# Distortion of Hausdorff measures under Orlicz-Sobolev maps

Andrea Cianchi
Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze
Viale Morgagni 67/A, 50134 Firenze, Italy
e-mail<EMAIL_ADDRESS>

Mikhail V. Korobkov
School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R.China
and
Sobolev Institute of Mathematics, pr-t Ac. Koptyug, 4, Novosibirsk, 630090,
Russia
e-mail<EMAIL_ADDRESS>

Jan Kristensen
Mathematical Institute, University of Oxford, Andrew Wiles Building
Oxford OX2 6GG, United Kingdom
e-mail<EMAIL_ADDRESS>
###### Abstract

A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean
spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary
Orlicz-Sobolev spaces embedded into the space of continuous function and
Hausdorff measures built upon general gauge functions are included in our
discussion. An explicit formula for the distortion of the relevant gauge
function under the action of these maps is exhibited in terms of the Young
function defining the Orlicz-Sobolev space. New phenomena and features,
related to the flexibility in the definition of the degree of integrability of
weak derivatives of maps and in the notion of measure of sets, are detected.
Classical results, dealing with standard Sobolev spaces and Hausdorff
measures, are recovered, and their optimality is shown to hold in a refined
stronger sense. Special instances available in the literature, concerning
Young functions and gauge functions of non-power type, are also reproduced
and, when not sharp, improved.

00footnotetext: Mathematics Subject Classifications: 46E35, 26B35.Key words
and phrases: Hausdorff measure, gauge function, Orlicz-Sobolev space.

##  1 Introduction and main results

A continuous map $u\colon\Omega\to\mathbb{R}^{n}$ in an open subset $\Omega$
of $\mathbb{R}^{n}$, with $n\in\mathbb{N}$, is said to satisfy the Lusin $N$
condition if it maps Lebesgue null sets into Lebesgue null sets. This amounts
to requiring that, for sets $E\subset\Omega$,

(1.1) if ${\mathscr{L}}^{n}(E)=0$, then ${\mathscr{L}}^{n}(u(E))=0$,

where ${\mathscr{L}}^{n}$ denotes the outer Lebesgue measure in
$\mathbb{R}^{n}$.

The Lusin $N$ property plays a fundamental role in various results from
classical real analysis and differentiation theory. Early applications
primarily concerning the case $n=1$ can be found in [Sa37]. Its crucial
importance in dimension $n>1$ is witnessed by its role for the validity of
various formulas in geometric measure theory and geometric function theory –
see e.g. [RR55, Th. 3, p. 364] and [MSZ02]. In particular, the Lusin $N$
condition is critical in the proof of central properties of quasiconformal
maps and, more generally, of maps with bounded distortion and of quasiregular
maps. The latter were introduced by Yu.G. Reshetnyak in [Re67] and
subsequently extensively analyzed and applied by several authors. A
comprehensive treatment of the pertaining theory and further references can be
found in the monographs [Re89, Vai71, Ri93, IM01]. More recently a natural
extension of the Lusin $N$ condition, where the Lebesgue measure is replaced
by lower dimensional Hausdorff measures, was decisive in works extending the
Morse-Sard theorem and related results to classes of Sobolev maps which need
neither be Lipschitz nor everywhere differentiable – see, in particular,
[BKK13, BKK15, KK18].

In this connection, note that a map $u\colon\Omega\to\mathbb{R}^{m}$, with
$m\in\mathbb{N}$, which is locally Lipschitz continuous, or everywhere
differentiable, is easily seen to possess the property that, for sets
$E\subset\Omega$,

(1.2) if $\mathscr{H}^{\alpha}(E)=0$, then $\mathscr{H}^{\alpha}(u(E))=0$,

where $\alpha\in(0,n]$ and $\mathscr{H}^{\alpha}$ denotes the
$\alpha$-dimensional Hausdorff measure. In particular, the Lusin $N$ property
(1.1) follows from (1.2), when $m=n$.

Information on the Hausdorff measure of the image of a set, which still
implies the Lusin $N$ condition (1.1), remains available if the map $u$,
though neither locally Lipschitz continuous nor everywhere differentiable, has
yet a suitable degree of regularity. In this connection, recall that the space
of locally Lipschitz continuous maps from $\Omega$ into $\mathbb{R}^{m}$
agrees with the space $\mathrm{V}^{1,\infty}_{\rm
loc}(\Omega,\mathbb{R}^{m})$. Here, $\mathrm{V}^{1,\infty}_{\rm
loc}(\Omega,\mathbb{R}^{m})$ stands for the local counterpart of the
homogeneous Sobolev space $\mathrm{V}^{1,\infty}(\Omega,\mathbb{R}^{m})$,
where $\mathrm{V}^{1,p}(\Omega,\mathbb{R}^{m})$ is defined, for
$p\in[1,\infty]$, as

(1.3)
$\mathrm{V}^{1,p}(\Omega,\mathbb{R}^{m})=\big{\\{}u\in\mathrm{L}^{1}_{\mathrm{loc}}(\Omega,\mathbb{R}^{m})\,\text{:
$u$ is weakly differentiable and $|\nabla
u|\in\mathrm{L}^{p}(\Omega)$}\big{\\}}.$

On extending earlier results from [GV73] about dimension distortion under
quasiconformal maps, in [Ka99] Kaufman established a modified version of
property (1.2) for maps in super-critical Sobolev spaces
$\mathrm{V}^{1,p}_{\rm loc}(\Omega,\mathbb{R}^{m})$, with $p>n$. Recall that
the latter assumption ensures that each map $u\in\mathrm{V}^{1,p}_{\rm
loc}(\Omega,\mathbb{R}^{m})$ has a (locally Hölder) continuous representative.
The result of [Ka99, Theorem 1] asserts that, with this choice of the
representative, if $\alpha\in(0,n)$, then for sets $E\subset\Omega$,

(1.4) if $\mathscr{H}^{\alpha}(E)<\infty$, then $\mathscr{H}^{\frac{\alpha
p}{\alpha+p-n}}(u(E))=0$.

The shift of the exponent from $\alpha$ to $\frac{\alpha p}{\alpha+p-n}$ is
usually referred to as distortion of Hausdorff measure under the map $u$. Let
us also notice that, by contrast, such a property does not hold for all maps
in $\mathrm{V}^{1,p}_{\rm loc}(\Omega,\mathbb{R}^{m})$ if $1\leq p\leq n$,
even under the additional assumption of continuity. With this regard, see
[Ce42, Po71, MM95] and, specifically, [HK14, Theorem 4.3], for a streamlined
presentation of Cesari’s example of a continuous map
$u\in\mathrm{V}^{1,n}(\mathbb{R}^{n},\mathbb{R}^{m})$, with $n\geq 2$ and
$m\in\mathbb{N}$, such that
$u\bigl{(}[-1,1]\times\\{0\\}^{n-1}\bigr{)}=[-1,1]^{m}$. This shows that the
Lusin $N$ condition fails in $\mathrm{V}^{1,n}_{\rm
loc}(\Omega,\mathbb{R}^{n})$, for $n\geq 2$. Let us also observe that the
distortion property is neither guaranteed by the mere membership of $u$ in the
space of locally Hölder continuous maps with exponent $1-\frac{n}{p}$, into
which $\mathrm{V}^{1,p}_{\rm loc}(\Omega,\mathbb{R}^{m})$ is embedded under
the assumption that $p>n$.

Several contributions on related questions are available in the literature –
see e.g. [ACTUTV13, BMT13, BKK15, FKR19, HH15, KKM99, KK14, KMZ15, KZ15,
LSUT10, RZZ11]. Specifically, [KKM99, Theorem C] ensures that the Lusin $N$
property holds for the continuous representative of every function from the
Sobolev type space $\mathrm{V}^{1}\mathrm{L}^{n,1}_{\rm
loc}(\Omega,\mathbb{R}^{n})$ of those weakly differentiable maps whose
gradient belongs to the Lorentz space $\mathrm{L}^{n,1}_{\rm
loc}(\Omega,\mathbb{R}^{n})$. This borderline Sobolev space, which coincides
with $\mathrm{V}^{1,1}_{\rm loc}(\Omega,\mathbb{R}^{n})$ when $n=1$, is in
fact optimal in the class of all Sobolev type spaces built upon a
rearrangement-invariant space for this property to hold. The same space is
even optimal for embeddings into the space of continuous functions [CP98]. On
the other hand, the space $\mathrm{V}^{1}\mathrm{L}^{n,1}_{\rm
loc}(\Omega,\mathbb{R}^{n})$ is not embedded into any space of locally
uniformly continuous functions (with a common modulus of continuity), and no
analogue of property (1.4) holds for maps in
$\mathrm{V}^{1}\mathrm{L}^{n,1}_{\rm loc}(\Omega,\mathbb{R}^{n})$, since they
can map sets of any positive Hausdorff dimension into sets of Hausdorff
dimension $n$. This means that they can map sets having zero
$\alpha$-dimensional Hausdorff measure, for any given $\alpha\in(0,n]$, into
sets of positive $n$-dimensional Hausdorff measure.

The present paper is aimed at a comprehensive description of the distortion of
Hausdorff measures in the broader class of Orlicz-Sobolev maps. These are maps
in the homogeneous Orlicz-Sobolev space

(1.5)
$\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})=\big{\\{}u\in\mathrm{L}^{1}_{\mathrm{loc}}(\Omega,\mathbb{R}^{m})\,\text{:
$u$ is weakly differentiable and $|\nabla
u|\in\mathrm{L}^{A}(\Omega)$}\big{\\}},$

or in its local counterpart $\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{m})$, where $A$ is a Young function and
$\mathrm{L}^{A}({{\mathbb{R}}^{n}})$ is the associated Orlicz space.

In view of the discussion above, it is natural to focus on Orlicz-Sobolev
spaces $\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$ which are embedded
into a space of locally uniformly continuous functions. A necessary and
sufficient condition on $A$ for embeddings of this kind to hold, as well as
the optimal modulus of continuity are well known [Ci96], and are recalled in
Section 2. As a matter of fact, if $n\geq 2$ the same condition is also
necessary and sufficient for the embedding of $\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{m})$ into the space of plainly continuous functions,
and even into $\mathrm{L}^{\infty}_{\rm loc}(\Omega,\mathbb{R}^{m})$. In what
follows, when dealing with a map from a space $\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{m})$ associated with a Young function $A$ satisfying
this condition, we shall always refer to its continuous representative.

A treatment of the relevant questions, tailored for the realm of general
Orlicz-Sobolev spaces, finds its proper environment in a wider range of
Hausdorff measures $\mathscr{H}^{\varphi}$, built upon gauge functions
$\varphi\colon[0,\infty)\to[0,\infty)$ which need not be just powers. In fact,
this extended class of measures already appears in Hausdorff’s original paper
[Ha1918]. The corresponding Hausdorff content, denoted by
$\mathscr{H}^{\varphi}_{\infty}$, will also come into play in our discussion.

Little seems to be known on the effect of Orlicz-Sobolev maps on general
Hausdorff measures of sets. The available results in this regard are
apparently limited to the very special case when $A(t)\sim t^{n}(\log t)^{q}$
near infinity for some $q>0$, and $\varphi(r)\sim
r^{\alpha}(\log\frac{1}{r})^{\beta}$ near $0$ for some $\beta>0$. They can be
found in [KZ15, KMZ15, RZZ11]. Here, and in what follows, the relation
“$\sim$” between two quantities means that they are bounded by each other, up
to positive multiplicative constants.

Our contribution approaches the question in full generality and offers a handy
recipe which, from any Young function $A$ and any gauge function $\varphi$ as
above, produces another gauge function $\psi$ with the property that if
$u\in\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$, then, for sets
$E\subset\Omega$,

(1.6) if $\mathscr{H}^{\varphi}(E)=0$ then $\mathscr{H}^{\psi}(u(E))=0$.

Besides capturing wider classes of Sobolev type maps and Hausdorff measures,
broadening the functional framework entails the surfacing of certain traits of
the theory, which, by contrast, remain hidden for customary Sobolev spaces and
Hausdorff measures defined in terms of plain power type functions. Allowing
for a more flexible choice of function spaces and measures also enables us to
supply sharper information about the optimality of well known results.

For instance, when $\varphi(r)=r^{n}$ property (1.6) is shown to hold with
$\psi(r)=r^{n}$. Hence, we infer that the Lusin $N$ property is supported by
any Orlicz-Sobolev space $\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{n})$
which is embedded into a space of locally uniformly continuous functions. This
conclusion reproduces [KRSS14, Corollary 2], and can also be reached via Lusin
$N$ property of the space $\mathrm{V}^{1}_{\rm
loc}\mathrm{L}^{n,1}(\Omega,\mathbb{R}^{n})$ recalled above, owing to the
inclusion $\mathrm{L}^{A}_{\rm loc}(\Omega)\subset\mathrm{L}^{n,1}_{\rm
loc}(\Omega)$, which is known to hold whenever $\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{n})$ is embedded into $\mathrm{L}^{\infty}_{\rm
loc}(\Omega,\mathbb{R}^{n})$.

Even more interestingly, the comprehensive environment of the present paper
provides us with the required precision to disclose a seemingly new stability
phenomenon, which is not visible when the Sobolev spaces and the Hausdorff
measures at disposal are the standard ones. It shows that the validity of
property (1.6) with $\mathscr{H}^{\psi}\sim\mathscr{H}^{\varphi}$ is not
peculiar to the Hausdorff measure $\mathscr{H}^{n}$, and hence, when $m=n$, to
the Lebesgue measure. Indeed, classes of Young functions $A$ and of gauge
functions $\varphi$ are identified for which such an equivalence holds.
Roughly speaking, this happens when the function $A(t)$ has an essentially
faster growth than the limiting power $t^{n}$ near infinity – e.g.
$A(t)=t^{p}$ with $p>n$ – and the gauge function $\varphi$ decays at zero
essentially more slowly than any power. One striking aspect of this result is
that it parallels the one holding for the Lebesgue measure, but at the
opposite endpoint, in a sense, of the scale of Hausdorff measures. Conversely,
for gauge functions $\varphi$ of intermediate strength, or for Young functions
$A$ with almost critical $n$-th power type growth, a gap always occurs between
the measures $\mathscr{H}^{\varphi}$ and $\mathscr{H}^{\psi}$.

Property (1.6) will follow as a corollary of Theorem 3.1. The latter amounts
to a robust form of this property, and tells us that, given any function
$u\in\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$, for every $\varepsilon>0$ there
exists $\delta>0$ such that, for sets $E\subset\Omega$,

(1.7) if $\mathscr{H}^{\varphi}(E)<\delta$ then
$\mathscr{H}^{\psi}(u(E))<\varepsilon$.

An analogue of property (1.7) is also established with Hausdorff measures
replaced by Hausdorff contents.

These results are refined in Theorems 3.4 and 3.5. The former provides us with
a criterion on the Young function $A$ and the gauge function $\varphi$ for an
augmented version of property (1.6) to hold. This version tells us that the
vanishing assumption on $\mathscr{H}^{\varphi}(E)$ can just be replaced by a
finiteness hypothesis, with the same conclusion. Namely, if
$u\in\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$, then, for sets
$E\subset\Omega$,

(1.8) if $\mathscr{H}^{\varphi}(E)<\infty$ then $\mathscr{H}^{\psi}(u(E))=0$.

The examples alluded to above, when
$\mathscr{H}^{\varphi}\sim\mathscr{H}^{\psi}$, demonstrate that property (1.8)
fails if $A$ and $\varphi$ do not meet the requirements of the relevant
criterion. However, in this case a quantitative form of property (1.6) is yet
offered, which tells us that

(1.9) $\mathscr{H}^{\psi}\bigl{(}u(E)\bigr{)}\leq c\,\mathscr{H}^{\varphi}(E)$

for maps $u\in\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$, for all sets
$E\subset\Omega$ and for some constant $c$ depending, in an explicit way, on
$n$, $\psi$ and $\|\nabla u\|_{\mathrm{L}^{A}(\Omega)}$.

Finally, Theorem 3.5 contains a variant of inequality (1.9) involving
Hausdorff contents instead of measures. The inequality in question holds for
maps in $\mathrm{V}^{1,A}(\mathbb{R}^{n},\mathbb{R}^{m})$. Moreover, the
dependence on $\mathscr{H}^{\varphi}_{\infty}(E)$ on the right-hand side is
nonlinear. A slightly more general version of this inequality can be
established, for maps defined in arbitrary open sets
$\Omega\subset\mathbb{R}^{n}$, in the case when
$\mathscr{H}^{\varphi}_{\infty}(E)$ agrees with the outer Lebesgue measure.
This result is enucleated in Theorem 3.7.

Precise statements of the results outlined above are given in Section 3, where
applications to special choices of the functions $A$ and $\varphi$ are also
proposed. In particular, they recover Kaufman’s theorem (1.4), improve a
theorem of [RZZ11], and specify the results highlighted above. Section 4 is
devoted to preliminary technical lemmas. The proofs of the main results are
then accomplished in Section 5. In the final Section 6, the sharpness of some
of the examples worked out is discussed. In particular, as hinted above,
thanks to the use of general Hausdorff measures, an enhanced optimality of the
results already available in the literature can be established.

##  2 Hausdorff measures and Orlicz-Sobolev spaces

In this section we collect a few definitions and properties about the
Hausdorff measures and the underlying gauge functions considered throughout,
and the Young functions for the admissible Orlicz-Sobolev spaces in our main
result.

###  2.1 Gauge functions and Hausdorff measures

A Hausdorff measure can be built upon any gauge function, an expression
whereby we understand any function $\varphi\colon[0,\infty)\to[0,\infty)$
which is increasing, continuous and vanishes only at $0$. Set, for
$\sigma\in(0,\infty]$,

(2.1)
$\mathscr{H}_{\sigma}^{\varphi}(E)=\inf\left\\{\sum_{i=1}^{\infty}\varphi\bigl{(}d(E_{i})\bigr{)}\
:\ E_{i}\subset\mathbb{R}^{n},\,\,d(E_{i})\leq\sigma,\ \
E\subset\bigcup\limits_{i=1}^{\infty}E_{i}\right\\}$

for any set $E\subset\mathbb{R}^{n}$. Here, $d(\cdot)$ stands for diameter.
Then the relevant measure $\mathscr{H}^{\varphi}$ is defined as

(2.2) $\mathscr{H}^{\varphi}(E)=\lim_{\sigma\to
0^{+}}\mathscr{H}_{\sigma}^{\varphi}(E)$

The set function $\mathscr{H}_{\infty}^{\varphi}$ also enters our results and
is called Hausdorff content.
For sets $E\subset\mathbb{R}^{n}$, one has that

(2.3) $\mathscr{H}_{\infty}^{\varphi}(E)=0\quad\text{if and only
if}\quad\mathscr{H}^{\varphi}(E)=0,$

see [AH96, Proposition 5.1.5].
In order to rule out trivial cases, we shall assume that

(2.4) $\liminf_{r\to 0^{+}}\frac{\varphi(r)}{r^{n}}>0.$

Actually, if condition (2.4) fails, then $\mathscr{H}^{\varphi}(E)=0$ for
every set $E\subset\mathbb{R}^{n}$. This follows from [AH96, Proposition
5.1.8]. In fact, a minor variant of this proposition also ensures that, under
(2.4), one can additionally assume, without loss of generality, that the
function

(2.5) $\frac{\varphi(r)}{r^{n}}$ is non-increasing,

see Lemma 2.1 below.
We shall therefore suppose throughout, without explicit mentioning, that all
gauge functions $\varphi$ fulfill condition (2.5).
Clearly, choosing $\varphi(t)=t^{\alpha}$, with $\alpha\in(0,n]$, in the
definitions of $\mathscr{H}^{\varphi}$ and $\mathscr{H}_{\infty}^{\varphi}$
reproduces (up to a multiplicative constant) the customary Hausdorff measure
$\mathscr{H}^{\alpha}$ and Hausdorff content $\mathscr{H}^{\alpha}_{\infty}$.
One has that $\mathscr{H}^{\varphi}\neq\mathcal{L}^{n}$ is and only if

(2.6) $\lim_{r\to 0^{+}}\frac{\varphi(r)}{r^{n}}=\infty.$

Observe that, thanks to property (2.5),

(2.7) $\varphi(kr)\leq\max\\{1,k^{n}\\}\,\varphi(r)\qquad\text{for $k>0$ and
$r\geq 0$.}$

An alternate Hausdorff type measure $\Lambda^{\varphi}$, defined in terms of
coverings by dyadic closed cubes, will be needed in our proofs. These measures
are sometimes called net measures, see [Fa85, Chap. 5]. On setting

(2.8)
$\Lambda_{\sigma}^{\varphi}(E)=\inf\bigg{\\{}\sum_{i}\varphi(d(Q_{i})):E\subset\bigcup_{i}Q_{i},\,\,Q_{i}\,\text{is
a dyadic cube and }d(Q_{i})\leq\sigma\bigg{\\}}$

for any set $E\subset\mathbb{R}^{n}$, the relevant measure is defined as

(2.9) $\Lambda^{\varphi}(E)=\lim_{\sigma\to
0^{+}}\Lambda_{\sigma}^{\varphi}(E).$

Thanks to property (2.7),

(2.10) $\mathscr{H}^{\varphi}_{\sigma}(E)\leq\Lambda^{\varphi}_{\sigma}(E)\leq
c_{n}\mathscr{H}^{\varphi}_{\frac{\sigma}{2}}(E)$

for $\sigma\in(0,\infty]$ and for every $E\subset{{\mathbb{R}}^{n}}$, and
hence

(2.11) $\mathscr{H}^{\varphi}(E)\leq\Lambda^{\varphi}(E)\leq
c_{n}\mathscr{H}^{\varphi}(E)$

for every set $E\subset{{\mathbb{R}}^{n}}$, where the constant $c_{n}$ only
depends on the dimension $n$ (see, e.g., [BKK15, Lemma 2.2] ). Moreover, by
property (2.7) again, given any $k>0$, on defining the gauge function
$\varphi_{k}$ as

(2.12) $\varphi_{k}(t)=\varphi(kt)\quad\text{ for $t\geq 0$,}$

one has that

(2.13) $\Lambda^{\varphi_{k}}(E)\approx\Lambda^{\varphi}(E)$

for every set $E\subset{{\mathbb{R}}^{n}}$, up to multiplicative constants
depending only on $k$.
Since $\mathscr{H}^{\varphi}$ is a regular outer measure, it enjoys the
ascending sets property, which tells us that, if $\\{E_{j}\\}$ is any sequence
of sets in ${{\mathbb{R}}^{n}}$ such that $E_{j}\subset E_{j+1}$ for
$j\in\mathbb{N}$, then

(2.14)
$\mathscr{H}^{\varphi}\Big{(}\bigcup_{j}E_{j}\Big{)}=\lim_{j\to\infty}\mathscr{H}^{\varphi}(E_{j}),$

see [Ro70/98]. A deeper result from [Da70] (see also [Ro70/98]) ensures that
the same conclusion holds for the Hausdorff content. Namely, if $\\{E_{j}\\}$
is a sequence as above, then

(2.15)
$\mathscr{H}^{\varphi}_{\infty}\Big{(}\bigcup_{j}E_{j}\Big{)}=\lim_{j\to\infty}\mathscr{H}^{\varphi}_{\infty}(E_{j}).$

Notice that we shall only make use of (2.15) in the case when the gauge
function $\varphi$ is a power. In fact, we could, at the expense of an
additional multiplicative constant, use the much more elementary result [Fa85,
Lemma 5.3]. Instead of (2.15), the latter yields the bound

$\mathscr{H}^{\varphi}_{\infty}\Big{(}\bigcup_{j}E_{j}\Big{)}\leq
c\lim_{j\to\infty}\mathscr{H}^{\varphi}_{\infty}(E_{j})$

for some constant that only depends on $\varphi$ and $n$.

###### Lemma 2.1

Let $\varphi\colon[0,\infty)\to[0,\infty)$ be a continuous increasing function
such that $\varphi(0)=0$ and satisfying condition (2.4). Then, there exist a
continuous increasing function $\varphi^{\circ}\colon[0,\infty)\to[0,\infty)$
and a positive constant $c=c(n)$ such that $\varphi^{\circ}(0)=0$,

(2.16) $\frac{\varphi^{\circ}(r)}{r^{n}}$ is non-increasing,

and

(2.17)
$c\,\mathscr{H}^{\varphi}_{\delta}(E)\leq\mathscr{H}^{\varphi^{\circ}}_{\delta}(E)\leq\mathscr{H}^{\varphi}_{\delta}(E)$

for every $\delta\in(0,\infty]$ and every set $E\subset\mathbb{R}^{n}$.

Proof. Define the function $\varphi^{\circ}\colon[0,\infty)\to[0,\infty)$ as

$\varphi^{\circ}(r)=\left\\{\begin{array}[]{ll}\displaystyle{r^{n}\inf_{0<t\leq
r}\frac{\varphi(t)}{t^{n}}}&\mbox{ if }r>0\\\ 0&\mbox{ if
}r=0.\end{array}\right.$

One can verify that the function $\varphi^{\circ}$ is increasing and fulfills
property (2.16). These monotonicity properties imply that $\varphi^{\circ}$ is
continuous in $(0,\infty)$, and, since $\varphi$ is continuous in $[0,\infty)$
and vanishes at $0$, the function $\varphi^{\circ}$ is continuous in
$[0,\infty)$ as well.
Inasmuch as $\varphi^{\circ}\leq\varphi$, the second inequality in equation
(2.17) is trivial. In order to prove the first one, fix $\delta\in(0,\infty]$
and $E\subset\mathbb{R}^{n}$. We may assume that
$\mathscr{H}^{\varphi^{\circ}}_{\delta}(E)<\infty$, otherwise there is nothing
to prove. Given any $\varepsilon>0$, pick a covering
$\bigcup_{i=1}^{\infty}E_{i}\supset E$, with $0<d(E_{i})<\delta$ and such that

(2.18)
$\sum_{i=1}^{\infty}\varphi^{\circ}\bigl{(}d(E_{i})\bigr{)}<\mathscr{H}^{\varphi^{\circ}}_{\delta}(E)+\varepsilon.$

For each $i$ we choose $d_{i}\in(0,d(E_{i})]$ with the property that
$\varphi^{\circ}(d(E_{i}))(1+\varepsilon)>\bigl{(}d(E_{i})/d_{i}\bigr{)}^{n}\varphi(d_{i})$.
Now, observe that there exists a dimensional constant $c$ such that, given
$0<d\leq D$, any subset of $\mathbb{R}^{n}$ of diameter $D$ can be covered by
a family of sets with cardinality less that $c\bigl{(}D/d\bigr{)}^{n}$, and
diameter at most $d$. This fact can be verified by considering a suitable
translation and dilation of the grid defined by $\mathbb{Z}^{n}$. Hence, for
each $i$ the set $E_{i}$ can be decomposed as $E_{i}=\bigcup_{j}E_{i,j}$ where
the cardinality of the family of the sets $E_{i,j}$ does not exceed
$c\bigl{(}d(E_{i})/d_{i}\bigr{)}^{n}$, and $d(E_{i,j})\leq d_{i}$.
Consequently,

(2.19)
$(1+\varepsilon)\sum_{i=1}^{\infty}\varphi^{\circ}\bigl{(}d(E_{i})\bigr{)}>\sum_{i=1}^{\infty}\left(\frac{d(E_{i})}{d_{i}}\right)^{n}\varphi(d_{i})\geq\frac{1}{c}\sum_{i=1}^{\infty}\sum_{j}\varphi\bigl{(}d(E_{i,j})\bigr{)}\geq\frac{1}{c}\mathscr{H}^{\varphi}_{\delta}(E).$

The conclusion follows from inequalities (2.18) and (2.19).

###  2.2 Young functions and Orlicz-Sobolev spaces

A Young function $A\colon[0,\infty)\to[0,\infty]$ is a (non trivial) convex
function such that $A(0)=0$. As a consequence, the function

(2.20) $\frac{A(t)}{t}$ is non-decreasing in $(0,\infty)$.

The Young conjugate $\widetilde{A}$ of $A$ is also a Young function and is
defined by

(2.21) $\widetilde{A}(t)=\sup\bigl{\\{}\tau t-A(\tau):\,\tau\geq
0\bigr{\\}}\quad\text{for $t\geq 0$.}$

One has that $\widetilde{\widetilde{A}}=A$.
A Young function $A$ is said to dominate another Young function $A_{0}$
globally if there exists a positive constant $c$ such that

(2.22) $A_{0}(t)\leq A(ct)\quad\text{for $t\geq 0.$}$

The function $A$ is said to dominate $A_{0}$ near infinity if there exists
$t_{0}>0$ such that (2.22) holds for $t\geq t_{0}$. The functions $A$ and
$A_{0}$ are called equivalent globally [near infinity] if they dominate each
other globally [near infinity].

Let $\Omega$ be a measurable set in $\mathbb{R}^{n}$ and let $A$ be a Young
function. The Orlicz space $\mathrm{L}^{A}(\Omega)$ is the Banach space of
measurable functions $u\colon\Omega\to\mathbb{R}$ for which the Luxemburg norm

(2.23)
$\|u\|_{\mathrm{L}^{A}(\Omega)}=\inf\left\\{\lambda>0:\int_{\Omega}A\left(\frac{|u|}{\lambda}\right)dx\leq
1\right\\}\,$

is finite.
If a Young function $A$ dominates another Young function $A_{0}$ globally
[near infinity], then

(2.24) $\mathrm{L}^{A}(\Omega)\hookrightarrow\mathrm{L}^{A_{0}}(\Omega)$

for every measurable set $\Omega$ [for every measurable set $\Omega$ with
${\mathscr{L}}^{n}(\Omega)<\infty$]. Moreover, embedding (2.24) holds as an
equality (up to equivalent norms), if equivalence holds between the Young
function $A$ and $A_{0}$ instead of domination.
When $\Omega$ is an open set, the corresponding homogeneous Orlicz-Sobolev
space $\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$ is defined by (1.5). The local
Orlicz-Sobolev space $\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$ is
defined accordingly.

As explained in Section 1, our analysis applies to Orlicz-Sobolev spaces
$\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$ supporting an embedding of
the form

(2.25) $\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{m})\hookrightarrow\mathrm{C}^{\omega}_{\rm
loc}(\Omega,\mathbb{R}^{m})$

for some space of (locally) uniformly continuous functions
$\mathrm{C}^{\omega}_{\rm loc}(\mathbb{R}^{n},\mathbb{R}^{m})$, with modulus
of continuity $\omega\colon[0,\infty)\to[0,\infty)$. By [Ci96, Theorems 1a and
3], such an embedding holds if and only if

(2.26)
$\begin{cases}\displaystyle\lim_{t\to\infty}\frac{t}{A(t)}=0&\quad\text{if
$n=1$,}\\\ \\\
\displaystyle\int^{\infty}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-1}}\,dt<\infty&\quad\text{if
$n\geq 2$.}\end{cases}$

When $n\geq 2$ the condition in (2.26) is in fact also necessary and
sufficient for the embedding of $\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{m})$ into the space of plainly continuous functions,
and even into $\mathrm{L}^{\infty}_{\rm loc}(\Omega,\mathbb{R}^{m})$. The
sufficiency of this condition was earlier established in [C51].
Specifically, in connection with (2.25), denote by $B$ the Young function
given by

(2.27) $B(t)=\begin{cases}A(t)&\quad\text{if $n=1$}\\\ \\\
\Bigg{(}t^{n^{\prime}}\displaystyle\int_{t}^{\infty}\frac{\widetilde{A}(s)}{s^{1+n^{\prime}}}\,ds\Bigg{)}^{\widetilde{\quad}}&\quad\text{if
$n\geq 2$,}\end{cases}$

where the notation $(\dots)^{\widetilde{\quad}}$ stands for the Young
conjugate of the function in parentheses. Observe that, by [Ci04, Lemma 4.4],
the convergence of the integral in (2.27) is equivalent to that in (2.26).
Then there exists a constant $\kappa=\kappa(n,m)$ such that the Morrey type
inequality

(2.28)
$\frac{d(Q)^{n}}{\kappa}B\bigg{(}\frac{d(u(Q))}{\kappa\,\lambda\,d(Q)}\bigg{)}\leq\int_{Q}A\bigg{(}\frac{|\nabla
u|}{\lambda}\bigg{)}\,dx,$

holds for the continuous representative of any map in
$u\in\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$. Here $Q$ denotes any
cube compactly contained in $\Omega$ and $\lambda$ any positive number which
makes the integral finite – see [AC05, Proof of Theorem 1.1] (and also [CC16,
Theorem 4.1]). Moreover, if $u\in\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$,
inequality (2.28) holds also for every open cube $Q\subset\Omega$.
In inequality (2.28), the continuous representative of $u$ is considered. As
already mentioned above, such a representative will always be considered also
in what follows, without explicitly mentioning, when dealing with maps in the
space $\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$ with a Young
function $A$ fulfilling assumption (2.26).

In order to avoid additional unnecessary technicalities, in our discussion of
the distortion of Hausdorff measures we shall assume that the Young function
$A$ defining the ambient Orlicz-Sobolev space is such that

(2.29) $0<A(t)<\infty\quad\text{for }t>0,$

and

(2.30) $\begin{cases}\displaystyle\lim_{t\to
0^{+}}\frac{t}{A(t)}=\infty&\quad\text{if $n=1$,}\\\ \\\
\displaystyle\int_{0}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-1}}\,dt=\infty&\quad\text{if
$n\geq 2$.}\end{cases}$

A few comments on assumptions (2.29) and (2.30) are in order. The assumptions
that $A(t)>0$ for $t>0$ and that $A$ satisfies (2.29) are immaterial when
dealing with the space $\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$.
The same is true for the space $\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$,
provided that ${\mathscr{L}}^{n}(\Omega)<\infty$. Indeed, the function $A$ can
be replaced, if necessary, by a Young function $A_{0}$ equivalent near
infinity such that $A_{0}(t)>0$ for $t>0$ and satisfying condition (2.30).
Thanks to equation (2.24), after this replacement, $\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{m})=\mathrm{V}^{1,A_{0}}_{\rm
loc}(\Omega,\mathbb{R}^{m})$, and, if ${\mathscr{L}}^{n}(\Omega)<\infty$, then
$\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})=\mathrm{V}^{1,A_{0}}(\Omega,\mathbb{R}^{m})$
(up to equivalent norms). By contrast, such a replacement does affect the
space $\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$ when
${\mathscr{L}}^{n}(\Omega)=\infty$. However, a close inspection of our proofs
will reveal versions of our main results stated for maps in the space
$\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$ continue to hold in the broader
space of those weakly differentiable maps $u:\Omega\to\mathbb{R}^{m}$ such
that $|\nabla u|\in\mathrm{L}^{A}(E)$ for every set $E\subset\Omega$ such that
${\mathscr{L}}^{n}(E)<\infty$. The latter space is again invariant under the
replacement of $A$ by a function $A_{0}$ as above.
As for the requirement that $A(t)<\infty$ for $t>0$, it can actually been
dropped in our results about properties (1.6) and (1.7), and in their
counterparts about Hausdorff contents. This is due to the fact that, if
$A(t)=\infty$ for large values of $t$, then
$\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})\hookrightarrow\mathrm{V}^{1,\infty}(\Omega,\mathbb{R}^{m})$.
Since the proofs of the properties in question only make use of the local
behaviour of maps in $\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$, our results
follow via the classical results for Lipschitz continuous maps recalled above,
inasmuch as maps in $\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$ are locally
Lipschitz continuous. On the other hand, the case of Lipschitz continuous
functions demonstrates that the finiteness assumption on $A$ cannot be
dispensed with when the stronger property (1.8) is in question.

Let us finally observe that, when $n\geq 2$, the function $A$ dominates
globally the function $B$ given by (2.27) [CC16, Lemma 3.2]. Moreover, the
functions $A$ and $B$ are equivalent near infinity if and only if

(2.31) $i_{\infty}(A)>n,$

where $i_{\infty}(A)$ denotes the lower Matuszewska–Orlicz index of $A$ at
infinity. For Young functions $A$ fulfilling condition (2.29), this index is
defined as

(2.32)
$i_{\infty}(A)=\lim_{\lambda\to\infty}\frac{\log\Big{(}\liminf_{t\to\infty}\tfrac{A(\lambda
t)}{A(t)}\Big{)}}{\log\lambda}.$

The functions $A$ and $B$ are equivalent globally if and only if

(2.33) $i(A)>n\,$

where $i(A)$ denotes the global lower Matuszewska–Orlicz index of $A$, given
by

(2.34)
$i(A)=\lim_{\lambda\to\infty}\frac{\log\Big{(}\inf_{t>0}\tfrac{A(\lambda
t)}{A(t)}\Big{)}}{\log\lambda}.$

See [St79, Lemma 2.3] and, for more details, [CM19, Proposition 4.1].

##  3 Main results and examples

The gauge function $\psi$ which renders implication (1.6) true is associated
with any Young function $A$ satisfying assumptions (2.26), (2.29), (2.30), and
any gauge function $\varphi$ as follows. Consider the function
$J\colon[0,\infty)\to(0,\infty)$ defined as

(3.1)
$J(s)=\begin{cases}s\,B^{-1}\Big{(}\displaystyle\frac{\varphi(s)}{s^{n}}\Big{)}&\quad\text{for
$s>0$}\\\ 0&\quad\text{for $s=0$,}\end{cases}$

where $B$ is defined at (2.27). Then the function
$\psi\colon[0,\infty)\to[0,\infty)$ is given by

(3.2) $\psi(r)=\varphi(J^{-1}(r))\qquad\text{ for $r>0$.}$

The function $\psi$ will be shown to be a well-defined gauge function in the
next section.

Let us point out that, in view of the definition of the Hausdorff measures
$\mathscr{H}^{\varphi}$ and $\mathscr{H}^{\psi}$, only the asymptotic
behaviour of $B$, and hence of $A$, near infinity, and the asymptotic
behaviours of $\varphi$, $J$ and $\psi$ near zero are relevant. In particular,
if $A$ fulfills property (2.31), then, owing to inequality (4.18) below,
replacing $B$ simply by $A$ in definitions (3.1) and (3.2) results in an
equivalent function $\psi$, and hence in an equivalent Hausdorff measure
$\mathscr{H}^{\psi}$. Analogously, under assumption (2.33), the Hausdorff
content $\mathscr{H}_{\infty}^{\psi}$ can be equivalently defined just via $A$
instead of $B$.

We are now in a position to state our first main result, which concerns
property (1.7).

###### Theorem 3.1

Let $n$, $m\in\mathbb{N}$, let $\Omega$ be an open set in $\mathbb{R}^{n}$,
and let $A$ be a Young function satisfying conditions (2.26), (2.29) and
(2.30). Assume that $\varphi$ is a gauge function, and let $\psi$ be the gauge
function defined by (3.2).
(i) Let $u\in\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$. For every
$\varepsilon>0$ there exists $\delta>0$ such that, for sets $E\subset\Omega$,

(3.3) if $\mathscr{H}^{\varphi}(E)<\delta$, then
$\mathscr{H}^{\psi}(u(E))<\varepsilon$.

(ii) Let $u\in\mathrm{V}^{1,A}(\mathbb{R}^{n},\mathbb{R}^{m})$. For every
$\varepsilon>0$ there exists $\delta>0$ such that, for sets
$E\subset\mathbb{R}^{n}$,

(3.4) if $\mathscr{H}^{\varphi}_{\infty}(E)<\delta$, then
$\mathscr{H}^{\psi}_{\infty}(u(E))<\varepsilon$.

###### Remark 3.2

Observe that, if $\varphi(r)=r^{n}$, then $\psi(r)=r^{n}$ for any Young
function $A$ as in the statement of Theorem 3.1. Hence, one infers for
$u\in\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{n})$ that for every $\varepsilon>0$
there exists $\delta>0$ such that, for sets $E\subset\Omega$,

(3.5) if ${\mathscr{L}}^{n}(E)<\delta$ then
${\mathscr{L}}^{n}(u(E))<\varepsilon$.

Thus the classical Lusin $N$-property for maps in the Orlicz-Sobolev space
$\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{n})$, with $A$ fulfilling condition
(2.26), announced in Section 1 is in particular recovered from Theorem 3.1.

Property (1.6) is the content of the next corollary and follows from Theorem
3.1.

###### Corollary 3.3

Let $n$, $m\in\mathbb{N}$, let $\Omega$ be an open set in $\mathbb{R}^{n}$,
and let $A$ be a Young function satisfying conditions (2.26), (2.29) and
(2.30). Assume that $\varphi$ is a gauge function, and let $\psi$ be the gauge
function defined by (3.2). Let $u\in\mathrm{V}^{1,A}_{\rm
loc}(\Omega,\mathbb{R}^{m})$. Then for sets $E\subset\Omega$,

(3.6) if $\mathscr{H}^{\varphi}(E)=0$, then $\mathscr{H}^{\psi}(u(E))=0$.

Formulating the qualifications on the Young function $A$ and the gauge
function $\varphi$ which entail the validity of the enhanced property (1.8)
involves the “scaling functions” defined as follows.
With any function $h\colon(0,\infty)\to(0,\infty)$, we associate the function
$\Theta_{h}^{0}\colon(0,\infty)\to[0,\infty]$, defined as

(3.7) $\Theta_{h}^{0}(r)=\limsup_{t\to 0^{+}}\frac{h(rt)}{h(t)}\quad\text{for
$r>0$,}$

the function $\Theta_{h}^{\infty}\colon(0,\infty)\to[0,\infty]$, defined as

(3.8)
$\Theta_{h}^{\infty}(r)=\limsup_{t\to\infty}\frac{h(rt)}{h(t)}\quad\text{for
$r>0$,}$

and the function $\Theta_{h}\colon(0,\infty)\to[0,\infty]$, defined as

(3.9) $\Theta_{h}(r)=\sup_{t>0}\frac{h(rt)}{h(t)}\quad\text{for $r>0$.}$

In particular, if the function $h$ is non-decreasing, then these three
functions are non-decreasing as well. They naturally arise in connection with
scaling properties of Hausdorff measures and Hausdorff contents.

###### Theorem 3.4

Let $n$, $m\in\mathbb{N}$ and let $\Omega$ be an open subset of
$\mathbb{R}^{n}$. Let $A$ be a Young function satisfying conditions (2.26),
(2.29), (2.30), and let $B$ be the function defined by (2.27). Assume that
$\varphi$ is a gauge function fulfilling condition (2.6), and let $\psi$ be
the gauge function defined by (3.2).
(i) Assume that

(3.10) $\lim_{r\to 0^{+}}\Theta_{\psi}^{0}(r)=0$ and $\lim_{r\to
0^{+}}\Theta_{B^{-1}}^{\infty}(r)=0$.

Let $u\in\mathrm{V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$. Then, for sets
$E\subset\Omega$,

(3.11) if $\mathscr{H}^{\varphi}(E)<\infty$, then
$\mathscr{H}^{\psi}\bigl{(}u(E)\bigr{)}=0$.

(ii) Assume that

(3.12) either $\lim_{r\to 0^{+}}\Theta_{\psi}^{0}(r)>0$ or $\lim_{r\to
0^{+}}\Theta_{B^{-1}}^{\infty}(r)>0$.

Let $u\in\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$. Then

(3.13) $\mathscr{H}^{\psi}\bigl{(}u(E)\bigr{)}\leq
c\,\Theta_{\psi}^{0}\bigl{(}\kappa\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{+}\bigr{)}\mathscr{H}^{\varphi}(E)$

for every set $E\subset\Omega$, where $\kappa$ is the constant appearing in
inequality (2.28), $c=c_{n}\lim_{r\to
0^{+}}\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)\bigr{)}$, and
$c_{n}$ is the constant appearing in equation (2.11). On the right-hand side
of (3.13), and in similar expressions below, the notation
$\Theta_{\psi}^{0}\bigl{(}\kappa\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{+}\bigr{)}$ stands for the one-sided limit of
$\Theta_{\psi}^{0}(r)$ as $r\to(\kappa\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)})^{+}$.

Let us note that the constant $c$ in inequality (3.13) is strictly positive
under assumption (3.12). This is a consequence of the fact, by property (4.17)
and Lemma 4.5, $\Theta_{\psi}^{0}(r)>0$ if $r>0$

A counterpart of inequality (3.13) for Hausdorff contents exhibits a nonlinear
dependence on the latter via the function
$\Xi_{\psi,B}\colon(0,\infty)\to[0,\infty)$ defined as

(3.14)
$\Xi_{\psi,B}(r)=r\,\Theta_{\psi}\big{(}\Theta_{B^{-1}}\big{(}1/r)^{+}\big{)}\quad\text{for
$r>0$.}$

Observe that, as will be shown Lemma 4.10, the function $\Xi_{\psi,B}$ is non-
decreasing. However, it need neither tend to $0$ as $r\to 0^{+}$, nor to
$\infty$ as $r\to\infty$ – see Example 3.9.

###### Theorem 3.5

Let $n$, $m\in\mathbb{N}$. Let $A$ be a finite-valued Young function
satisfying conditions (2.26), (2.29), (2.30), and let $B$ be the function
defined by (2.27). Assume that $\varphi$ is a gauge function fulfilling
condition (2.5), and let $\psi$ be the gauge function defined by (3.2). Let
$u\in\mathrm{V}^{1,A}(\mathbb{R}^{n},\mathbb{R}^{m})$. Then

(3.15) $\mathscr{H}^{\psi}_{\infty}\bigl{(}u(E)\bigr{)}\leq
2\kappa\Theta_{\psi}\bigl{(}\kappa\|\nabla
u\|_{\mathrm{L}^{A}(\mathbb{R}^{n})}^{+}\bigr{)}\Xi_{\psi,B}\bigg{(}\frac{c_{n}\mathscr{H}^{\varphi}_{\infty}(E)}{\kappa}\bigg{)}$

for every set $E\subset\mathbb{R}^{n}$, where $\kappa$ is the constant
appearing in inequality (2.28), and $c_{n}$ is the constant appearing in
(2.11).

###### Example 3.6

In the case $u\in\mathrm{V}^{1,p}(\mathbb{R}^{n},\mathbb{R}^{m})$ for $p>n$
and $\varphi(t)=t^{\alpha}$ for $\alpha\in(0,n]$ we have that

(3.16) $\mathscr{H}^{\frac{\alpha
p}{\alpha+p-n}}_{\infty}\bigl{(}u(E)\bigr{)}\leq
2c_{n}p^{\frac{\alpha}{\alpha+p-n}}\left(\frac{p-1}{n^{\prime}-p^{\prime}}\right)^{\frac{\alpha(p-1)}{\alpha+p-n}}\|\nabla
u\|_{p}^{\frac{\alpha
p}{\alpha+p-n}}\mathscr{H}_{\infty}^{\alpha}(E)^{\frac{p-n}{\alpha+p-n}}.$

Although Theorem 3.5 as stated applies to the case $\varphi(t)=t^{n}$ too, a
sharper and localized result holds in this case. This is the content of the
next theorem that is stated in terms of the auxiliary function
$\Phi_{B}\colon[0,\infty)\to[0,\infty)$ given by

(3.17) $\Phi_{B}(r)=\begin{cases}rB^{-1}(1/r)^{n}&\quad\text{if $r>0$}\\\
0&\quad\text{if $r=0$.}\end{cases}$

Here, $B$ denotes the function defined in terms of $A$ and $n$ in (2.27).
Under our assumptions on $A$, the function $\Phi_{B}$ is increasing,
continuous in $[0,\infty)$ and $\lim_{r\to 0^{+}}\Phi_{B}(r)=0$ – see the
proof of Lemma 4.2.

###### Theorem 3.7

Let $n,m\in\mathbb{N}$, with $m\geq n$, and let $\Omega$ be an open subset of
$\mathbb{R}^{n}$. Let $A$ be a Young function satisfying conditions (2.26),
(2.29), (2.30), and let $B$ be the function defined by (2.27). Let
$u\in\mathrm{V}^{1,A}(\Omega,\mathbb{R}^{m})$. Then

(3.18) $\mathscr{H}^{n}_{\infty}\bigl{(}u(E)\bigr{)}\leq 2\kappa^{n+1}\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{n}\Phi_{B}\big{(}{\mathscr{L}}^{n}(E)n^{\frac{n}{2}}/\kappa\big{)}$

for every set $E\subset\Omega$. Here, $\kappa$ denotes the constant from
inequality (2.28).

Since $\Phi_{B}\big{(}{\mathscr{L}}^{n}(E)n^{\frac{n}{2}}/\kappa\big{)}\to
0^{+}$ as ${\mathscr{L}}^{n}(E)\to 0^{+}$, Theorem 3.7 in particular
recaptures a stability version of the Lusin $N$ property for maps taking
values in a target space whose dimension is at least that of the domain space.
Namely, it tells us that for each $\varepsilon>0$ there exists a $\delta>0$
such that, for sets $E\subset\Omega$,

(3.19) if ${\mathscr{L}}^{n}(E)<\delta$, then
$\mathscr{H}^{n}_{\infty}(u(E))<\varepsilon$.

Let us notice that inequality (3.18) is also related to the coarea formula in
Sobolev spaces [MSZ02].

The remaining part of this section is devoted to applications of Theorem 3.1
to specific choices of the Young function $A$ and of the gauge function
$\varphi$.

###### Example 3.8

Assume that

(3.20) $A(t)\,\,\text{is equivalent to}\,\ t^{p}(\log t)^{q}\qquad\text{ near
infinity,}$

where $p>n$ and $q\in\mathbb{R}$, and let

(3.21) $\varphi(r)\sim r^{\alpha}\big{(}\log\tfrac{1}{r})^{\beta}\qquad\text{
near zero,}$

where $\alpha$ and $\beta$ fulfill one of the following alternatives

(3.22) $\begin{cases}0<\alpha<n\,\,\text{and}\,\,\beta\in\mathbb{R}\\\
\alpha=0\,\,\text{and}\,\,\beta<0\\\ \alpha=n\,\,\text{and}\,\,\beta\geq
0.\end{cases}$

Then property (3.3) holds with

(3.23) $\psi(r)\sim\begin{cases}r^{\frac{\alpha
p}{p+\alpha-n}}\big{(}\log\tfrac{1}{r}\big{)}^{\frac{\alpha(q-\beta)}{p+\alpha-n}+\beta}&\quad\text{if
$0\leq\alpha<n$}\\\
r^{n}\big{(}\log\tfrac{1}{r}\big{)}^{\frac{\beta(p-n)}{p}}\big{(}\log(\log\tfrac{1}{r})\big{)}^{\frac{qn}{p}}&\quad\text{if
$\alpha=n$ and $\beta>0$}\\\ r^{n}&\quad\text{if $\alpha=n$ and
$\beta=0$}.\end{cases}\quad\text{near zero.}$

In particular, if $q=0$ and $\beta=0$, then

(3.24) $A(t)=t^{p},\quad\varphi(r)=r^{\alpha},\quad\psi(r)=r^{\frac{\alpha
p}{p+\alpha-n}},$

and we recover Kaufman’s result [Ka99, Theorem 1].
Furthermore, if $q=0$, $\alpha=0$, then

(3.25)
$A(t)=t^{p},\quad\varphi(r)\sim\big{(}\log\tfrac{1}{r}\big{)}^{\beta},\quad\psi(r)\sim\big{(}\log\tfrac{1}{r}\big{)}^{\beta}.$

This is an example where property (3.11) fails, assumption (3.10) not being
satisfied. More instances in the same vein are exhibited in Example 3.11
below.
In order to verify equation (3.23), observe that, since $i(A)=p$, condition
(2.31) is fulfilled. Therefore,

$B(t)\,\,\text{is equivalent to}\,\ t^{p}(\log t)^{q}\qquad\text{ near
infinity.}$

Hence,

$B^{-1}(t)\sim t^{\frac{1}{p}}(\log t)^{-\frac{q}{p}}\qquad\text{ near
infinity,}$

and

$J(s)\sim\begin{cases}s^{\frac{p+\alpha-n}{p}}\big{(}\log\tfrac{1}{s}\big{)}^{\frac{\beta-q}{p}}&\quad\text{if
$0\leq\alpha<n$}\\\
s\big{(}\log\tfrac{1}{s}\big{)}^{\frac{\beta}{p}}\big{(}\log(\log\tfrac{1}{s})\big{)}^{-\frac{q}{p}}&\quad\text{if
$\alpha=n$ and $\beta>0$}\\\ s&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near zero.}$

Consequently,

$J^{-1}(r)\sim\begin{cases}r^{\frac{p}{p+\alpha-n}}\big{(}\log\tfrac{1}{r}\big{)}^{\frac{q-\beta}{p+\alpha-n}}&\quad\text{if
$0\leq\alpha<n$}\\\
r\big{(}\log\tfrac{1}{r}\big{)}^{-\frac{\beta}{p}}\big{(}\log(\log\tfrac{1}{r})\big{)}^{\frac{q}{p}}&\quad\text{if
$\alpha=n$ and $\beta>0$}\\\ r&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near zero,}$

whence equation (3.23) follows.

###### Example 3.9

Assume that

$A(t)\,\,\text{is equivalent to}\,\ t^{n}(\log t)^{q}\qquad\text{ near
infinity,}$

where $q>n-1$, and let $\varphi$ be as in (3.21).
Then property (1.6) holds with

(3.26) $\psi(r)\sim\begin{cases}r^{\frac{n\beta}{\beta+(n-1)-q}}&\quad\text{if
$\alpha=0$}\\\ r^{n}\big{(}\log\tfrac{1}{r}\big{)}^{q-(n-1)}&\quad\text{if
$0<\alpha<n$}\\\
r^{n}\big{(}\log(\log\tfrac{1}{r})\big{)}^{q-(n-1)}&\quad\text{if $\alpha=n$
and $\beta>0$}\\\ r^{n}&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near zero.}$

We emphasize that the result concerning the case when $0<\alpha<n$ enhances a
result from [RZZ11, Theorem 1 (i)], where the weaker conclusion with the gauge
function $r^{n}\big{(}\log\tfrac{1}{r}\big{)}^{q-(n-1)}$ replaced by any gauge
function $r^{n}\big{(}\log\tfrac{1}{r}\big{)}^{\gamma}$, with $\gamma<q-n+1$,
was established.
In order to prove equation (3.26), notice that

$\widetilde{A}(t)\,\,\text{is equivalent to}\,\ t^{n^{\prime}}(\log
t)^{-\frac{q}{n-1}}\qquad\text{ near infinity,}$

whence

$\widetilde{B}(t)\,\,\text{is equivalent to}\,\ t^{n^{\prime}}(\log
t)^{1-\frac{q}{n-1}}\qquad\text{ near infinity.}$

Therefore

$B(t)\,\,\text{is equivalent to}\,\ t^{n}(\log t)^{q-(n-1)}\qquad\text{ near
infinity,}$

and

(3.27) $B^{-1}(t)\sim t^{\frac{1}{n}}(\log t)^{\frac{(n-1)-q}{n}}\qquad\text{
near infinity.}$

Hence,

$J(s)\sim\begin{cases}s^{\frac{\alpha}{n}}\big{(}\log\tfrac{1}{s}\big{)}^{\frac{\beta+(n-1)-q}{n}}&\quad\text{if
$0\leq\alpha<n$}\\\
s\big{(}\log\tfrac{1}{s}\big{)}^{\frac{\beta}{n}}\big{(}\log(\log\tfrac{1}{s})\big{)}^{\frac{(n-1)-q}{n}}&\quad\text{if
$\alpha=n$ and $\beta>0$}\\\ s&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near infinity,}$

and

(3.28)
$J^{-1}(r)\sim\begin{cases}e^{-r^{\frac{n}{\beta+(n-1)-q}}}&\quad\text{if
$\alpha=0$}\\\
r^{\frac{n}{\alpha}}\big{(}\log\tfrac{1}{r}\big{)}^{\frac{q-(n-1)-\beta}{\alpha}}&\quad\text{if
$0<\alpha<n$}\\\
r\big{(}\log\tfrac{1}{r}\big{)}^{-\frac{\beta}{n}}\big{(}\log(\log\tfrac{1}{r})\big{)}^{\frac{q-(n-1)}{n}}&\quad\text{if
$\alpha=n$ and $\beta>0$ }\\\ r&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near zero.}$

Equation (3.26) follows from (3.21) and (3.28).
Let us notice that, if either $0<\alpha<n$ and $\beta\in\mathbb{R}$, or
$\alpha=n$ and $\beta>0$, then the function given by (3.14) does not tend to
$0$ as $r\to 0^{+}$. Indeed, the function $\varphi$ fulfills condition (2.6).
Moreover, by equation (3.27),

$\Theta_{B^{-1}}(1/r)\geq\Theta_{B^{-1}}^{\infty}(1/r)=r^{-\frac{1}{n}}\quad\text{for
$r>0$,}$

and, by equation (3.26),

$\Theta_{\psi}(s)\geq\Theta_{\psi}^{0}(s)=s^{n}\quad\text{for $s>0$.}$

Hence, by the monotonicity of the functions $\Theta_{B^{-1}}$ and
$\Theta_{\psi}$,

$\Xi_{\psi,B}(r)=r\,\Theta_{\psi}\left(\Theta_{B^{-1}}(1/r)^{+}\right)\geq
1\quad\text{for $r>0$.}$

###### Example 3.10

Assume that

$A(t)\,\,\text{is equivalent to}\,\,e^{t^{\gamma}}\qquad\text{ near
infinity,}$

for some $\gamma>0$, and let $\varphi$ be as in (3.21). Then equation (3.3)
holds with

(3.29)
$\psi(r)\sim\begin{cases}r^{\alpha}\big{(}\log\tfrac{1}{r}\big{)}^{\beta-\frac{\alpha}{\gamma}}&\quad\text{if
$0\leq\alpha<n$}\\\
r^{n}\big{(}\log\tfrac{1}{r}\big{)}^{\beta}\big{(}\log(\log\tfrac{1}{r})\big{)}^{-\frac{n}{\gamma}}&\quad\text{if
$\alpha=n$ and $\beta>0$}\\\ r^{n}&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near zero.}$

Indeed, since $i(A)=\infty$, condition (2.31) holds, whence

$B(t)\,\,\text{is equivalent to}\,\,e^{t^{\gamma}}\qquad\text{ near
infinity.}$

Hence,

$B^{-1}(t)\sim(\log t)^{\frac{1}{\gamma}}\qquad\text{ near infinity.}$

Thus,

$J(s)\sim\begin{cases}s(\log\tfrac{1}{s})^{\frac{1}{\gamma}}&\quad\text{if
$0\leq\alpha<n$}\\\
s\big{(}\log(\log\tfrac{1}{s})\big{)}^{\frac{1}{\gamma}}&\quad\text{if
$\alpha=n$ and $\beta>0$}\\\ s&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near infinity,}$

and

(3.30)
$J^{-1}(r)\sim\begin{cases}r(\log\tfrac{1}{r})^{-\frac{1}{\gamma}}&\quad\text{if
$0\leq\alpha<n$}\\\
r\big{(}\log(\log\tfrac{1}{r})\big{)}^{-\frac{1}{\gamma}}&\quad\text{if
$\alpha=n$ and $\beta>0$}\\\ r&\quad\text{if $\alpha=n$ and
$\beta=0$}\end{cases}\quad\text{near zero.}$

Coupling equation (3.21) with (3.30) yields (3.29).

###### Example 3.11

Here, we exhibit classes of Young functions $A$ and gauge functions $\varphi$
for which the gauge function $\psi$ appearing in property (3.3) is equivalent
to $\varphi$. Loosely speaking, this is the case when the function $A$ grows
essentially faster than the $n$-th power near infinity, and the function
$\varphi$ decays more slowly than any power near zero. These instances provide
us with a generalization of example (3.25). Clearly, they also show that
property (3.11) may actually fail if assumption (3.10) is not fulfilled.
Precisely, assume that $A$ is any Young function such that

$i_{\infty}(A)>n,$

and let $\varphi$ be any gauge function such that

(3.31) $\liminf_{r\to
0^{+}}\frac{\varphi(r^{\gamma})}{\varphi(r)}>0\quad\text{for every
$\gamma>0$.}$

Then property (3.3) holds with

(3.32) $\psi(r)\sim\varphi(r)\quad\text{near zero.}$

To verify this assertion, recall that the current assumption on $A$ ensures
that $B$ is equivalent to $A$ near infinity. Moreover, it also implies that
there exists $\epsilon>0$ such that $A(t)\geq t^{n+\epsilon}$ near infinity.
Altogether, one has that

(3.33) $B(t)\geq t^{n+\epsilon}\quad\text{near infinity.}$

Now, in order to establish equation (3.32) it suffices to show that

(3.34)
$\psi(r)\geq{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}\,\varphi(r)\quad\text{near
zero}$

for some positive constant $c$, the reverse inequality being always fulfilled
(up to a multiplicative constant). Owing to assumption (3.31), inequality
(3.34) will in turn follow if we prove that there exists $\gamma>0$ such that

(3.35) $J^{-1}(r)\geq r^{\gamma}\quad\text{near zero,}$

or, equivalently,

(3.36) $J(s)\leq s^{\frac{1}{\gamma}}\quad\text{near zero.}$

The last inequality reads

(3.37) $s\,B^{-1}\Big{(}\frac{\varphi(s)}{s^{n}}\Big{)}\leq
s^{\frac{1}{\gamma}}\quad\text{near zero,}$

namely

(3.38) $\varphi(s)\leq
s^{n}B\big{(}s^{\frac{1}{\gamma}-1}\big{)}\quad\text{near zero.}$

Choosing
$\gamma>{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1+\frac{n}{\epsilon}}$
and making use of equation (3.33) tell us that

(3.39) $s^{n}B\big{(}s^{\frac{1}{\gamma}-1}\big{)}\geq
s^{\frac{n+\epsilon}{\gamma}-\epsilon}\quad\text{near zero.}$

By our choice of $\gamma$, one has that
$\frac{n+\epsilon}{\gamma}-\epsilon<0$. Inequality (3.38) hence follows.

##  4 Technical lemmas

The results collected in this section concern properties of the functions $B$,
$J$ and $\psi$, defined by equations (2.27), (3.1) and (3.2), of the scaling
functions given by (3.7)– (3.9), and of the function $\Xi_{\psi,B}$ defined as
in (3.14).

The first result summarizes certain basic properties of the function $A$ which
follow from our assumptions, and tells us that they are inherited by
$\widetilde{A}$ and by $B$.

###### Lemma 4.1

Assume that $A$ is a Young function fulfilling conditions (2.26), (2.29) and
(2.30).
(i) We have that

(4.1) $\lim_{t\to
0^{+}}\frac{A(t)}{t}=0\qquad\text{and}\qquad\lim_{t\to\infty}\frac{A(t)}{t}=\infty.$

Moreover, the function

(4.2) $\frac{A(t)}{t}$ is increasing in $(0,\infty)$.

(ii) The Young conjugate $\widetilde{A}$ is such that
$0<\widetilde{A}(t)<\infty$ for $t>0$, and

(4.3) $\lim_{t\to
0^{+}}\frac{\widetilde{A}(t)}{t}=0\quad\text{and}\qquad\lim_{t\to\infty}\frac{\widetilde{A}(t)}{t}=\infty.$

(iii) The Young function $B$ given by (2.27) is such that $0<B(t)<\infty$ for
$t>0$, and

(4.4) $\lim_{t\to
0^{+}}\frac{B(t)}{t}=0\quad\text{and}\qquad\lim_{t\to\infty}\frac{B(t)}{t}=\infty.$

In particular, the function $B:[0,\infty)\to[0,\infty)$ is bijective.

Proof (i) The first limit in equation (4.1) follows from assumption (2.30) and
property (2.20). The second limit is a consequence of the same property and of
assumption (2.26). The strengthening of property (2.20) stated in equation
(4.2) holds owing to the first limit in (4.1).
(ii) The properties of the function $\widetilde{A}$ claimed here are
classically known to follow, via its very definition, from parallel properties
enjoyed by $A$.
(iii) The statement is trivial when $n=1$, thanks to Part (i). Assume that
$n\geq 2$. As observed in the proof of Part (ii), it suffices to prove the
statement with $B$ replaced by $\widetilde{B}$. Since

$\widetilde{B}(t)=t^{n^{\prime}}\displaystyle\int_{t}^{\infty}\frac{\widetilde{A}(s)}{s^{1+n^{\prime}}}\,ds\quad\text{for
$t>0$,}$

one clearly has that $0<\widetilde{B}(t)<\infty$ for $t>0$. Moreover, thanks
to equation (4.3),

$\lim_{t\to 0^{+}}\frac{\widetilde{B}(t)}{t}=\lim_{t\to
0^{+}}t^{n^{\prime}-1}\displaystyle\int_{t}^{\infty}\frac{\widetilde{A}(s)}{s^{1+n^{\prime}}}\,ds=0,$

and

$\lim_{t\to\infty}\frac{\widetilde{B}(t)}{t}=\lim_{t\to\infty}t^{n^{\prime}-1}\displaystyle\int_{t}^{\infty}\frac{\widetilde{A}(s)}{s^{1+n^{\prime}}}\,ds=\infty.$

The following result ensures that the inverse $J^{-1}$ of $J$ is well defined.

###### Lemma 4.2

Let $n\in\mathbb{N}$, let $A$ be a Young function fulfilling conditions
(2.26), (2.29), (2.30), and let $\varphi$ be a gauge function. Then the
function $J\colon(0,\infty)\to(0,\infty)$, given by (3.1), is increasing and
such that

(4.5) $\lim_{t\to
0^{+}}J(t)=0\qquad\text{and}\qquad\lim_{t\to\infty}J(t)=\infty.$

In particular, $J$ is bijective.

Proof To begin with, by Lemma 4.1 the Young function $B$ is bijective, and
hence its inverse $B^{-1}:[0,\infty)\to[0,\infty)$ is classically well
defined. Suppose first that $n\geq 2$. One has that the function

(4.6) $\frac{B(t)}{t^{n}}\quad\text{is non-decreasing}$

Indeed, property (4.6) is equivalent to the fact that the function

(4.7) $\frac{\widetilde{B}(t)}{t^{n^{\prime}}}\quad\text{is non-increasing}$

and the latter holds, since

(4.8)
$\frac{\widetilde{B}(t)}{t^{n^{\prime}}}=\int_{t}^{\infty}\frac{\widetilde{A}(s)}{s^{1+n^{\prime}}}\,ds\qquad\text{for
$t>0$.}$

As a consequence of property (4.6), the function

(4.9) $tB^{-1}(t^{-n})\quad\text{is non-decreasing.}$

Hence, the function

(4.10)
$J(t)=tB^{-1}\Big{(}\frac{\varphi(t)}{t^{n}}\Big{)}=\frac{t}{\varphi(t)^{\frac{1}{n}}}B^{-1}\Big{(}\frac{\varphi(t)}{t^{n}}\Big{)}\varphi(t)^{\frac{1}{n}}\quad\text{is
increasing,}$

inasmuch as the function $\varphi$ is increasing and satisfies condition
(2.5).
Next, owing to equations (2.4) and (4.10), and to the fact that the function
$\varphi$ is increasing, we have that

(4.11) $\lim_{t\to 0^{+}}J(t)\leq\lim_{t\to
0^{+}}tB^{-1}\Big{(}\frac{1}{t^{n}}\Big{)}.$

Moreover, a property of Young functions tells us that

(4.12) $t\leq B^{-1}(t)\widetilde{B}^{-1}(t)\leq 2t\qquad\text{for $t\geq
0$.}$

Thus,

$\lim_{t\to
0^{+}}J(t)=0\quad\text{if}\quad\lim_{t\to\infty}\frac{B^{-1}(t)}{t^{\frac{1}{n}}}=0.$

Now, notice that

(4.13)
$\displaystyle\lim_{t\to\infty}\frac{B^{-1}(t)}{t^{\frac{1}{n}}}=0\quad\text{if
and only
if}\quad\lim_{t\to\infty}\frac{t^{\frac{1}{n^{\prime}}}}{\widetilde{B}^{-1}(t)}=0\quad\text{
if}\quad\lim_{t\to\infty}\frac{\widetilde{B}(t)}{t^{n^{\prime}}}=0,$

and the last limit holds, owing to equation (4.8). The first limit in (4.5) is
thus established.
As for the second one, from equations (2.4) and (4.10) again, and the
monotonicity of the function $\varphi$, one has that

(4.14)
$\lim_{t\to\infty}J(t)\geq{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c}\lim_{t\to\infty}tB^{-1}\Big{(}\frac{1}{t^{n}}\Big{)},$

where $c=\min\bigl{\\{}1,\lim\limits_{t\to\infty}\varphi(t)\bigr{\\}}$.
Therefore, by an analogous chain as above,

(4.15) $\lim_{t\to\infty}J(t)=\infty\quad\text{if}\quad\lim_{t\to
0^{+}}\frac{B^{-1}(t)}{t^{\frac{1}{n}}}=\infty\quad\text{if and only
if}\quad\lim_{t\to
0^{+}}\frac{t^{\frac{1}{n^{\prime}}}}{\widetilde{B}^{-1}(t)}=\infty\quad\text{if}\quad\lim_{t\to
0^{+}}\frac{\widetilde{B}(t)}{t^{n^{\prime}}}=\infty.$

By [Ci04, Lemma 4.4], condition (2.30) is equivalent to

(4.16) $\int_{0}\frac{\widetilde{A}(t)}{t^{1+n^{\prime}}}\,dt=\infty.$

Hence, the last limit in chain (4.15) holds. This proves the second limit in
(4.5).
These pieces of information conclude the proof, since the function $J$ is
continuous, inasmuch as the function $B^{-1}$ is continuous in $(0,\infty)$.
Assume next that $n=1$, and hence $B=A$. Since $A$ is a Young function,
equation (4.9) still holds. Hence, we conclude as above that $J$ is
increasing. The limits

$\lim_{t\to 0^{+}}J(t)=0\quad\text{and}\quad\lim_{t\to\infty}J(t)=\infty$

follow from equations (4.11) and (4.14), with $B=A$, via property (4.1).

The next lemma substantiates the fact that $\psi$, defined as in (3.2), is
actually a gauge function.

###### Lemma 4.3

Let $n$, $A$ and $\varphi$ be as in Lemma 4.2. Then the function $\psi$
defined by (3.2) is continuous and increasing, and the function

(4.17) $\frac{\psi(r)}{r^{n}}$ is non-increasing.

In particular,

(4.18) $\psi(kr)\leq\max\\{1,k^{n}\\}\psi(r)\qquad\text{for $k>0$ and $r\geq
0$.}$

Proof The fact that $\psi$ is continuous and increasing is a consequence of
the fact that both $\varphi$ and $J$ are continuous and increasing. As for
property (4.17), note that it is equivalent to the fact that the function

(4.19) $\frac{\varphi(r)}{J(r)^{n}}$ is non-increasing.

On the other hand,

(4.20)
$\frac{\varphi(r)}{J(r)^{n}}=\frac{\frac{\varphi(r)}{r^{n}}}{B^{-1}\big{(}\frac{\varphi(r)}{r^{n}}\big{)}}\qquad\text{for
$r>0$.}$

By property (2.5) the function $\frac{\varphi(r)}{r^{n}}$ is non-increasing,
whereas the function $\frac{t}{B^{-1}(t)}$ is non-decreasing, inasmuch as $B$
is a Young function (in fact, it is increasing, as a consequence of Lemma 4.1,
Part (iii)). Hence, property (4.19) follows.
Inequality (4.18) is a consequence of inequality (2.7), which, thanks to
property (4.17), can be applied with $\varphi$ replaced by $\psi$.

The function $\psi$ owes its definition to the inequality established in the
following lemma.

###### Lemma 4.4

Let $n$, $A$ and $\varphi$ be as in Lemma 4.2, and let $B$ and $\psi$ be the
functions defined by (2.27) and (3.2), respectively. Then

(4.21) $\psi(st)\leq\varphi(t)+t^{n}B(s)\qquad\text{for $s,t>0$.}$

Proof Inequality (4.21) trivially holds if either $s=0$ or $t=0$. On the other
hand, if both $s>0$ and $t>0$, it is equivalent to

(4.22) $\psi(s)\leq\varphi(t)+t^{n}B(s/t).$

The function $\varphi$ is increasing, by our assumption. On the other hand, by
property (4.6), for each $s>0$, the function $t^{n}B(s/t)$ is non-increasing
in $t$. Now observe that

(4.23) $\varphi(J^{-1}(s))=J^{-1}(s)^{n}B(s/J^{-1}(s))\quad\text{for $s>0$.}$

The monotonicity properties mentioned above imply that

$\varphi(t)+t^{n}B(s/t)\geq\varphi(J^{-1}(s))\qquad\text{if $t\geq
J^{-1}(s)$,}$

and

$\varphi(t)+t^{n}B(s/t)\geq J^{-1}(s)^{n}B(s/J^{-1}(s))\qquad\text{if $0<t\leq
J^{-1}(s)$.}$

Hence, by equation (4.23) and the definition of the function $\psi$,

$\varphi(t)+t^{n}B(s/t)\geq\varphi(J^{-1}(s))=\psi(s)\qquad\text{for $t>0$.}$

Inequality (4.22) is thus established.

Our analysis of the scaling functions begins with the next result, which, in
particular, deals with a submultiplicativity property.

###### Lemma 4.5

Let $h\colon(0,\infty)\to(0,\infty)$ be a non-decreasing continuous function
such that the function

(4.24) $\frac{h(t)}{t^{\gamma}}$ is non-increasing for some $\gamma>0$.

Denote by $\Theta$ either of the functions $\Theta_{h}^{0}$,
$\Theta_{h}^{\infty}$ and $\Theta_{h}$ associated with $h$ as in (3.7), (3.8)
and (3.9), respectively, and by $\Theta_{*}$ the function defined analogously,
with $``\limsup"$ or $``\sup"$ replaced with $``\liminf"$ or $``\inf"$,
respectively. Then

(4.25)
$\min\\{1,r^{\gamma}\\}\leq\Theta(r)\leq\max\\{1,r^{\gamma}\\}\quad\text{for
$r>0$,}$

and

(4.26) $\Theta(rs)\leq\Theta(r)\Theta(s)\quad\text{for $r,s>0$.}$

Furthermore,

(4.27)
$\Theta_{*}(r)=\frac{1}{\Theta\bigl{(}\tfrac{1}{r}\bigr{)}}\quad\text{for
$r>0$.}$

Proof We consider the case when $\Theta=\Theta_{h}^{0}$, the proof in the
remaining cases being analogous. The monotonicity of the function $h$ and
property (4.24) ensure that $r^{\gamma}\leq\Theta_{h}^{0}(r)\leq 1$ for
$r\in(0,1]$. Indeed, if $r\in(0,1]$, then the upper bound follows since $h$ is
non-decreasing, whereas the lower bound is a consequence of the fact that

$\frac{h(rt)}{h(t)}=\frac{h(rt)}{(rt)^{\gamma}}\,\frac{t^{\gamma}}{h(t)}r^{\gamma}\geq
r^{\gamma}$

for $t>0$, whence $\Theta_{h}^{0}(r)\geq r^{\gamma}$. The proof of the
inequalities $1\leq\Theta_{h}^{0}(r)\leq r^{\gamma}$ for $r>1$ is analogous.
Equation (4.25) is therefore established.
As for inequality (4.26), fix $r>0$ and $s>0$, and choose a sequence
$\\{t_{k}\\}$ such that $t_{k}\to 0^{+}$ and
$h(rst_{k})/h(t_{k})\to\Theta_{h}^{0}(rs)$. Now

$\Theta_{h}^{0}(rs)=\lim_{k\to\infty}\frac{h(rst_{k})}{h(st_{k})}\,\frac{h(st_{k})}{h(t_{k})}\leq\Theta_{h}^{0}(r)\Theta_{h}^{0}(s),$

namely (4.26).
Finally, equation (4.27) follows from the chain:

$\frac{1}{\Theta_{h}^{0}\left(\tfrac{1}{r}\right)}=\left(\limsup_{t\to
0^{+}}\frac{h(t/r)}{h(t)}\right)^{-1}=\liminf_{t\to
0^{+}}\frac{h(t)}{h(t/r)}=\liminf_{t\to
0^{+}}\frac{h(rt)}{h(t)}=\big{(}\Theta^{0}_{h}\big{)}_{*}(r)\quad\text{for
$r>0$.}$

###### Corollary 4.6

Let $h\colon(0,\infty)\to(0,\infty)$ be a non-decreasing continuous function
and let $\Theta$ be any of the functions associated with $h$ as in
(3.7)–(3.9). Then

(4.28) either $\lim_{r\to 0^{+}}\Theta(r)=0$, or $\Theta(r)=1$ for every
$r\in(0,1]$.

Proof One has that $\Theta(r)\in(0,1]$ for $r\in(0,1]$. If there exists
$r_{0}\in(0,1)$ such that $\Theta(r_{0})<1$, then, by property (4.26),
$\Theta(r_{0}^{k})\leq\Theta(r_{0})^{k}$ for every $k\in\mathbb{N}$. Hence,
$\lim_{r\to 0^{+}}\Theta(r)=0$.

The next two lemmas are concerned with properties of a one-parameter family of
functions $J_{r}$ obtained by inserting a scaling factor $r$ in the definition
of function $J$. Specifically, the function
$J_{r}\colon[0,\infty)\to[0,\infty)$ is defined, for $r>0$, as

(4.29)
$J_{r}(s)=\begin{cases}sB^{-1}\left(\displaystyle\frac{r\varphi(s)}{s^{n}}\right)&\quad\text{for
$s>0$}\\\ 0&\quad\text{for $s=0$.}\end{cases}$

###### Lemma 4.7

Let $n\in\mathbb{N}$, let $A$ be a Young function fulfilling conditions
(2.26), (2.29) and (2.30), let $B$ be the function given by (2.27), and let
$\varphi$ be a gauge function. Then, for each fixed $r$, the function $J_{r}$
given by (4.29) is increasing and bijective, and for each fixed $s>0$ the
function

$(0,\infty)\ni r\mapsto J_{r}^{-1}(s)\quad\text{is non-increasing.}$

Moreover,

(4.30)
$\varphi\bigl{(}J_{r}^{-1}(st)\bigr{)}\leq\varphi(t)+\frac{t^{n}}{r}B(s)\quad\text{for
$s,t>0$.}$

Proof. The function $J_{r}$ is defined as the function $J$ given by (3.1),
with the gauge function $\varphi$ replaced by $r\varphi$. The asserted
properties of the function $J_{r}$ follow from Lemmas 4.2 and 4.4. In
particular, inequality (4.30) follows on replacing the function $\varphi$ with
$r\varphi$ in equation (4.21), and dividing through the resultant inequality
by $r$.

###### Lemma 4.8

Let $n\in\mathbb{N}$, let $A$ be a Young function fulfilling conditions
(2.26), (2.29), (2.30), and let $B$ be the function given by (2.27). Assume
that $\varphi$ is a gauge function satisfying property (2.6), and let $\psi$
be the gauge function defined by (3.2). Let $J_{r}$ be the function defined by
(4.29). Then

(4.31) $\liminf_{t\to
0^{+}}\frac{\varphi\bigl{(}J_{r}^{-1}(t)\bigr{)}}{\psi(t)}\geq\frac{1}{\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}}\quad\text{for
$r>0$,}$

and

(4.32)
$\inf_{t>0}\frac{\varphi\bigl{(}J_{r}^{-1}(t)\bigr{)}}{\psi(t)}\geq\frac{1}{\Theta_{\psi}\bigl{(}\Theta_{B^{-1}}(r)^{+}\bigr{)}}\quad\text{for
$r>0$.}$

Proof. We limit ourselves to proving equation (4.31), the proof of (4.32)
being analogous. Fix $r>0$. By the very definitions of the functions $\psi$,
$J$ and $J_{r}$, and the properties of the latter established in Lemma 4.8,
one has that

$\liminf_{t\to
0^{+}}\frac{\varphi\bigl{(}J_{r}^{-1}(t)\bigr{)}}{\psi(t)}=\liminf_{s\to
0^{+}}\frac{\varphi(s)}{\psi(J_{r}(s))}=\left(\limsup_{s\to
0^{+}}\frac{\psi\left(sB^{-1}\bigl{(}r\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}{\psi\left(sB^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}\right)^{-1}.$

Therefore, inequality (4.31) will follow if we show that

(4.33) $\limsup_{s\to
0^{z}{+}}\frac{\psi\left(sB^{-1}\bigl{(}r\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}{\psi\left(sB^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}\leq\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\infty}(r)^{+}\bigr{)}.$

To verify inequality (4.33), observe that, owing to assumption (2.6), for
every $\varepsilon>0$ there exists $\delta>0$ such that

$B^{-1}\bigl{(}r\tfrac{\varphi(s)}{s^{n}}\bigr{)}<\Theta_{B^{-1}}^{\infty}(r)(1+\varepsilon)B^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}\quad\text{for
$s\in(0,\delta)$.}$

Hence, inasmuch as $\psi$ is an increasing function,

$\frac{\psi\left(sB^{-1}\bigl{(}r\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}{\psi\left(sB^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}\leq\frac{\psi\left(\Theta_{B^{-1}}^{\infty}(r)(1+\varepsilon)sB^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}{\psi\left(sB^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}\quad\text{for
$s\in(0,\delta)$.}$

Since, by equation (4.5), $\lim_{s\to
0^{+}}sB^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}=0$, we deduce from this
inequality that

$\limsup_{s\to
0^{+}}\frac{\psi\left(sB^{-1}\bigl{(}r\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}{\psi\left(sB^{-1}\bigl{(}\tfrac{\varphi(s)}{s^{n}}\bigr{)}\right)}\leq\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)(1+\varepsilon)\bigr{)}.$

Inequality (4.33) hence follows, owing to the arbitrariness of $\varepsilon$.

###### Remark 4.9

Under the additional condition that $i_{\infty}(A)>n$, the Young function $B$
defined by (2.27) is equivalent to $A$ at infinity. Namely,

(4.34) $B\bigl{(}c_{1}t\bigr{)}\leq A(t)\leq B\bigl{(}c_{2}t\bigr{)}$

for every $t\geq t_{0}$, where $c_{1}$, $c_{2}$, $t_{0}$ are positive
constants. As a consequence,

(4.35) $\Theta_{A^{-1}}^{\infty}(0^{+})=0\,\mbox{ if and only if
}\,\Theta_{B^{-1}}^{\infty}(0^{+})=0.$

Indeed, inequalities (4.34) imply that

(4.36) $\frac{B^{-1}(s)}{c_{2}}\leq A^{-1}(s)\leq\frac{B^{-1}(s)}{c_{1}}$

for large $s$. Hence,
$\frac{c_{1}}{c_{2}}\Theta_{B^{-1}}^{\infty}(r)\leq\Theta_{A^{-1}}^{\infty}(r)\leq\frac{c_{2}}{c_{1}}\Theta_{B^{-1}}^{\infty}(r)$
for $r>0$, whence assertion (4.35) follows.
Analogously, if $i(A)>n$, then

(4.37) $\Theta_{A^{-1}}(0^{+})=0\,\mbox{ if and only if
}\,\Theta_{B^{-1}}(0^{+})=0.$

We conclude with a proof of the monotonicity of the function $\Xi_{\psi,B}$.

###### Lemma 4.10

Let $n\in\mathbb{N}$, let $A$ be a Young function fulfilling conditions
(2.26), (2.29),(2.30), and let $B$ be the function given by (2.27). Assume
that $\varphi$ is a gauge function and let $\psi$ be the function given by
(3.2). Then the function $\Xi_{\psi,B}$, defined by (3.14), is non-decreasing.

Proof. It clearly suffices to show that, for fixed $s,t>0$, the function

(4.38) $(0,\infty)\ni
r\mapsto\frac{\psi\left(\frac{B^{-1}\left(\frac{t}{r}\right)}{B^{-1}(t)}s\right)}{\psi(s)}r\quad\text{is
non-decreasing.}$

On the other hand, property (4.38) holds if the function

$(0,\infty)\ni
r\mapsto\psi\left(\frac{B^{-1}(rt)}{B^{-1}(t)}s\right)\,\frac{1}{r}\quad\text{is
non-increasing.}$

Since

$\psi\left(\frac{B^{-1}(rt)}{B^{-1}(t)}s\right)\,\frac{1}{r}=\frac{\psi\left(B^{-1}(rt)\displaystyle\frac{s}{B^{-1}(t)}\right)}{\left(B^{-1}(rt)\displaystyle\frac{s}{B^{-1}(t)}\right)^{n}}\cdot\frac{\left(B^{-1}(rt)\displaystyle\frac{s}{B^{-1}(t)}\right)^{n}}{r}\quad\text{for
$r>0$,}$

this property is a consequence of equations (4.17) and (4.9).

##  5 Proofs of the main results

Having the necessary preliminary material at disposal, we are now in a
position to accomplish the proofs of our main results.

Proof of Theorem 3.1. Let us begin with Part (i) and let us temporarily
assume, in addition, that the closure $\overline{E}$ of $E$ is compact and
$\overline{E}\subset\Omega$. Hence, there exists $\varrho_{0}>0$ such that the
metric neighbourhood of $E$ of radius $\varrho_{0}$, defined as

(5.1) $B_{\varrho_{0}}(E)=\\{x\in\mathbb{R}^{n}:{\rm
dist}(x,E)<\varrho_{0}\\},$

is compactly contained in $\Omega$. Owing to our assumptions on $A$, the
function $u$ is uniformly continuous in $B_{\varrho_{0}}(E)$. Let us denote by
$\omega$ its modulus of continuity.
Let $\kappa$ be the constant appearing in inequality (2.28) and let
$\varphi_{\kappa}$ be the function defined as in (2.12). By virtue of
equations (2.11) and (2.13), there exists a constant $c=c(n,\kappa)$ such that
$\Lambda^{\varphi_{\kappa}}(E)\leq c\mathscr{H}^{\varphi}(E)$. Now, suppose
that $\mathscr{H}^{\varphi}(E)<\delta$, where the number $\delta>0$ will be
specified in the course of the proof. Hence
$\Lambda^{\varphi_{\kappa}}(E)<c\delta$ and for any $\sigma\in(0,1)$ we may
select a family of non-overlapping dyadic cubes $\\{Q_{i}\\}$ such that
$E\subset\bigcup_{i}Q_{i}$,

(5.2) $\sup_{i}d(Q_{i})<\sigma,$

and

(5.3) $\sum_{i}\varphi(\kappa d(Q_{i}))<c\delta.$

We may clearly also assume that $Q_{i}\subset B_{\varrho_{0}}(E)$. Hence,

(5.4) $\sup_{i}d(u(Q_{i}))<\omega(\sigma).$

From inequalities (4.21) and (2.28) we infer that, if $\lambda>\|\nabla
u\|_{L^{A}(\Omega)}$, then

(5.5)
$\displaystyle\psi\bigg{(}\frac{d(u(Q_{i}))}{\lambda}\bigg{)}\leq\varphi(\kappa
d(Q_{i}))+\kappa^{n}d(Q_{i})^{n}B\bigg{(}\frac{d(u(Q_{i}))}{\kappa\lambda
d(Q_{i})}\bigg{)}\leq\varphi(\kappa
d(Q_{i}))+\kappa^{n+1}\int_{Q_{i}}A\bigg{(}\frac{|\nabla
u|}{\lambda}\bigg{)}\,dx$

for every $i$. Hence, by Lemma 4.3,

(5.6)
$\displaystyle\min\\{1,\lambda^{-n}\\}\sum_{i}\psi\big{(}d(u(Q_{i}))\big{)}$
$\displaystyle\leq\sum_{i}\psi\bigg{(}\frac{d(u(Q_{i}))}{\lambda}\bigg{)}$
$\displaystyle\leq\sum_{i}\varphi(\kappa
d(Q_{i}))+\kappa^{n+1}\int_{\bigcup_{i}Q_{i}}A\bigg{(}\frac{|\nabla
u|}{\lambda}\bigg{)}\,dx.$

Owing to equations (2.5), (5.2) and (5.3),

(5.7) $\displaystyle{\mathscr{L}}^{n}\Big{(}\bigcup_{i}Q_{i}\Big{)}$
$\displaystyle=n^{-\frac{n}{2}}\sum_{i}d(Q_{i})^{n}=n^{-\frac{n}{2}}\kappa^{-n}\sum_{i}\varphi\big{(}\kappa
d(Q_{i})\big{)}\frac{(\kappa d(Q_{i}))^{n}}{\varphi\big{(}\kappa
d(Q_{i})\big{)}}$
$\displaystyle\leq\frac{c\delta}{n^{\frac{n}{2}}\kappa^{n}}\frac{\bigl{(}\kappa\sigma\bigr{)}^{n}}{\varphi\bigl{(}\kappa\sigma\bigr{)}}\leq\frac{c\delta}{n^{\frac{n}{2}}\varphi(\kappa)}.$

Hence, there exists
$\delta^{\prime}=\delta^{\prime}(n,m,A,\varphi,u,\varepsilon)>0$ such that

(5.8) $\kappa^{n+1}\int_{\bigcup_{i}Q_{i}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,dx<\frac{\varepsilon}{2\max\\{1,\lambda^{n}\\}}$

if $\delta\leq\delta^{\prime}$. Combining inequalities (5.6), (5.3) and (5.8)
yields

(5.9)
$\sum_{i}\psi\big{(}d(u(Q_{i}))\big{)}<c\delta\max\\{1,\lambda^{n}\\}+\frac{\varepsilon}{2}$

provided $\delta\leq\delta^{\prime}$. Therefore, if
$\delta<\min\big{\\{}\delta^{\prime},\frac{\varepsilon}{2c(1+\lambda^{n})}\big{\\}}$,
then, owing to inequality (5.4), one has that

$\mathscr{H}_{\omega(\sigma)}^{\psi}\bigl{(}u(E)\bigr{)}\leq\sum_{i}\psi\bigl{(}d(u(Q_{i}))\bigr{)}<\varepsilon$

for all $\sigma\in(0,1)$. Hence, property (3.3) follows, since
$\lim_{\sigma\to 0^{+}}\omega(\sigma)=0$.
It remains to show that this property continues to hold for an arbitrary set
$E\subset\Omega$. For any set $E$ of this kind, define the increasing sequence
of sets $\\{E_{j}\\}$ as

(5.10) $E_{j}=\bigl{\\{}x\in E:\,|x|<j\mbox{ and
}\mathrm{dist}(x,\partial\Omega)>\tfrac{1}{j}\bigr{\\}}$

for $j\in\mathbb{N}$. Clearly each set $E_{j}$ has a compact closure in
$\Omega$, and $E=\bigcup_{j}E_{j}$. Given $\varepsilon>0$, let $\delta>0$ be
such that property (3.3) holds for every set whose closure is compact and
contained in $\Omega$. Assume that $\mathscr{H}^{\varphi}(E)<\delta$. Hence,
$\mathscr{H}^{\varphi}(E_{j})<\delta$ as well, and, by property (3.3) applied
with $E$ replaced by $E_{j}$, we have that
$\mathscr{H}^{\psi}\big{(}u(E_{j})\big{)}<\varepsilon$. Inasmuch as
$u(E)=\bigcup_{j}u(E_{j})$, from property (2.14) we can conclude that
$\mathscr{H}^{\psi}\big{(}u(E)\big{)}\leq\lim_{j\to\infty}\mathscr{H}^{\psi}\big{(}u(E_{j})\big{)}\leq\varepsilon$.
The proof of (3.3) is complete.

Let us next consider Part (ii). Assume that
$\mathscr{H}^{\varphi}_{\infty}(E)<\delta$, where the number $\delta>0$ will
be fixed later. Hence, there exists $\sigma>0$ such that
$\mathscr{H}^{\varphi}_{\sigma}(E)<\delta$ as well. Furthermore, owing to
property (2.10), we have that $\Lambda_{2\sigma}^{\varphi}(E)\leq
c_{n}\mathscr{H}^{\varphi}_{\sigma}(E)$. Thus, by equation (2.13), there
exists a constant $c=c(n,\kappa)$ such that
$\Lambda_{2\sigma}^{\varphi_{\kappa}}(E)<c\delta$, and hence we may select a
family $\\{Q_{i}\\}$ of non-overlapping dyadic cubes such that
$E\subset\bigcup_{i}Q_{i}$,

(5.11) $\sup_{i}d(Q_{i})<2\sigma,$

and

(5.12) $\sum_{i}\varphi\bigl{(}\kappa d(Q_{i})\bigr{)}<c\delta.$

Inequalities (5.5) and (5.6) continue to hold also in this case, whence

(5.13)
$\displaystyle\sum_{i}\psi\big{(}d(u(Q_{i}))\big{)}\leq\max\\{1,\lambda^{n}\\}\bigg{(}\sum_{i}\varphi(\kappa
d(Q_{i}))+\kappa^{n+1}\int_{\bigcup_{i}Q_{i}}A\bigg{(}\frac{|\nabla
u|}{\lambda}\bigg{)}\,dx\bigg{)},$

provided that $\lambda>\|\nabla u\|_{L^{A}(\mathbb{R}^{n})}$. Also, the same
chain as in equation (5.7) tells us that

(5.14)
$\displaystyle{\mathscr{L}}^{n}\Big{(}\bigcup_{i}Q_{i}\Big{)}\leq\frac{c\delta}{n^{\frac{n}{2}}}\frac{\sigma^{n}}{\varphi\bigl{(}\kappa\sigma\bigr{)}}\leq\frac{c\delta}{n^{\frac{n}{2}}\varphi\bigl{(}\kappa\bigr{)}}.$

Consequently, there exists
$\delta^{\prime}=\delta^{\prime}(n,m,A,\varphi,u,\varepsilon)>0$ such that

(5.15) $\kappa^{n+1}\int_{\bigcup_{i}Q_{i}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,dx<\frac{\varepsilon}{2\max\\{1,\lambda^{n}\\}}$

if $\delta<\delta^{\prime}$. On making use inequalities (5.13) and (5.15) one
can infer that

$\mathscr{H}_{\infty}^{\psi}\bigl{(}u(E)\bigr{)}\leq\sum_{i}\psi\bigl{(}d(u(Q_{i}))\bigr{)}<\varepsilon.$

Property (3.4) is thus established.

Proof of Corollary 3.3. Let $E$ be any subset of $\Omega$ such that
$\mathscr{H}^{\varphi}(E)=0$. Consider any increasing sequence
$\\{\Omega_{j}\\}$ of open subsets of $\Omega$, such that
$\overline{\Omega_{j}}\subset\Omega$ for $j\in\mathbb{N}$, and
$\Omega=\bigcup_{j}\Omega_{j}$. Define, for each $j\in\mathbb{N}$, the set
$E_{j}=E\cap\Omega_{j}$. Obviously, $\mathscr{H}^{\varphi}(E_{j})=0$ for every
$j\in\mathbb{N}$. Since $u\in{\rm V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$,
one has that $u\in{\rm V}^{1,A}(\Omega_{j},\mathbb{R}^{m})$ for every
$j\in\mathbb{N}$. An application of Theorem 3.1, Part (i), with $\Omega$ and
$E$ replaced by $\Omega_{j}$ and $E_{j}$, respectively, tells us that
$\mathscr{H}^{\psi}\bigl{(}u(E_{j})\bigr{)}=0$ for every $j\in\mathbb{N}$.
Inasmuch as $\\{u(E_{j})\\}$ is an increasing sequence of sets, and
$u(E)=\bigcup_{j}u(E_{j})$, we deduce through property (2.14) that
$\mathscr{H}^{\psi}(u(E))=0$.

Proof of Theorem 3.4 The main step consists in proving that, if $u\in{\rm
V}^{1,A}(\Omega,\mathbb{R}^{m})$, then

(5.16) $\mathscr{H}^{\psi}\bigl{(}u(E)\bigr{)}\leq c_{n}\lim_{r\to
0^{+}}\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)\bigr{)}\Theta_{\psi}^{0}\bigl{(}\kappa\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{+}\bigr{)}\mathscr{H}^{\varphi}(E),$

for every set $E\subset\Omega$. This will evidently establish Part (ii).
In order to prove inequality (5.16), we may clearly assume that
$\mathscr{H}^{\varphi}(E)<\infty$. We also assume in addition, for the time
being, that $\overline{E}$ is compact and $\overline{E}\subset\Omega$, and we
choose $\varrho_{0}$ such that the set $B_{\varrho_{0}}(E)$, defined as in
(5.1), is compactly contained in $\Omega$. Our assumptions on $A$ ensure that
the function $u$ is uniformly continuous in $B_{\varrho_{0}}(E)$, with a
modulus of continuity $\omega$, say.
Now, fix $\varepsilon>0$. In view of (2.11), for each
$\delta\in(0,\varrho_{0})$ we can select a family $\\{Q_{i}\\}$ of non-
overlapping dyadic cubes such that $E\subset\bigcup_{i}Q_{i}$,
$d\bigl{(}Q_{i}\bigr{)}<\delta$ and

(5.17)
$\sum_{i}\varphi\bigl{(}d(Q_{i})\bigr{)}<c_{n}\mathscr{H}^{\varphi}\bigl{(}E\bigr{)}+\varepsilon.$

We may also assume, without loss of generality, that $Q_{i}\subset
B_{\varrho_{0}}(E)$. Next, fix any $r\in(0,1)$. Thanks to inequality (4.30),
one can deduce that

(5.18)
$\varphi\left(J_{r}^{-1}\left(\frac{d\bigl{(}u(Q_{i})\bigr{)}}{\kappa\lambda}\right)\right)\leq\varphi\bigl{(}d(Q_{i})\bigr{)}+\frac{d(Q_{i})^{n}}{r}B\left(\frac{d\bigl{(}u(Q_{i})\bigr{)}}{\kappa\lambda
d(Q_{i})}\right)$

for every $i$. Since $d\bigl{(}u(Q_{i})\bigr{)}\leq\omega(\delta)$, by
inequality (4.31) we may choose $\delta<\varrho_{0}$ such that, if
$d\bigl{(}u(Q_{i})\bigr{)}>0$, then

(5.19)
$\varphi\left(J_{r}^{-1}\left(\frac{d\bigl{(}u(Q_{i})\bigr{)}}{\kappa\lambda}\right)\right)>\frac{\psi\left(\frac{d(u(Q_{i}))}{\kappa\lambda}\right)}{\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}+\varepsilon}.$

On the other hand, all cubes for which $d\bigl{(}u(Q_{i})\bigr{)}=0$ can be
disregarded in estimating $\mathscr{H}^{\psi}(u(E))$, since $\psi(0)=0$. On
decreasing $\delta$, if necessary, we infer from inequality (5.19) and the
definition of the function $\Theta_{\psi}^{0}$ that

(5.20)
$\varphi\left(J_{r}^{-1}\left(\frac{d\bigl{(}u(Q_{i})\bigr{)}}{\kappa\lambda}\right)\right)>\frac{\psi\bigl{(}d\bigl{(}u(Q_{i})\bigr{)}\bigr{)}}{\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}+\varepsilon}\,\frac{1}{\Theta_{\psi}^{0}(\kappa\lambda)+\varepsilon}$

for every index $i$ such that $d\bigl{(}u(Q_{i})\bigr{)}>0$. Combining
inequalities (5.18) and (5.20) with inequality (2.28) yields

(5.21)
$\psi\bigl{(}d\bigl{(}u(Q_{i})\bigr{)}\bigr{)}\leq\big{(}\Theta_{\psi}^{0}(\kappa\lambda)+\varepsilon\big{)}\biggl{(}\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}+\varepsilon\biggr{)}\left(\varphi\bigl{(}d(Q_{i})\bigr{)}+\frac{\kappa}{r}\int_{Q_{i}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,\mathrm{d}x\right)$

for every index $i$ such that $d\bigl{(}u(Q_{i})\bigr{)}>0$. Summing this
inequality over all these indices $i$ and recalling inequality (5.17) tell us
that

(5.22)
$\mathscr{H}^{\psi}_{\omega(\delta)}\bigl{(}u(E)\bigr{)}\leq\big{(}\Theta_{\psi}^{0}(\kappa\lambda)+\varepsilon\big{)}\biggl{(}\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}+\varepsilon\biggr{)}\left(c_{n}\mathscr{H}^{\varphi}(E)+\varepsilon+\frac{\kappa}{r}\int_{\bigcup
Q_{i}}\\!A\left(\frac{|\nabla u|}{\lambda}\right)\,\mathrm{d}x\right).$

We claim that the integral in this inequality tends to $0$ as $\delta\to
0^{+}$. Indeed,

$\displaystyle{\mathscr{L}}^{n}\Big{(}\bigcup_{i}Q_{i}\Big{)}$
$\displaystyle=$ $\displaystyle
n^{-\frac{n}{2}}\sum_{i}d(Q_{i})^{n}=n^{-\frac{n}{2}}\sum_{i}\varphi\bigl{(}d(Q_{i})\bigr{)}\frac{d(Q_{i})^{n}}{\varphi\bigl{(}d(Q_{i})\bigr{)}}$
$\displaystyle\leq$ $\displaystyle
n^{-\frac{n}{2}}\sum_{i}\varphi\bigl{(}d(Q_{i})\bigr{)}\frac{\delta^{n}}{\varphi(\delta)}\leq
n^{-\frac{n}{2}}\bigl{(}c_{n}\mathscr{H}^{\varphi}(E)+\varepsilon\bigr{)}\frac{\delta^{n}}{\varphi(\delta)},$

where the first inequality holds thanks to property (2.5). Our claim hence
follows, owing to assumption (2.6). Consequently, from inequality (5.22) we
obtain that

$\mathscr{H}^{\psi}\bigl{(}u(E)\bigr{)}\leq\big{(}\Theta_{\psi}^{0}(\kappa\lambda)+\varepsilon\big{)}\biggl{(}\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}+\varepsilon\biggr{)}\bigl{(}c_{n}\mathscr{H}^{\varphi}(E)+\varepsilon\bigr{)}.$

Inequality (5.16) follows, owing to the arbitrariness of $r\in(0,1]$, of
$\varepsilon>0$, and of $\lambda>\|\nabla u\|_{\mathrm{L}^{A}(\Omega)}$.
Our next task is to show that inequality (5.16) continues to hold for an
arbitrary subset of $\Omega$. Of course, we can still assume that
$\mathscr{H}^{\varphi}(E)<\infty.$ Consider the increasing sequence of sets
$\\{E_{j}\\}$ defined in (5.10). Note that $\overline{E_{j}}$ is a compact set
contained in $\Omega$, and $E=\bigcup_{j}E_{j}$. Inequality (5.16) can thus be
applied with $E$ replaced by $E_{j}$ for each $j\in\mathbb{N}$, and tells us
that

(5.23) $\displaystyle\mathscr{H}^{\psi}\bigl{(}u(E_{j})\bigr{)}$
$\displaystyle\leq c_{n}\Theta_{\psi}^{0}\bigl{(}\kappa\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{+}\bigr{)}\lim_{r\to
0^{+}}\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}\mathscr{H}^{\varphi}(E_{j})$
$\displaystyle\leq c_{n}\Theta_{\psi}^{0}\bigl{(}\kappa\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{+}\bigr{)}\lim_{r\to
0^{+}}\Theta_{\psi}^{0}\bigl{(}\Theta_{B^{-1}}^{\infty}(r)^{+}\bigr{)}\mathscr{H}^{\varphi}(E).$

Inasmuch as $u(E)=\bigcup_{j}u(E_{j})$, inequality (5.16) for the set $E$
follows thanks to property (2.14). Part (ii) is thus fully proved.
Part (i) is a straightforward consequence of inequality (5.16) under the
assumption that $u\in{\rm V}^{1,A}(\Omega,\mathbb{R}^{m})$. It remains to show
that the latter assumption can be relaxed by just requiring that $u\in{\rm
V}^{1,A}_{\rm loc}(\Omega,\mathbb{R}^{m})$. This can be accomplished again by
an invasion argument for $E$ by subsets. Specifically, suppose that $E$ is any
subset of $\Omega$ such that $\mathscr{H}^{\varphi}(E)<\infty$. Let
$\\{\Omega_{j}\\}$ be an increasing sequence of open subsets of $\Omega$,
whose closure is also contained in $\Omega$ and such that
$\Omega=\bigcup_{j}\Omega_{j}$. Set $E_{j}=E\cap\Omega_{j}$ for
$j\in\mathbb{N}$ and note that $E_{j}$ is a subset of $\Omega_{j}$ with
$\mathscr{H}^{\varphi}(E_{j})<\infty$. Furthermore, $u\in{\rm
V}^{1,A}(\Omega_{j},\mathbb{R}^{m})$. Owing to inequality (5.16), applied with
$\Omega$ and $E$ replaced by $\Omega_{j}$ and $E_{j}$, and to assumption
(3.10), one has that $\mathscr{H}^{\psi}\bigl{(}u(E_{j})\bigr{)}=0$ for every
$j\in\mathbb{N}$. Assertion (3.11) hence follows via property (2.14), since
$u(E)=\bigcup_{j}u(E_{j})$.

Proof of Theorem 3.5. We may assume that
$\mathscr{H}^{\varphi}_{\infty}(E)>0$, the case when
$\mathscr{H}^{\varphi}_{\infty}(E)=0$ being covered by Theorem 3.4. Let us
suppose, for the time being, that $\mathscr{H}^{\varphi}_{\infty}(E)<\infty$.
Thanks to the second inequality in (2.10), one has that
$\Lambda_{\infty}^{\varphi}(E)\leq c_{n}\mathscr{H}^{\varphi}_{\infty}(E)$.
Thus, given any $\varepsilon>0$, there exists a family $\\{Q_{i}\\}$ of non-
overlapping dyadic cubes such that $E\subset\bigcup_{i}Q_{i}$ and

(5.24)
$\sum_{i}\varphi\bigl{(}d(Q_{i})\bigr{)}<c_{n}\mathscr{H}^{\varphi}_{\infty}(E)+\varepsilon.$

From inequalities (5.18), (4.32), the definition of the function
$\Theta_{\psi}$ and inequality (2.28) we deduce (in analogy with inequality
(5.21)) that, for each index $i$,

(5.25)
$\psi\bigl{(}d\bigl{(}u(Q_{i})\bigr{)}\bigr{)}\leq\big{(}\Theta_{\psi}(\kappa\lambda){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+\varepsilon}\big{)}\bigl{(}\Theta_{\psi}\bigl{(}\Theta_{B^{-1}}(r)^{+}\bigr{)}+\varepsilon\bigr{)}\left(\varphi\bigl{(}d(Q_{i})\bigr{)}+\frac{\kappa}{r}\int_{Q_{i}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,\mathrm{d}x\right)$

for every $r>0$ and $\lambda>\|\nabla u\|_{\mathrm{L}^{A}(\mathbb{R}^{n})}$.
Summing over all indices $i$ in inequality (5.25), and making use of (5.24)
enable one to infer that

$\mathscr{H}^{\psi}_{\infty}\bigl{(}u(E)\bigr{)}\leq\big{(}\Theta_{\psi}(\kappa\lambda)+\varepsilon\big{)}\bigl{(}\Theta_{\psi}\bigl{(}\Theta_{B^{-1}}(r)^{+}\bigr{)}+\varepsilon\bigr{)}\left(c_{n}\mathscr{H}^{\varphi}_{\infty}(E)+\varepsilon+\frac{\kappa}{r}\int_{\mathbb{R}^{n}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,\mathrm{d}x\right).$

Now we let $\lambda\to\|\nabla u\|_{\mathrm{L}^{A}(\Omega)}^{+}$ and
$\varepsilon\to 0^{+}$ to deduce that

$\mathscr{H}^{\psi}_{\infty}\bigl{(}d\bigl{(}u(E)\bigr{)}\bigr{)}\leq\Theta_{\psi}\bigl{(}\kappa\|\nabla
u\|_{\mathrm{L}^{A}}^{+}\bigr{)}\Theta_{\psi}\left(\Theta_{B^{-1}}(r)^{+}\right)\biggl{(}c_{n}\mathscr{H}^{\varphi}_{\infty}(E)+\frac{\kappa}{r}\biggr{)}.$

Choosing $r=\tfrac{\kappa}{c_{n}\mathscr{H}^{\varphi}_{\infty}(E)}$ in this
inequality yields inequality (3.15).
It remains to consider the case when
$\mathscr{H}^{\varphi}_{\infty}(E)=\infty$. One can then just apply inequality
(3.15) to each set $E_{j}$ of the sequence $\\{E_{j}\\}$ defined by
$E_{j}=E\cap\\{x\in{{\mathbb{R}}^{n}}:|x|<j\\}$ for $j\in\mathbb{N}$. Passing
to the limit as $j\to\infty$ and making use of property (2.15) yields the
conclusion also in this case.

Proof of Theorem 3.7. As a first step, we establish inequality (3.18) for sets
$E$ whose closure is compact and contained in $\Omega$. Fix any such set $E$
and set $\delta=\mathrm{dist}\bigl{(}E,\partial\Omega\bigr{)}$. Let
$\lambda>\|\nabla u\|_{L^{A}(\Omega)}$. Given any $\varepsilon\in(0,\delta)$,
we can select a family $\bigl{\\{}Q_{i}\bigr{\\}}$ of non-overlapping dyadic
cubes such that $E\subset\bigcup_{i}Q_{i}\subset
B_{\varepsilon}(E)\subset\overline{B_{\varepsilon}(E)}\subset\Omega$,
$d(Q_{i})<\varepsilon$ and
${\mathscr{L}}^{n}\bigl{(}\bigcup_{i}Q_{i}\bigr{)}<{\mathscr{L}}^{n}(E)+\varepsilon$.
Here, $B_{\varepsilon}(E)$ denotes the set defined as in (5.1). Let $r>0$ be a
number to be specified later. The function $J_{r}$, associated with $B$ and
with the function $\varphi(s)=s^{n}$ as in (4.29), obeys

(5.26) $J_{r}(s)=sB^{-1}(r)\quad\text{for $s\geq 0$.}$

Thus, from inequalities (4.30) and (2.28) one infers that

$\displaystyle\frac{r}{B^{-1}(r)^{n}}\bigg{(}\frac{d\bigl{(}u(Q_{i})\bigr{)}}{\kappa\lambda}\bigg{)}^{n}\leq
rd(Q_{i})^{n}+d(Q_{i})^{n}B\bigg{(}\frac{d\bigl{(}u(Q_{i})\bigr{)}}{\kappa\lambda
d(Q_{i})}\bigg{)}\leq rd(Q_{i})^{n}+\kappa\int_{Q_{i}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,dx$

for every $i$. Hence,

(5.27) $\displaystyle d\bigl{(}u(Q_{i})\bigr{)}^{n}$
$\displaystyle\leq\kappa^{n}\lambda^{n}B^{-1}(r)^{n}\bigg{(}{\mathscr{L}}^{n}(Q_{i})n^{\frac{n}{2}}+\frac{\kappa}{r}\int_{Q_{i}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,dx\bigg{)}$

for every $i$. Summing this inequality over all indices $i$ yields

$\displaystyle{\mathscr{L}}^{n}\bigl{(}u(E)\bigr{)}$
$\displaystyle\leq\kappa^{n}\lambda^{n}B^{-1}(r)^{n}\bigg{(}({\mathscr{L}}^{n}(E)+\varepsilon)n^{\frac{n}{2}}+\frac{\kappa}{r}\int_{\bigcup_{i}Q_{i}}\\!A\left(\frac{|\nabla
u|}{\lambda}\right)\,dx\bigg{)}$
$\displaystyle\leq\kappa^{n}\lambda^{n}B^{-1}(r)^{n}\left(({\mathscr{L}}^{n}(E)+\varepsilon)n^{\frac{n}{2}}+\frac{\kappa}{r}\right),$

whence, by the arbitrariness of $\lambda>\|\nabla u\|_{L^{A}(\Omega)}$ and of
$\varepsilon\in(0,\delta)$, one deduces that

(5.28) ${\mathscr{L}}^{n}\bigl{(}u(E)\bigr{)}\leq\bigl{(}\kappa\|\nabla
u\|_{L^{A}(\Omega)}B^{-1}(r)\bigr{)}^{n}\left({\mathscr{L}}^{n}(E)n^{\frac{n}{2}}+\frac{\kappa}{r}\right).$

Inequality (3.18) follows from (5.28), by either choosing
$r=\frac{\kappa}{n^{\frac{n}{2}}{\mathscr{L}}^{n}(E)}$ or letting $r\to
0^{+}$, according to whether ${\mathscr{L}}^{n}(E)>0$ or
${\mathscr{L}}^{n}(E)=0$. In the latter case, the fact that $\lim_{r\to
0^{+}}\Phi_{B}(r)=0$ plays a role.
Assume now that $E$ is an arbitrary subset of $\Omega$ and let $E_{j}$ be the
set defined, for $j\in\mathbb{N}$, as in (5.10). The sequence of sets
$\\{E_{j}\\}$ is increasing and consists of sets with compact closure
contained in $\Omega$. Furthermore, $E=\bigcup E_{j}$. We already know that
inequality (3.18) holds with $E$ replaced by each $E_{j}$. Namely,

$\mathscr{H}^{n}_{\infty}\bigl{(}u(E_{j})\bigr{)}\leq 2\kappa^{n+1}\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{n}\Phi_{B}\big{(}{\mathscr{L}}^{n}(E_{j})n^{\frac{n}{2}}/\kappa\big{)}$

for every $j\in\mathbb{N}$. Since the function $\Phi_{B}$ is increasing, this
inequality implies that

$\mathscr{H}^{n}_{\infty}\bigl{(}u(E_{j})\bigr{)}\leq 2\kappa^{n+1}\|\nabla
u\|_{\mathrm{L}^{A}(\Omega)}^{n}\Phi_{B}\big{(}{\mathscr{L}}^{n}(E)n^{\frac{n}{2}}/\kappa\big{)}$

for every $j\in\mathbb{N}$. Inasmuch as $\bigcup_{j}u(E_{j})=u(E)$, property
(2.15) ensures that
$\lim_{j\to\infty}\mathscr{H}^{n}_{\infty}\bigl{(}u(E_{j})\bigr{)}=\mathscr{H}^{n}_{\infty}\bigl{(}u(E)\bigr{)}$.
Inequality (3.18) is fully proved.

##  6 Sharpness

This final section is devoted to discussing the sharpness of the conclusions
of Theorems 3.1 for some examples from Section 3.

#### I.

Let $A$ and $\varphi$ be as in Example 3.11. Namely, let $A$ be any Young
function such that $i_{\infty}(A)>n$ and let $\varphi$ be a gauge function
fulfilling $\liminf_{r\to 0^{+}}\frac{\varphi(r^{\gamma})}{\varphi(r)}>0$ for
every $\gamma>0$. We have shown that

$\psi(r)\sim\varphi(r)\quad\text{near zero.}$

Thus the result is sharp, since this is the best possible choice of the gauge
function $\psi$ one can hope for. In particular, this includes the case
exhibited in equation (3.25), i.e. maps
$u\in\mathrm{V}^{1,p}(\mathbb{R}^{n},\mathbb{R}^{n})$ with $p>n$, and gauge
functions satisfying $\varphi(r)\sim\big{(}\log\tfrac{1}{r}\big{)}^{\beta}$
near zero, for some $\beta<0$.

#### II.

As displayed in equation (3.24), when the classical Sobolev space
$\mathrm{V}^{1,p}(\mathbb{R}^{n},\mathbb{R}^{n})$, with $p>n$, and the
standard Hausdorff measure $\mathscr{H}^{\alpha}$, with $\alpha\in(0,n]$, are
in question, a special instance of Example 3.8 reproduces Kaufman’s theorem.
In Kaufman’s paper, the exponent in the Hausdorff measure
$\mathscr{H}^{\frac{\alpha p}{p+\alpha-n}}$ is shown to be optimal.
The next result augments this conclusion and ensures that the gauge function
$\psi(r)=r^{\frac{\alpha p}{p+\alpha-n}}$, which defines the Hausdorff measure
$\mathscr{H}^{\frac{\alpha p}{p+\alpha-n}}$ in the target space, does not even
admit an improvement in the scale of powers of a logarithm, at least if these
powers are not too small.

###### Theorem 6.1

Assume that $n\in\mathbb{N}$, $p>n$ and $\alpha\in(0,n)$. Set
$\sigma=\frac{p\alpha}{p+\alpha-n}$ and let $\gamma>1+\sigma$. Let $\phi$ be a
gauge function such that

(6.1) $\phi(r)\sim
r^{\sigma}\big{(}\log\tfrac{1}{r}\big{)}^{\gamma}\quad\text{near zero.}$

Then there exist a mapping
$u\in\mathrm{V}^{1,p}(\mathbb{R}^{n},\mathbb{R}^{n})$ and a compact set
$E\subset\mathbb{R}^{n}$ such that

(6.2) $0<\mathscr{H}^{\alpha}(E)<\infty\qquad\mbox{\rm and
}\qquad\mathscr{H}^{\phi}(u(E))=\infty.$

#### III.

Here we focus on maps from Example 3.9, dealing with Young functions

$A(t)$ equivalent to $t^{n}(\log t)^{q}$ near infinity,

in the special case when the gauge function

$\varphi(r)\sim\big{(}\log\tfrac{1}{r}\big{)}^{\beta}\quad\text{ near zero,}$

for some $\beta<0$. The result presented in that example asserts that the
corresponding gauge function in the target space is of merely power type, and
fulfills

$\psi(r)\sim r^{\frac{n\beta}{n-q-1+\beta}}\quad\text{ near zero.}$

The following theorem tells us that not only is this gauge function the best
possible among all standard Hausdorff measures of power type, but it is also
optimal in the scale of “powers times powers of a logarithm” type gauge
functions, at least if the power of the logarithm is not too small.

###### Theorem 6.2

Assume that $n\in\mathbb{N}$,

(6.3) $A(t)\sim t^{n}(\log t)^{q}\quad\text{\rm near infinity }$

for some $q>n-1$, and

(6.4) $\varphi(r)\sim\big{(}\log\tfrac{1}{r}\big{)}^{\beta}\quad\text{near
zero}$

for some $\beta<0$. Set $\sigma=\frac{n\beta}{n-q-1+\beta}$, let
$\mu>1+\sigma$, and let $\phi$ be a gauge function such that

(6.5) $\phi(r)\sim
r^{\sigma}\big{(}\log\tfrac{1}{r}\big{)}^{\mu}\quad\text{near zero.}$

Then there exist a mapping
$u\in\mathrm{V}^{1,A}(\mathbb{R}^{n},\mathbb{R}^{n})$ and a compact set
$E\subset\mathbb{R}^{n}$ such that

(6.6) $0<\mathscr{H}^{\varphi}(E)<\infty\qquad\mbox{\rm and
}\qquad\mathscr{H}^{\phi}(u(E))=\infty.$

The proofs of Theorems 6.1 and 6.2 are reminiscent of arguments from [Ka99]
and their modification in [HH15, section 4.2]. In particular, they involve a
construction resting upon probabilistic techniques. Additional difficulties
arise in the present situation, which are specific of the Orlicz ambient
space. Theorem 6.2 will be established in detail. The proof of Theorem 6.1
follows along the same lines, and it is in fact simpler, since the Orlicz-
Sobolev space agrees with a classical Sobolev space in this case. For brevity,
this proof will be omitted.

The question of whether the lower bounds on the exponents $\gamma$ and $\mu$
in Theorems 6.1 and 6.2, respectively, can be removed is open. This is a
natural interesting problem, whose solution seems however to require
approaches and techniques substantially different from those employed in the
present paper and in previous contributions in the literature. An even more
challenging question is of course whether $\psi$ is best possible among all
gauge functions, for any given $A$ and $\varphi$.

In preparation for the proof of Theorem 6.2, we recall a few definitions and
establish a few preliminary lemmas.
The Hausdorff dimension $\dim_{\mathscr{H}}(E)$ of a set
$E\subset{{\mathbb{R}}^{n}}$ is defined as

$\dim_{\mathscr{H}}(E)=\inf\\{\alpha>0:\mathscr{H}^{\alpha}(E)=0\\}.$

One also has that
$\dim_{\mathscr{H}}(E)=\sup\\{\alpha>0:\mathscr{H}^{\alpha}(E)=\infty\\}$.
Given a Radon measure $\mu$ on a compact set $E\subset\mathbb{R}^{n}$ and a
number $\alpha>0$, we set

$I_{\alpha}(\mu)=\int\limits_{E}\int\limits_{E}|x-y|^{-\alpha}\,d\mu(x)\,d\mu(y).$

It is well known that

(6.7) $\dim_{\mathscr{H}}(E)=\sup\\{\alpha>0:\text{there exists $\mu\neq 0$
such that $I_{\alpha}(\mu)<\infty$}\\},$

see, e.g., [Ma95, Theorem 8.9].
The following result (see [Ma95, Theorem 8.7]) will also be needed in our
proof of Theorem 6.2.

###### Lemma 6.3

Let $\phi$ be a gauge function, and let $E$ is compact set in
${{\mathbb{R}}^{n}}$ . Assume that there exists a nonnegative Radon measure
$\mu$ such that $0<\mu(E)<\infty$ and

$\int\limits_{E}\int\limits_{E}\frac{\mathrm{d}\mu(x)\,\mathrm{d}\mu(y)}{\phi\bigl{(}|x-y|\bigr{)}}<\infty.$

Then $\mathscr{H}^{\phi}(E)=\infty$.

The following probabilistic lemma extends, with analogous proof, [BMT13, Lemma
4.4]. In its statement, $\mathbf{E}$ denotes expectation, and
$\|\\{a_{i}\\}\|_{\infty}$ the $\ell^{\infty}$ norm of a sequence
$\\{a_{i}\\}$. Henceforth, $\log$ stands for $\log_{2}$.

###### Lemma 6.4

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be a countable sequence of independent random
variables, identically distributed according to the uniform distribution on
the unit ball $B$ in $\mathbb{R}^{n}$. Let $\phi:[0,\infty)\to[0,\infty)$ be
the function defined by

$\phi(r)=r^{\sigma}\log^{\mu}\bigl{(}b+\frac{1}{r}\bigr{)}\quad\text{for
$r\geq 0$,}$

with $\sigma\in(0,n)$, $\mu\in\mathbb{R}$, and $b$ large enough for the
function $\phi$ to be increasing. Assume that $\\{a_{i}\\}\in\ell_{1}$. Then
there exists a constant $c$, depending on $n$, $\sigma$, $\mu$, and $b$, such
that

(6.8)
$\mathbf{E}\biggl{[}\biggl{(}\phi\biggl{(}\biggr{|}\sum\limits_{i=1}^{\infty}a_{i}X_{i}\biggr{|}\biggr{)}\biggr{)}^{-1}\biggr{]}\leq
c\,\frac{1}{\phi\bigl{(}\|\\{a_{i}\\}\|_{\infty}\bigr{)}}$

Let us denote by $L^{n}\log^{q}L({{\mathbb{R}}^{n}})$ the Orlicz space built
upon the Young function $A(t)=t^{n}\log^{q}(t+2)$, with $q>n-1$, and equipped
with the Luxemburg norm. The corresponding Orlicz-Sobolev space will be
denoted by $\mathrm{V}^{1}L^{n}\log^{2}L({{\mathbb{R}}^{n}})$.
We use the notation $Q$ for a cube in ${{\mathbb{R}}^{n}}$ with sides parallel
to the coordinate axes, and $\ell(Q)$ for its sidelength. By $2Q$ we denote
the cube with double sidelength and the same center as $Q$. Finally, $Q(r)$
stands for the cube, centered at the origin, with $\ell(Q)=r$.

###### Lemma 6.5

For every $j\in\mathbb{N}$, there exists a function
$\eta_{j}\in\mathrm{V}^{1}L^{n}\log^{2}L({{\mathbb{R}}^{n}})$ such that

$0\leq\eta(x)\leq 1\quad\text{for $x\in{{\mathbb{R}}^{n}}$,}$
$\eta_{j}(x)=\begin{cases}1\qquad\text{if \, $x\in Q(2^{-2^{j}})$}\\\\[10.0pt]
0\qquad\text{if \, $x\notin Q(2^{-2^{j-1}})$,}\end{cases}and$ (6.9)
$\|\nabla\eta_{j}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}\leq
c\,2^{j(q-n+1)/n},$

where $c$ is a constant independent of $j$.

Proof. Let us define, for $j\in\mathbb{N}$,

$\eta_{j}(x)=\begin{cases}1\qquad&\text{if $2|x|_{\infty}\leq 2^{-2^{j}}$}\\\
\frac{1}{2^{(j-1)}}\biggl{(}-2^{(j-1)}+\log\frac{1}{2|x|_{\infty}}\biggr{)}\qquad&\text{if
$2^{-2^{j}}<2|x|_{\infty}<2^{-2^{(j-1)}}$}\\\ 0\qquad&\text{if
$2|x|_{\infty}\geq 2^{-2^{(j-1)}}$,}\end{cases}$

where $|x|_{\infty}=\max\limits_{i=1,\dots,n}|x_{i}|$ and
$x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}$. In particular, given $r>0$,

$x\in Q(r)\quad\text{if and only if}\quad 2|x|_{\infty}\leq r.$

Thus, the function $\eta_{j}$ satisfies the first two properties claimed in
the statement.
As for inequality (6.9), notice that

$|\nabla\eta_{j}(x)|\begin{cases}\sim\frac{1}{2^{j}\,|x|}\quad&\text{if
$2^{-2^{j}}<2|x|_{\infty}<2^{-2^{(j-1)}}$ }\\\
=0\quad&\text{otherwise.}\end{cases}$

Now, set $\lambda=2^{j(q-n+1)/n}$. Thus, on passing to polar coordinates, one
obtains that

$\displaystyle\int\limits_{\mathbb{R}^{n}}A\bigg{(}\frac{|\nabla\eta_{j}|}{\lambda}\bigg{)}\,dx$
$\displaystyle\leq$
$\displaystyle\frac{c}{2^{jn}\lambda^{n}}\int\limits_{2^{-2^{j}}}^{2^{-2^{(j-1)}}}\frac{1}{r}\log^{q}\Big{(}2+\frac{1}{\lambda
2^{j}r}\Big{)}\,\mathrm{d}r$ $\displaystyle\leq$
$\displaystyle\frac{c}{2^{j(q+1)}}\int\limits_{2^{-2^{j}}}^{2^{-2^{(j-1)}}}\frac{1}{r}\log^{q}\Big{(}\frac{1}{r}\Big{)}\,\mathrm{d}r\leq\frac{c}{2^{j(q+1)}}2^{j(q+1)}\leq
c.$

for some constant $c$ independent of $j$. Hence, inequality (6.9) follows.

###### Lemma 6.6

Let $N\in\mathbb{N}$ and let $v_{i}\colon\mathbb{R}^{n}\to\mathbb{R}$,
$i=1,\dots,N$, be a family of functions in
$\mathrm{V}L^{n}\log^{q}L({{\mathbb{R}}^{n}})$, with disjoint supports, and
such that

$\|\nabla v_{1}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}=\|\nabla
v_{2}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}=\dots=\|\nabla
v_{N}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}.$

Then

(6.10) $\biggl{\|}\sum\limits_{i=1}^{N}\nabla
v_{i}\biggr{\|}_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}\leq
N^{\frac{1}{n}}\|\nabla v_{1}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}.$

Proof. Without loss of generality, we can assume that $\|\nabla
v_{i}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}=1$ for $i=1,\dots,N$, namely,

$\int\limits_{\mathbb{R}^{n}}\\!|\nabla v_{i}|^{n}\log^{q}\bigl{(}|\nabla
v_{i}|+2\bigr{)}\,\mathrm{d}x=1\qquad\text{for $i=1,\dots,N$.}$

On setting $\lambda=N^{\frac{1}{n}}$, one has that

$\displaystyle\int\limits_{\mathbb{R}^{n}}\\!A\biggl{(}\frac{1}{\lambda}\biggr{|}\sum\limits_{i=1}^{N}\nabla
v_{i}\biggr{|}\biggr{)}\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{n}}\sum\limits_{i=1}^{N}\int\limits_{\mathbb{R}^{n}}|\nabla
v_{i}|^{n}\log^{q}\biggl{(}\frac{1}{\lambda}|\nabla
v_{i}|+2\biggr{)}\,\mathrm{d}x$ $\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{n}}\sum\limits_{i=1}^{N}\int\limits_{\mathbb{R}^{n}}|\nabla
v_{i}|^{n}\log^{q}\bigl{(}|\nabla
v_{i}|+2\bigr{)}\,\mathrm{d}x=\frac{N}{\lambda^{n}}=1.$

Hence, inequality (6.10) follows.

We are now ready to provide a proof of Theorem 6.2.

Proof of Theorem 6.2. For simplicity of notation, throughout the proof we
rename the parameter $\beta$, and set $\nu=-\beta$. Hence, $\nu>0$ and

(6.11) $\varphi(r)=\bigl{(}\log\tfrac{1}{r}\bigr{)}^{-\nu}\quad\text{for
$r>0$.}$

For each $j\in\mathbb{N}$, consider a system of dyadic cubes $\mathcal{A}_{j}$
in $[0,1]^{n}$ enjoying the following properties:

$\ell(Q)=2^{-2^{j}}\qquad\text{for every $Q\in\mathcal{A}_{j}$},$

the cardinality $\\#\bigl{(}\mathcal{A}_{j}\bigr{)}$ of $\mathcal{A}_{j}$
satisfies

(6.12) $\\#\bigl{(}\mathcal{A}_{j}\bigr{)}=[2^{j\nu}],$

where $[\,\cdot\,]$ denotes integer part, and

$\mbox{ for every }Q\in\mathcal{A}_{j+1}\quad\mbox{ there exists
}\,\widetilde{Q}\in\mathcal{A}_{j}\quad\mbox{ such that }\,Q\subset\tilde{Q}.$

The construction of such a family is elementary and standard.
Let us set

$\mathcal{A}=\bigcup\limits_{j\in\mathbb{N}}\mathcal{A}_{j}\quad\text{and}\quad
M=\bigcap\limits_{j\in\mathbb{N}}\bigcup\limits_{Q\in\mathcal{A}_{j}}Q.$

One can verify that

(6.13) $0<\mathscr{H}^{\varphi}(M)<\infty.$

Moreover, we can assume that the cubes are evenly distributed, and hence that
there exists a universal constant $c>0$ such that, for every $j\in\mathbb{N}$,

(6.14) if $Q\in\mathcal{A}_{j}$, then $\mathscr{H}^{\varphi}(Q\cap M)\leq
c\,2^{-j}$,

and

(6.15) if $\widetilde{Q}\in\mathcal{A}_{j}$, then
$\\#\\{Q\in\mathcal{A}_{j+1}:Q\subset\widetilde{Q}\\}\leq c$.

Set

(6.16) $\sigma=\frac{n\nu}{q-n+1+\nu},$

and fix $\mu>1+\sigma$. Define the function $\phi:[0,\infty)\to[0,\infty)$ by

$\phi(r)=\begin{cases}r^{\sigma}\log^{\mu}\bigl{(}b+\tfrac{1}{r})\quad&\text{if
$r>0$}\\\ 0\quad&\text{if $r=0$,}\end{cases}$

where the constant $b$ is large enough for the function $\phi$ to be
increasing.

For each $Q\in\mathcal{A}_{j}$ let $\eta_{Q}$ be a function defined as an
appropriate translate of the function $\eta_{j}$ from Lemma 6.5, which
fulfills the following properties:
(i) $\eta_{Q}$ is supported in the cube $Q^{\prime}\supset\supset Q$ with
$\ell(Q^{\prime})=2^{-2^{j-1}}$;
(ii) $0\leq\eta_{Q}\leq 1$;
(iii) $\eta_{Q}=1$ on $Q$;
(iv) $\eta_{Q}=0$ outside $Q^{\prime}$;
(v) $\|\nabla\eta_{Q}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}\leq
c\,2^{j(q-n+1)/n}$.

Let $\xi=\\{\xi_{Q}\\}_{Q\in\mathcal{A}}$ be a countable sequence of points in
the unit ball $B$ of $\mathbb{R}^{n}$. Fix $\delta>1$ such that

(6.17) $1+\delta\sigma<\mu.$

For each $j\in\mathbb{N}$, define the function
$u_{\xi,j}:{{\mathbb{R}}^{n}}\to{{\mathbb{R}}^{n}}$ as

$u_{\xi,j}(x)=\frac{1}{j^{\delta}}2^{-\frac{j\nu}{\sigma}}\sum\limits_{Q\in\mathcal{A}_{j}}\eta_{Q}(x)\xi_{Q}\quad\text{for
$x\in{{\mathbb{R}}^{n}}$,}$

and the function $u_{\xi}:{{\mathbb{R}}^{n}}\to{{\mathbb{R}}^{n}}$ as

$u_{\xi}(x)=\sum\limits_{j}u_{\xi,j}(x)\quad\text{for
$x\in{{\mathbb{R}}^{n}}$.}$

We claim that there exists a universal constant $c>0$ such that

(6.18) $\\#\bigl{\\{}Q\in\mathcal{A}_{j}:(\mathop{\rm
supp}\eta_{Q})\cap(\mathop{\rm
supp}\eta_{\widetilde{Q}})\neq\emptyset\bigr{\\}}\leq c$

for every $j\in\mathbb{N}$ and every $\widetilde{Q}\in\mathcal{A}_{j}$.
Indeed, if $Q,\widetilde{Q}\in\mathcal{A}_{j}$ and $(\mathop{\rm
supp}\eta_{Q})\cap(\mathop{\rm supp}\eta_{\widetilde{Q}})\neq\emptyset$, then
$Q$ and $\widetilde{Q}$ either lie in the same cube or in adjacent cubes from
$\mathcal{A}_{j-1}$. Moreover, any dyadic cube has exactly $3^{n}-1$ adjacent
dyadic cubes of the same size. These pieces of information, combined with
property (6.15), enable one to complete the proof of (6.18).
From equations (6.12), (6.18) and Lemmas 6.5–6.6 we deduce that there exists a
constant $c$, independent of $\xi$ and $j$, such that

$\|\nabla u_{\xi,j}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}\leq
c\,\frac{1}{j^{\delta}}2^{-\frac{j\nu}{\sigma}}\ 2^{j\nu/n}\
2^{j(q-n+1)/n}=c\frac{1}{j^{\delta}}.$

Therefore,

$\|\nabla u_{\xi}\|_{L^{n}\log^{q}L({{\mathbb{R}}^{n}})}\leq
c\sum\limits_{j}\frac{1}{j^{\delta}}<\infty.$

This shows that the mapping
$u_{\xi}\in\mathrm{V}^{1}L^{n}\log^{q}L({{\mathbb{R}}^{n}})$, and hence, in
particular, $u_{\xi}$ is continuous, since we are assuming that $q>n-1$. Also,
we have that $\mathop{\rm supp}u_{\xi}\subset[0,1]^{n}$.

The remaining part of the proof is devoted to showing that, for a generic
choice of $\xi$, the map $u_{\xi}$ enjoys the property

(6.19) $\mathscr{H}^{\phi}\bigl{(}u_{\xi}(M)\bigr{)}=\infty.$

The desired conclusion will then follow by coupling equation (6.19) with
(6.13).
In order to establish property (6.19), let us regard
$\xi=\\{\xi\\}_{Q\in\mathcal{A}}$ as a sequence of independent random
variables, which are identically distributed according to the uniform
probability distribution on $B$. Consider the measure
$\mu_{\xi}=(u_{\xi})_{\\#}(\mathscr{H}^{\varphi}\begin{picture}(10.0,8.0)\put(2.0,0.0){\line(0,1){7.0}}\put(1.8,0.0){\line(1,0){7.0}}\end{picture}M)$
supported on the compact set $u_{\xi}(M)$, i.e., $\mu_{\xi}$ is the push-
forward of the measure $\mathscr{H}^{\varphi}$ restricted to $M$ via the map
$u_{\xi}$. We claim that its expectation satisfies the condition

(6.20)
$\mathbf{E}_{\xi}\biggl{(}\int\limits_{u_{\xi}(M)}\int\limits_{u_{\xi}(M)}\frac{\mathrm{d}\mu_{\xi}(x)\,\mathrm{d}\mu_{\xi}(y)}{\phi\bigl{(}|x-y|\bigr{)}}\biggr{)}<\infty.$

In order to prove inequality (6.20), we make use Fubini’s theorem and rewrite
the left-hand side of inequality (6.20) as

(6.21)
$\int\limits_{M}\int\limits_{M}\mathbf{E}_{\xi}\biggl{(}\frac{1}{\phi\bigl{(}\bigl{|}u_{\xi}(x)-u_{\xi}(y)\bigr{|}\bigr{)}}\biggr{)}\,\mathrm{d}\mathscr{H}^{\varphi}(x)\,\mathrm{d}\mathscr{H}^{\varphi}(y).$

We have that

$u_{\xi}(x)-u_{\xi}(y)=\sum\limits_{Q\in\mathcal{A}}a_{Q}(x,y)\xi_{Q}\quad\text{for
$x,y\in{{\mathbb{R}}^{n}}$,}$

where

(6.22) $a_{Q}(x,y)=\frac{1}{j^{\delta}}2^{-\frac{\nu
j}{\sigma}}\bigl{(}\eta_{Q}(x)-\eta_{Q}(y)\bigr{)}\quad\text{ for
$Q\in\mathcal{A}_{j}$.}$

The sequence $\\{a_{Q}\\}$ clearly belongs to $\ell^{\infty}$. Thus, thanks to
inequality (6.8), inequality (6.21) will follow if we show that

(6.23)
$\int\limits_{M}\frac{1}{\phi\bigl{(}\|a(x,y)\|_{\infty}\bigr{)}}\,\mathrm{d}\mathscr{H}^{\varphi}(x)\leq
c,$

for some constant $c$ is independent of $y$. Let us fix $y\in M$. For each
$x\in M$, let us denote by $j(x)$ the largest integer such that both $x$ and
$y$ either lie in the same cube or in adjacent cubes from
$\mathcal{A}_{j(x)}$. Hence, there exist cubes $\widetilde{Q}_{1}\ni x$ and
$\widetilde{Q}_{2}\ni y$ such that
$\widetilde{Q}_{1},\widetilde{Q}_{2}\in\mathcal{A}_{j(x)+1}$ and
$2\tilde{Q}_{1}\cap Q_{2}=\emptyset$. Moreover, there exist cubes $Q_{1}\ni x$
and $Q_{2}\ni y$ such that $Q_{1},Q_{2}\in\mathcal{A}_{j(x)+2}$ and

$\bigl{(}\mathop{\rm supp}\eta_{Q_{1}}\bigr{)}\cap\bigl{(}\mathop{\rm
supp}\eta_{Q_{2}}\bigr{)}=\emptyset.$

Consequently, $\eta_{Q_{1}}(x)=1$ and $\eta_{Q_{1}}(y)=0$. Therefore, there
exists a nonzero coefficient $a_{Q}$ of size
$\frac{2^{-(j(x)+2)\frac{\nu}{\sigma}}}{(j(x)+2)^{\delta}}$. Hence

$\|a(x,y)\|_{\infty}\geq\frac{2^{-j(x)\frac{\nu}{\sigma}}}{j(x)^{\delta}}.$

The monotonicity of the function $\phi$ guarantees that

(6.24)
$\frac{1}{\phi\bigl{(}\|a(x,y)\|_{\infty}\bigr{)}}=\frac{1}{\|a(x,y)\|^{\sigma}_{\infty}\log^{\mu}\bigl{(}b+\frac{1}{\|a(x,y)\|_{\infty}}\bigr{)}}\leq
c\,j(x)^{\delta\sigma-\mu}\,\,2^{\nu j(x)}$

for some constant $c$.
Now, for each $y\in M$ and $j\in\mathbb{N}$, there exist at most $3^{n}$
dyadic cubes $Q\in\mathcal{A}_{j}$ with the property that there exists $x\in
Q\cap M$ with $j(x)=j$. Thereby, equation (6.14) implies that

$\mathscr{H}^{\varphi}\bigl{(}\bigl{\\{}x\in M:j(x)=j\bigr{\\}}\bigr{)}\leq
3^{n}\mathscr{H}^{\varphi}(Q\cap M)\leq c\ 2^{-j\nu}.$

Finally, owing to inequality (6.24) and assumption (6.17),

$\int\limits_{M}\frac{1}{\phi\bigl{(}\|a(x,y)\|_{\infty}\bigr{)}}\,\mathrm{d}\mathscr{H}^{\varphi}(x)\leq
c\sum\limits_{j=1}^{\infty}2^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-j\nu}}\biggl{(}j^{\delta\sigma-\mu}\,2^{\nu
j}\biggr{)}=c\sum\limits_{j=1}^{\infty}j^{\delta\sigma-\mu}<\infty.$

Inequality (6.23), and hence property (6.20), are thus established.
On the other hand, property (6.20) clearly entails that, almost surely with
respect to $\xi$,

$\int\limits_{u_{\xi}(M)}\int\limits_{u_{\xi}(M)}\frac{\mathrm{d}\mu_{\xi}(x)\,\mathrm{d}\mu_{\xi}(y)}{\phi\bigl{(}|x-y|\bigr{)}}<\infty.$

Hence, equation (6.19) follows via Lemma 6.3. The proof is complete.

## Compliance with Ethical Standards

Funding. This research was partly funded by:
(i) Research Project 201758MTR2 of the Italian Ministry of Education,
University and Research (MIUR) Prin 2017 “Direct and inverse problems for
partial differential equations: theoretical aspects and applications”;
(ii) GNAMPA of the Italian INdAM - National Institute of High Mathematics
(grant number not available).

Conflict of Interest. The authors declare that they have no conflict of
interest.

## References

  * [AH96] D.R. Adams & L.I. Hedberg, Function spaces and potential theory, Springer, Berlin, 1996.
  * [AC05] A. Alberico & A. Cianchi, Differentiability properties of Orlicz-Sobolev functions, _Ark. Mat._ 43 (2005), 1–28.
  * [ACTUTV13] K. Astala, A. Clop, X. Tolsa, I. Uriarte-Tuero & J. Verdera, Quasiconformal distortion of Riesz capacities and Hausdorff measures in the plane, Amer. J. Math.  135 (2013), 17–52.
  * [BMT13] Z.M. Balogh, R. Monti & J.T. Tyson, Frequency of Sobolev and quasiconformal dimension distortion, J. Math. Pures Appl. 99 (2013), 125–149.
  * [BKK13] J. Bourgain, M.V. Korobkov & J. Kristensen, On the Morse-Sard property and level sets of Sobolev and BV functions, Rev. Mat. Iberoam. 29 (2013), no. 1, 1–23.
  * [BKK15] J. Bourgain, M.V. Korobkov & J. Kristensen, On the Morse–Sard property and level sets of $W^{n,1}$ Sobolev functions on $\mathbb{R}^{n}$, J. Reine Angew. Math. 700 (2015), 93–112. http://dx.doi.org/10.1515/crelle-2013-0002
  * [C51] A.P. Calderon, On the differentiability of absolutely continuous functions, Riv. Math. Univ. Parma 2 (1951), 203–213.
  * [Ca67] L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Mathematical Studies No. 13, D. Van Nostrand Company, INC., Princeton, New Jersey, 1967.
  * [CC16] M. Carozza & A. Cianchi, Smooth approximation of Orlicz-Sobolev maps between manifolds, Potential Anal. 45 (2016), 557–578.
  * [Ce42] L. Cesari, Sulle trasformazioni continue, Ann. Mat. Pura Appl. 21 (1942), 157–188.
  * [Ci96] A. Cianchi, Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 575–608.
  * [Ci97] A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Diff. Equat.  22 (1997), 1629–1646.
  * [Ci04] A. Cianchi, Optimal Orlicz–Sobolev embeddings, Rev. Mat. Iberoamericana 20 (2004), 427–474.
  * [CM19] A. Cianchi & V. Musil, Optimal domain spaces in Orlicz–Sobolev embeddings, Indiana Univ. Math. J. 68 (2019), 925–966.
  * [CP98] A. Cianchi & L. Pick, Sobolev embeddings into $BMO$, $VMO$, and $L^{\infty}$, Ark. Mat. 36 (1998), 317–340.
  * [CM01] M. Csörnyei & R.D. Mauldin, Scaling properties of Hausdorff and packing measures, Math. Ann. 319 (2001), 817–836.
  * [Da70] R.O. Davies, Increasing sequences of sets and Hausdorff measure, Proc. London Math. Soc.(3) 20 (1970), 222-236.
  * [Fa85] K.J. Falkoner, The Geometry of Fractal Sets, Cambridge Tracts in Maths. 85, Cambridge University Press, 1986 Edition.
  * [FKR19] A. Ferone, M.V. Korobkov & A. Roviello, On the Luzin $N$-property and the uncertainty principle for Sobolev mappings, Analysis & PDEs, 12:5 (2019), 1149–1175.
  * [GV73] F.W. Gehring & J. Väisälä, Hausdorff dimension and quasiconformal mappings, J. London Math. Soc. 2(6) (1973), 504–512.
  * [GP77] J. Gustavsson & J. Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1977), 33–59.
  * [Ha1918] F. Hausdorff, Dimension und äusseres Mass, (German) Math. Ann. 79 (1918), 157–179.
  * [HH15] S. Hencl & P. Honzík, Dimension distortion of images of sets under Sobolev mappings, Ann. Acad. Sci. Fenn. Math. 40(1) (2015), 427–442.
  * [HK14] S. Hencl & P. Koskela, Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics 2096, Springer 2014.
  * [IM01] T. Iwaniec & G. Martin, Geometric Function Theory and Nonlinear Analysis. Oxford Mathematical Monographs. Clarendon Press, Oxford 2001.
  * [Ja80] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47(4) (1980), 959–982.
  * [Ka99] R. Kaufman, Sobolev spaces, dimensions, and random series, Proc. Amer. Math. Soc. 128 (1999), 427–431.
  * [KKM99] J. Kauhanen, P. Koskela & J. Malý, On functions with derivatives in a Lorentz space, Manuscripta Math., 100 (1999), no. 1, 87–101.
  * [KK14] M.V. Korobkov & J. Kristensen, On the Morse-Sard theorem for the sharp case of Sobolev mappings, Indiana Univ. Math. J. 63 (2014), no. 6, 1703–1724.
  * [KK18] M.V. Korobkov & J. Kristensen, The trace theorem, the Luzin N- and Morse-Sard properties for the sharp case of Sobolev-Lorentz mappings, J. Geom. Anal. 28 (2018), no. 3, 2834–-2856.
  * [KMZ15] P. Koskela, J. Malý & T. Zürcher, Luzin’s condition (N) and modulus of continuity, Adv. Calc. Var. 8(2) (2015), 155–171.
  * [KZ15] P. Koskela & A. Zapadinskaya, Dimension gap under Sobolev mappings, Adv. Math. 274 (2015), 385–403.
  * [KRSS14] D.A. Kovtonyuk, V.I. Ryazanov, R.R. Salimov & E.A. Sevost’yanov, Toward the theory of Orlicz–Sobolev classes, St. Petersburg Math. J., 25 (2014), 929–963.
  * [LSUT10] M.T. Lacey, E.T. Sawyer & I. Uriarte-Tuero, Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps of the plane, Acta Math. 204 (2010), 273–292.
  * [MM95] J. Malý & O. Martio, Lusin’s condition (N) and mappings of class $W^{1,n}$, J. Reine Angew. Math. 458 (1995), 19–36.
  * [MSZ02] J. Malý, D. Swanson & W.P. Ziemer, The co-area formula for Sobolev mappings, Trans. Amer. Math. Soc. 355(2) (2002), 477–492.
  * [MSZ09] J. Malý, D. Swanson & W.P. Ziemer, Fine behavior of functions whose gradient are in an Orlicz space, Studia Math. 190(1) (2009), 33–71.
  * [Ma95] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, 1995.
  * [Pe97] Y.B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lecture in Mathematics, The University of Chicago Press 1997.
  * [Po71] S.P. Ponomarev, An example of an $ACTL^{p}$ homeomorphism that is not absolutely continuous in the sense of Banach, Sov. Math., Dokl. 12 (1971), 1788–1790.
  * [RR55] T. Rado & P.V. Reichelderfer: Continuous Transformations in Analysis, Grundlehren der Math. Wissenschaften 75, Springer Berlin-Göttingen-Heidelberg, 1955.
  * [RZZ11] T. Rajala, A. Zapadinskaya & T. Zürcher, Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings, J. Math. Anal. Appl. 384(2) (2011), 468–477.
  * [Re67] Yu.G. Reshetnyak, Space mappings with bounded distortion. Sibirsk. Mat. Z. 8 (1967), 629–658.
  * [Re89] Yu.G. Reshetnyak, Space mappings with bounded distortion. Translations of Mathematical Monographs, 73. American Mathematical Society, Providence, RI, 1989.
  * [Ri93] S. Rickman, Quasiregular Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)), vol. 26. Springer, Berlin 1993.
  * [Ro70/98] C.A. Rogers, Hausdorff Measures, Cambridge Mathematical Library, Cambridge University Press, 1998 edition.
  * [Sa37] S. Saks, Theory of the Integral. ”Monografie Matematyczne” series, vol. 7, Polish Mathematical Society, Warszawa–Lwów, 1937.
  * [St79] J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511–544.
  * [Vai71] J. Väisälä, Lectures on $n$-Dimensional Quasiconformal Mappings, Lect. Notes in Maths., vol. 229, Springer, Berlin-Heidelberg-New York, 1971.