C}}}\frac{f(w)}{(z-w)^{2}}dA(w).$

is the complex Beurling-Ahlfors transform. In ${\mathbb{R}}^{2}$ the even part
of $\Lambda{\mathbb{R}}^{2}$, i.e. the linear subspace
$\Lambda^{0}{\mathbb{R}}^{2}\oplus\Lambda^{2}{\mathbb{R}}^{2}$ is a subalgebra
of the euclidean Clifford algebra $\Delta({\mathbb{R}}^{2})$ isomorphic to
${\mathbb{C}}$. In particular, Clifford multiplication by $e_{1}$ define a map
$e_{1}:V\to\Lambda^{0}{\mathbb{R}}^{2}\oplus\Lambda^{2}{\mathbb{R}}^{2}$ where
$e_{12}\cong i$. Thus, with this identification we have