# INEQUALITIES FOR WEIGHTED SPACES WITH VARIABLE EXPONENTS

Pablo Rocha

###### Abstract

In this article we obtain an ”off-diagonal” version of the Fefferman-Stein
vector-valued maximal inequality on weighted Lebesgue spaces with variable
exponents. As an application of this result and the atomic decomposition
developed in [12] we prove, for certain exponents $q(\cdot)$ in
$\mathcal{P}^{\log}(\mathbb{R}^{n})$ and certain weights $\omega$, that the
Riesz potential $I_{\alpha}$, with $0<\alpha<n$, can be extended to a bounded
operator from $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$ into
$L^{q(\cdot)}_{\omega}(\mathbb{R}^{n})$, for
$\frac{1}{p(\cdot)}:=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$.

†† 2020 Mathematics Subject Classification. Primary 42B30; Secondary 42B25,
42B35, 46E30. †† Key words and phrases: Fefferman-Stein inequalities, weighted
variable Hardy spaces, atomic decomposition, Riesz potential.

##  1 Introduction

The theory of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb{R}^{n})$
was developed independently by Nakai and Sawano in [15] and by Cruz-Uribe and
Wang in [5]. Both theories prove equivalent definitions in terms of maximal
operators using different approaches. In [15, 5], one of their main goals is
the atomic decomposition of elements in $H^{p(\cdot)}$, as an application of
the atomic decomposition they proved that singular integrals are bounded on
$H^{p(\cdot)}$. Later in [18], the author jointly with Urciuolo proved the
$H^{p(\cdot)}-L^{q(\cdot)}$ boundedness of certain generalized Riesz
potentials and the $H^{p(\cdot)}-H^{q(\cdot)}$ boundedness of Riesz potential
via the infinite atomic and molecular decomposition developed in [15]. In
[17], the author gave another proof of the results obtained in [18], but by
using the finite atomic decomposition developed in [5].

Recently, Kwok-Pun Ho in [12] developed the weighted theory for variable Hardy
spaces on $\mathbb{R}^{n}$. He established the atomic decompositions for the
weighted Hardy spaces with variable exponents
$H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$ and also revealed some intrinsic
structures of atomic decomposition for Hardy type spaces. His results
generalize the infinite atomic decompositions obtained in [8, 1, 22, 15].

The main results of this work are contained in Theorems 4.1 and 5.1 below. The
first result concerns to an ”off-diagonal” version of the Fefferman-Stein
vector-valued maximal inequality on weighted variable Lebesgue spaces
(originally due to Harboure, Macías and Segovia [10]). Our second result
states, for certain exponent functions $q(\cdot)$ and certain weights
$\omega$, the boundedness of Riesz potential $I_{\alpha}$ from
$H^{p(\cdot)}_{\omega}$ into $L^{p(\cdot)}_{\omega}$ where $0<\alpha<n$ and
$\frac{1}{p(\cdot)}:=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$. This result is
achieved via the off-diagonal Fefferman-Stein inequality (Theorem 4.1), the
atomic decomposition established in [12], and the following additional
assumption: for every cube $Q\subset\mathbb{R}^{n}$

(1)
$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega}}\approx|Q|^{-\alpha/n}\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}.$

If $\omega\equiv 1$ and $q(\cdot)$ is log-Hölder continuous (locally and at
infinite), then (1) holds. This case was proved in [18]. In this article we
will give non trivial examples of power weights satisfying (1). So, (1) is an
admissible hypothesis.

This paper is organized as follows. Section 2 gives the definition of weighted
variable Lebesgue spaces $L^{p(\cdot)}_{\omega}$ and weighted variable Hardy
spaces $H^{p(\cdot)}_{\omega}$ and some of their preliminary results. Section
3 presents some basics properties of the set of weights
$\mathcal{W}_{p(\cdot)}$ used to define $H^{p(\cdot)}_{\omega}$ and some
results about the size of the cubes in the $L^{p(\cdot)}_{\omega}$-norm. The
off-diagonal version of the Fefferman-Stein inequality on weighted variable
Lebesgue spaces is established in Section 4. The
$H^{p(\cdot)}_{\omega}-L^{q(\cdot)}_{\omega}$ boundedness of Riesz potential
$I_{\alpha}$ is proved in Section 5. Finally, Section 6 gives non trivial
examples of power weights satisfying (1).

Notation. The symbol $A\lesssim B$ stands for the inequality $A\leq cB$ for
some constant $c$. The symbol $A\approx B$ stands for $B\lesssim A\lesssim B$.
We denote by $Q\left(z,r\right)$ the cube centered at $z\in\mathbb{R}^{n}$
with side lenght $r$. Given a cube $Q=Q(z,r)$, we set $kQ=Q(z,kr)$ and
$l(Q)=r$. For a measurable subset $E\subseteq\mathbb{R}^{n}$ we denote by
$|E|$ and $\chi_{E}$ the Lebesgue measure of $E$ and the characteristic
function of $E$ respectively. Given a real number $s\geq 0$, we write $\lfloor
s\rfloor$ for the integer part of $s$. As usual we denote with
$\mathcal{S}(\mathbb{R}^{n})$ the space of smooth and rapidly decreasing
functions and with $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ the dual space. If
$\beta$ is the multiindex $\beta=(\beta_{1},...,\beta_{n})$, then
$|\beta|=\beta_{1}+...+\beta_{n}.$

##  2 Preliminaries

Let $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a measurable function. Given a
measurable set $E$, let

$p_{-}(E)=\mathop{\rm ess\,inf}_{x\in
E}p(x),\,\,\,\,\text{and}\,\,\,\,p_{+}(E)=\mathop{\rm ess\,sup}_{x\in E}p(x).$

When $E=\mathbb{R}^{n}$, we will simply write $p_{-}:=p_{-}(\mathbb{R}^{n})$
and $p_{+}:=p_{+}(\mathbb{R}^{n})$.

Given a measurable function $f$ on $\mathbb{R}^{n}$, define the modular $\rho$
associated with $p(\cdot)$ by

$\rho(f)=\int_{\mathbb{R}^{n}}|f(x)|^{p(x)}dx.$

We define the variable Lebesgue space
$L^{p(\cdot)}=L^{p(\cdot)}(\mathbb{R}^{n})$ to be the set of all measurable
functions $f$ such that, for some $\lambda>0$,
$\rho\left(f/\lambda\right)<\infty$. This becomes a quasi normed space when
equipped with the Luxemburg norm

$\|f\|_{L^{p(\cdot)}}=\inf\left\\{\lambda>0:\rho\left(f/\lambda\right)\leq
1\right\\}.$

Given a weight $\omega$, i.e.: a locally integrable function on
$\mathbb{R}^{n}$ such that $0<\omega(x)<\infty$ almost everywhere, we define
the weighted variable Lebesgue space $L^{p(\cdot)}_{\omega}$ as the set of all
measurable functions $f:\mathbb{R}^{n}\to\mathbb{C}$ such that
$\|f\omega\|_{L^{p(\cdot)}}<\infty$. If $f\in L^{p(\cdot)}_{\omega}$, we
define its ”norm” by

$\|f\|_{L^{p(\cdot)}_{\omega}}:=\|f\omega\|_{L^{p(\cdot)}}.$

######  Lemma 2.1.

Given a measurable function $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ with
$0<p_{-}\leq p_{+}<\infty$ and a weight $\omega$, then for every $s>0$,

$\|f\|_{L^{p(\cdot)}_{\omega}}^{s}=\||f|^{s}\|_{L^{p(\cdot)/s}_{\omega^{s}}}.$

###### Proof..

The condition $p_{+}<\infty$ implies that $|\\{x:p(x)=\infty\\}|=0$. Then, for
every $s>0$,

$\||f|^{s}\|_{L^{p(\cdot)/s}_{\omega^{s}}}=\inf\left\\{\lambda>0:\int\left(\frac{|f(x)|\,\omega(x)}{\lambda^{1/s}}\right)^{p(x)}dx\leq
1\right\\}$
$=\inf\left\\{\mu^{s}>0:\int\left(\frac{|f(x)|\,\omega(x)}{\mu}\right)^{p(x)}dx\leq
1\right\\}=\|f\|_{L^{p(\cdot)}_{\omega}}^{s}.$

∎

For a measurable function $p(\cdot):\mathbb{R}^{n}\to[1,\infty)$, its
conjugate function $p^{\prime}(\cdot)$ is defined by
$\frac{1}{p(x)}+\frac{1}{p^{\prime}(x)}=1$. We have the following
generalization of Hölder’s inequality and an equivalent expression for the
$L^{p(\cdot)}_{\omega}$-norm.

######  Lemma 2.2.

(Hölder’s inequality) Let $p(\cdot):\mathbb{R}^{n}\to[1,\infty)$ be a
measurable function and $\omega$ be a locally integrable function such that
$0<\omega(x)<\infty$ almost everywhere. Then, there exists a constant $C>0$
such that

$\int_{\mathbb{R}^{n}}|f(x)g(x)|dx\leq
C\|f\|_{L^{p(\cdot)}_{\omega}}\|g\|_{L^{p^{\prime}(\cdot)}_{\omega^{-1}}}.$

###### Proof..

The lemma follows from [6, Lemma 3.2.20]. ∎

######  Proposition 2.3.

Let $p(\cdot):\mathbb{R}^{n}\to[1,\infty)$ be a measurable function and
$\omega$ be a locally integrable function such that $0<\omega(x)<\infty$
almost everywhere. Then

$\|f\|_{L^{p(\cdot)}_{\omega}}\approx\sup\left\\{\int_{\mathbb{R}^{n}}|f(x)g(x)|dx:\|g\|_{L^{p^{\prime}(\cdot)}_{\omega^{-1}}}\leq
1\right\\}.$

###### Proof..

The proposition follows from [6, Corollary 3.2.14]. ∎

Now, we briefly present the basics of weighted variable Hardy spaces theory
and recall the atomic decomposition developed by K.-P. Ho in [12].

We introduce the weights used in [12] to define weighted Hardy spaces with
variable exponents.

######  Definition 2.4.

Let $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a measurable function with
$0<p_{-}\leq p_{+}<\infty$. We define $\mathcal{W}_{p(\cdot)}$ as the set of
all weights $\omega$ such that

$(i)\,\,\text{there exists}\,\,0<p_{\ast}<\min\\{1,p_{-}\\}\,\,\text{such
that}\,\,\|\chi_{Q}\|_{L^{p(\cdot)/p_{\ast}}_{\omega^{p_{\ast}}}}<\infty,\,\,\,\text{and}$
$\|\chi_{Q}\|_{L^{(p(\cdot)/p_{\ast})^{\prime}}_{\omega^{-p_{\ast}}}}<\infty,\,\,\,\text{for
all cube}\,\,Q;$ $(ii)\,\,\text{there exist $\kappa>1$ and $s>1$ such that
Hardy-Littlewood maximal}$ $\text{operator $M$ is bounded on
$L^{(sp(\cdot))^{\prime}/\kappa}_{\omega^{-\kappa/s}}$}.$

######  Remark 2.5.

In [12, Definition 2.3], the author considers $p_{\ast}=\min\\{1,p_{-}\\}$. We
observe that the whole theory is still valid if we take
$0<p_{\ast}<\min\\{1,p_{-}\\}$ instead of $p_{\ast}=\min\\{1,p_{-}\\}$. This
slight modification allows us to compare the sets $\mathcal{W}_{q(\cdot)}$ and
$\mathcal{W}_{p(\cdot)}$ related by
$\frac{1}{p(\cdot)}-\frac{1}{q(\cdot)}=\frac{\alpha}{n}$ with $0<\alpha<n$
(see Proposition 3.1 below).

For a measurable function $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ such that
$0<p_{-}\leq p_{+}<\infty$ and $\omega\in\mathcal{W}_{p(\cdot)}$, in [12] the
author give a variety of distinct approaches, based on differing definitions,
all lead to the same notion of weighted variable Hardy space
$H^{p(\cdot)}_{\omega}$.

We recall some terminologies and notations from the study of maximal
functions. Given $N\in\mathbb{N}$, define

$\mathcal{F}_{N}=\left\\{\varphi\in\mathcal{S}(\mathbb{R}^{n}):\sum\limits_{\left|\mathbf{\beta}\right|\leq
N}\sup\limits_{x\in\mathbb{R}^{n}}\left(1+\left|x\right|\right)^{N}\left|\partial^{\mathbf{\beta}}\varphi(x)\right|\leq
1\right\\}.$

For any $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$, the grand maximal function
of $f$ is given by

$\mathcal{M}f(x)=\sup\limits_{t>0}\sup\limits_{\varphi\in\mathcal{F}_{N}}\left|\left(\varphi_{t}\ast
f\right)(x)\right|,$

where $\varphi_{t}(x)=t^{-n}\varphi(t^{-1}x)$.

######  Definition 2.6.

Let $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$, $0<p_{-}\leq p_{+}<\infty$, and
$\omega\in\mathcal{W}_{p(\cdot)}$. The weighted variable Hardy space
$H^{p(.)}_{\omega}(\mathbb{R}^{n})$ is the set of all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ for which
$\|\mathcal{M}f\|_{L^{p(\cdot)}_{\omega}}<\infty$. In this case we define
$\|f\|_{H^{p(\cdot)}_{\omega}}:=\|\mathcal{M}f\|_{L^{p(\cdot)}_{\omega}}$.

######  Definition 2.7.

Let $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$, $0<p_{-}\leq p_{+}<\infty$,
$p_{0}>1$, and $\omega\in\mathcal{W}_{p(\cdot)}$. Fix an integer $d\geq 1$. A
function $a(\cdot)$ on $\mathbb{R}^{n}$ is called a
$\omega-(p(\cdot),p_{0},d)$ atom if there exists a cube $Q$ such that

$a_{1})$ $\mathop{\rm supp}(a)\subset Q$,

$a_{2})$
$\|a\|_{L^{p_{0}}}\leq\frac{|Q|^{\frac{1}{p_{0}}}}{\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}}$,

$a_{3})$ $\displaystyle{\int}x^{\beta}a(x)\,dx=0$ for all $|\beta|\leq d$.

Next, we introduce two indices, which are related to the intrinsic structure
of the atomic decomposition of $H^{p(\cdot)}_{\omega}$. Given
$\omega\in\mathcal{W}_{p(\cdot)}$, we write

$s_{\omega,\,p(\cdot)}:=\inf\left\\{s\geq 1:M\,\,\text{is bounded
on}\,\,L^{(sp(\cdot))^{\prime}}_{\omega^{-1/s}}\right\\}$

and

$\mathbb{S}_{\omega,\,p(\cdot)}:=\left\\{s\geq 1:M\,\,\text{is bounded
on}\,\,L^{(sp(\cdot))^{\prime}/\kappa}_{\omega^{-\kappa/s}}\,\,\text{for
some}\,\,\kappa>1\right\\}.$

Then, for every fixed $s\in\mathbb{S}_{\omega,\,p(\cdot)}$, we define

$\kappa_{\omega,\,p(\cdot)}^{s}:=\sup\left\\{\kappa>1:M\,\,\text{is bounded
on}\,\,L^{(sp(\cdot))^{\prime}/\kappa}_{\omega^{-\kappa/s}}\right\\}.$

The index $\kappa_{\omega,\,p(\cdot)}^{s}$ is used to measure the left-
openness of the boundedness of $M$ on the family
$\left\\{L^{(sp(\cdot))^{\prime}/\kappa}_{\omega^{-\kappa/s}}\right\\}_{\kappa>1}$.
The index $s_{\omega,\,p(\cdot)}$ is related to the vanishing moment condition
and the index $\kappa_{\omega,\,p(\cdot)}^{s}$ is related to the size
condition of the atoms (see [12, Theorems 5.3 and 6.3]).

The following three results are crucial in obtaining the
$H^{p(\cdot)}_{\omega}-L^{q(\cdot)}_{\omega}$ boundedness of the Riesz
potential. The first is a supporting result [12, Lemma 5.4], the second is a
combination of [12, Theorem 6.2] and [16, Theorem 3.1], the last one was
proved in [13, Proposition 2.1].

######  Proposition 2.8.

([12, Lemma 5.4]) Let $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a measurable
function with $0<p_{-}\leq p_{+}<\infty$ and
$\omega\in\mathcal{W}_{p(\cdot)}$. Let $s\in\mathbb{S}_{\omega,\,p(\cdot)}$
and $\\{\lambda_{j}\\}_{j\in\mathbb{N}}$ be a sequence of scalars. If
$r>(\kappa_{\omega,\,p(\cdot)}^{s})^{\prime}$, $\\{b_{j}\\}_{j\in\mathbb{N}}$
is a sequence in $L^{r}$ such that, for every $j$, $\mathop{\rm supp}(b_{j})$
is contained in a cube $Q_{j}$ and

$\|b_{j}\|_{L^{r}}\leq A_{j}|Q_{j}|^{1/r},$

where the $A_{j}$’s are positive scalars for all $j\in\mathbb{N}$, then

$\left\|\sum_{j\in\mathbb{N}}\lambda_{j}b_{j}\right\|_{L^{sp(\cdot)}_{\omega^{1/s}}}\leq
C\left\|\sum_{j\in\mathbb{N}}A_{j}|\lambda_{j}|\chi_{Q_{j}}\right\|_{L^{sp(\cdot)}_{\omega^{1/s}}}$

for some $C>0$ independent of $\\{\lambda_{j}\\}_{j\in\mathbb{N}}$,
$\\{b_{j}\\}_{j\in\mathbb{N}}$ and $\\{A_{j}\\}_{j\in\mathbb{N}}$.

######  Theorem 2.9.

Let $1<p_{0}<\infty$, $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a measurable
function with $0<p_{-}\leq p_{+}<\infty$ and
$\omega\in\mathcal{W}_{p(\cdot)}$. Then, for every $s>1$ fixed, and every
$f\in H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$ there
exist a sequence of scalars $\\{\lambda_{j}\\}$, a sequence of cubes
$\\{Q_{j}\\}$ and $\omega-(p(\cdot),p_{0},\lfloor
ns_{\omega,\,p(\cdot)}-n\rfloor)$ atoms $a_{j}$ supported on $Q_{j}$ such that

(2)
$\left\|\sum_{j}\left(\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\chi_{Q_{j}}\right\|_{L^{p(\cdot)/\theta}_{\omega^{\theta}}}^{1/\theta}\leq
C\|f\|_{H^{p(\cdot)}_{\omega}},\,\,\,\text{for all}\,\,\,0<\theta<\infty,$

and $f=\sum_{j}\lambda_{j}a_{j}$ converges in $L^{s}(\mathbb{R}^{n})$.

###### Proof..

The existence of a such atomic decomposition as well as the validity of
inequality in (2) are guaranteed by [12, Theorem 6.2]. The construction of a
such atomic decomposition is the same that the given for the classical Hardy
spaces (see [19, Chapter III]). So, following the proof in [16, Theorem 3.1],
we obtain the convergence in $L^{s}(\mathbb{R}^{n})$. ∎

We define the set $\mathcal{S}_{0}(\mathbb{R}^{n})$ by

$\mathcal{S}_{0}(\mathbb{R}^{n})=\left\\{\varphi\in\mathcal{S}(\mathbb{R}^{n}):\int
x^{\beta}\varphi(x)dx=0,\,\,\text{for
all}\,\beta\in\mathbb{N}_{0}^{n}\right\\}.$

We say that an exponent function $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ such
that $0<p_{-}\leq p_{+}<\infty$ belongs to
$\mathcal{P}^{log}(\mathbb{R}^{n})$, if there exist two positive constants $C$
and $C_{\infty}$ such that $p(\cdot)$ satisfies the local log-Hölder
continuity condition, i.e.:

$|p(x)-p(y)|\leq\frac{C}{-\log(|x-y|)},\,\,\,|x-y|\leq\frac{1}{2},$

and is log-Hölder continuous at infinity, i.e.:

$|p(x)-p_{\infty}|\leq\frac{C_{\infty}}{\log(e+|x|)},\,\,\,x\in\mathbb{R}^{n},$

for some $p_{-}\leq p_{\infty}\leq p_{+}$.

######  Theorem 2.10.

([13, Proposition 2.1]) Let $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$
with $0<p_{-}\leq p_{+}<\infty$. If $\omega\in\mathcal{W}_{p(\cdot)}$, then
$\mathcal{S}_{0}(\mathbb{R}^{n})\subset H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$
densely.

##  3 Auxiliary Results

In this section we prove some basics properties of the set
$\mathcal{W}_{p(\cdot)}$ and some results about the size of the cubes in the
$L^{p(\cdot)}_{\omega}$-norm.

######  Proposition 3.1.

Let $0<\alpha<n$ and let $q(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a
measurable function such that $0<q_{-}\leq q_{+}<\infty$. If
$\omega\in\mathcal{W}_{q(\cdot)}$ and
$\frac{1}{p(\cdot)}:=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$, then
$\omega\in\mathcal{W}_{p(\cdot)}$. Moreover, $s_{\omega,\,p(\cdot)}\leq
s_{\omega,\,q(\cdot)}+\frac{\alpha}{n}$.

###### Proof..

By Definition 2.4, $\omega\in\mathcal{W}_{q(\cdot)}$ if and only if

$(i)\,\,\text{there exists}\,\,0<q_{\ast}<\min\\{1,q_{-}\\}\,\,\text{such
that}\,\,\|\chi_{Q}\|_{L^{q(\cdot)/q_{\ast}}_{\omega^{q_{\ast}}}}<\infty,\,\,\,\text{and}$
$\|\chi_{Q}\|_{L^{(q(\cdot)/q_{\ast})^{\prime}}_{\omega^{-q_{\ast}}}}<\infty,\,\,\,\text{for
all cube}\,\,Q;$ $(ii)\,\,\text{there exist $\kappa>1$ and $s>1$ such that
Hardy-Littlewood maximal}$ $\text{operator $M$ is bounded on
$L^{(sq(\cdot))^{\prime}/\kappa}_{\omega^{-\kappa/s}}$}.$

We define $\frac{1}{p_{\ast}}:=\frac{1}{q_{\ast}}+\frac{\alpha}{n}$. Since
$\frac{1}{p(\cdot)}-\frac{1}{q(\cdot)}=\frac{1}{p_{\ast}}-\frac{1}{q_{\ast}}$,
it follows that $0<p_{\ast}<\min\\{1,p_{-}\\}$ and
$\frac{p_{\ast}}{q_{\ast}}\left(\frac{p(\cdot)}{p_{\ast}}\right)^{\prime}=\left(\frac{q(\cdot)}{q_{\ast}}\right)^{\prime}$.
So, from Lemma 2.1 and (i) above, we have

$\|\chi_{Q}\|_{L^{\left(\frac{p(\cdot)}{p_{\ast}}\right)^{\prime}}_{\omega^{-p_{\ast}}}}=\|\chi_{Q}\|_{L^{\frac{p_{\ast}}{q_{\ast}}\left(\frac{p(\cdot)}{p_{\ast}}\right)^{\prime}}_{\omega^{-q_{\ast}}}}^{\frac{p_{\ast}}{q_{\ast}}}=\|\chi_{Q}\|_{L^{\left(\frac{q(\cdot)}{q_{\ast}}\right)^{\prime}}_{\omega^{-q_{\ast}}}}^{\frac{p_{\ast}}{q_{\ast}}}<\infty.$

Being $p(\cdot)\leq q(\cdot)$, by [4, Corollary 2.48], Lemma 2.1 and (i), we
have

$\|\chi_{Q}\|_{L^{p(\cdot)/p_{\ast}}_{\omega^{p_{\ast}}}}^{q_{\ast}/p_{\ast}}=\|\chi_{Q}\,\omega^{p_{\ast}}\|_{L^{p(\cdot)/p_{\ast}}}^{q_{\ast}/p_{\ast}}\leq(1+|Q|)^{q_{\ast}/p_{\ast}}\|\chi_{Q}\,\omega^{p_{\ast}}\|_{L^{q(\cdot)/p_{\ast}}}^{q_{\ast}/p_{\ast}}$
$=(1+|Q|)^{q_{\ast}/p_{\ast}}\|\chi_{Q}\|_{L^{q(\cdot)/q_{\ast}}_{\omega^{q_{\ast}}}}<\infty.$

From $(ii)$ follows that the maximal operator $M$ is bounded on
$L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}$, with $s>1$. We fix
$r>s+\frac{\alpha}{n}$, and define $q_{0}:=\frac{r}{s}$ and
$\frac{1}{p_{0}}:=\frac{1}{q_{0}}+\frac{\alpha}{nr}$. It is clear that
$\frac{r}{p_{0}}>1$, $\frac{q_{0}}{p_{0}}>1$ and
$\frac{p_{0}}{q_{0}}\left(\frac{r}{p_{0}}p(\cdot)\right)^{\prime}=\left(\frac{r}{q_{0}}q(\cdot)\right)^{\prime}=(sq(\cdot))^{\prime}$.
Thus, for $\tilde{s}:=\frac{r}{p_{0}}$ and
$\tilde{\kappa}:=\frac{q_{0}}{p_{0}}$, we obtain

$\|Mf\|_{L^{(\tilde{s}p(\cdot))^{\prime}/\tilde{\kappa}}_{\omega^{-\tilde{\kappa}/\tilde{s}}}}=\|Mf\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}\leq
C\|f\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}=C\|f\|_{L^{(\tilde{s}p(\cdot))^{\prime}/\tilde{\kappa}}_{\omega^{-\tilde{\kappa}/\tilde{s}}}}.$

So, $\omega\in\mathcal{W}_{p(\cdot)}$ and $s_{\omega,\,p(\cdot)}\leq
s_{\omega,\,q(\cdot)}+\frac{\alpha}{n}$. ∎

The following necessary condition is due to Cruz-Uribe, Fiorenza and
Neugebauer (see [3, Theorem 1.5]). It should be compared to the Muckenhoupt
$\mathcal{A}_{p}$ condition from the study of weighted norm inequalities (see
[9, Chapter 7]).

######  Lemma 3.2.

Given a weight $\omega$ and an exponent function
$p(\cdot):\mathbb{R}^{n}\to(1,\infty)$ such that the Hardy-Littlewood maximal
operator is bounded on $L^{p(\cdot)}_{w}(\mathbb{R}^{n})$, then, there exists
a positive constant $C$ such that for every cube $Q\subset\mathbb{R}^{n}$,

(3)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\|\chi_{Q}\|_{L^{p^{\prime}(\cdot)}_{\omega^{-1}}}\leq
C|Q|.$

######  Definition 3.3.

Given an exponent function $p(\cdot):\mathbb{R}^{n}\to(1,\infty)$ and a weight
$\omega$, we write $\omega\in\mathcal{A}_{p(\cdot)}$ if $\omega$ satisfies
(3).

######  Lemma 3.4.

Let $p(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a measurable function with
$0<p_{-}\leq p_{+}<\infty$. If $\omega\in\mathcal{W}_{p(\cdot)}$, then, for
every cube $Q\subset\mathbb{R}^{n}$,

$\|\chi_{2Q}\|_{L^{p(\cdot)}_{\omega}}\approx\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}.$

###### Proof..

By the order preserving property of the norm
$\|\,\cdot\,\|_{L^{p(\cdot)}_{\omega}}$, we have that

(4)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\leq\|\chi_{2Q}\|_{L^{p(\cdot)}_{\omega}}.$

On the other hand, since $\omega\in\mathcal{W}_{p(\cdot)}$, the maximal
operator is bounded on $L^{(sp(\cdot))^{\prime}}_{\omega^{-1/s}}$. Then, by
Lemma 3.2 (exchanging the roles of $p(\cdot)$ and $p^{\prime}(\cdot)$), (4)
above, and Hölder’s inequality applied to $|Q|=\int\chi_{Q}(x)dx$ (see Lemma
2.2), we have that

$\|\chi_{2Q}\|^{1/s}_{L^{p(\cdot)}_{\omega}}=\|\chi_{2Q}\|_{L^{sp(\cdot)}_{\omega^{1/s}}}\leq
C|Q|\|\chi_{2Q}\|^{-1}_{L^{(sp(\cdot))^{\prime}}_{\omega^{-1/s}}}\leq
C|Q|\|\chi_{Q}\|^{-1}_{L^{(sp(\cdot))^{\prime}}_{\omega^{-1/s}}}$ $\leq
C\|\chi_{Q}\|_{L^{sp(\cdot)}_{\omega^{1/s}}}=C\|\chi_{Q}\|^{1/s}_{L^{p(\cdot)}_{\omega}}.$

This completes the proof. ∎

A weight $\omega$ satisfies the reverse Hölder inequality with exponent $s>1$,
denoted by $\omega\in RH_{s}$, if there exists a constant $C>0$ such that for
every cube $Q$,

$\left(\frac{1}{|Q|}\int_{Q}[\omega(x)]^{s}dx\right)^{\frac{1}{s}}\leq
C\frac{1}{|Q|}\int_{Q}\omega(x)dx;$

the best possible constant is denoted by $[\omega]_{RH_{s}}$. We observe that
if $\omega\in RH_{s}$, then by Hölder’s inequality, $\omega\in RH_{t}$ for all
$1<t<s$, and $[\omega]_{RH_{t}}\leq[\omega]_{RH_{s}}$.

######  Lemma 3.5.

Let $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ with $0<p_{-}\leq
p_{+}<\infty$, $\gamma\in\mathbb{R}$, and let $f\in
L^{1}_{loc}(\mathbb{R}^{n})$ be a function such that
$|f(x)|\lesssim(1+|x|)^{\gamma}$. Then
$\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\approx\|f\|_{L^{p_{\infty}}(\mathbb{R}^{n})}$.

###### Proof..

We take $s>1/p_{-}$. Then, by applying [7, Lemma 2.7] with $\omega\equiv 1$,
we get
$\|f\|^{1/s}_{L^{p(\cdot)}}=\||f|^{1/s}\|_{L^{sp(\cdot)}}\approx\||f|^{1/s}\|_{L^{sp_{\infty}}}=\|f\|^{1/s}_{L^{p_{\infty}}}$.
∎

######  Lemma 3.6.

Let $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ with $1<p_{-}\leq
p_{+}<\infty$ and $p_{\infty}=p_{-}$, and let $\omega$ be a weight such that
$\omega(x)\lesssim(1+|x|)^{\gamma}$ for some $\gamma\in\mathbb{R}$. If
$\omega\in RH_{p_{+}}$, then for every cube $Q$ we have

(5)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\approx\left\\{\begin{array}[]{c}\,\,\,\left[\omega^{p_{-}}(Q)\right]^{1/p_{-}},\hskip
28.45274pt\text{if}\,\,\ell(Q)>1\\\ \\\
\left[\omega^{p_{-}(Q)}(Q)\right]^{1/p_{-}(Q)},\,\,\text{if}\,\,\ell(Q)\leq
1\end{array}\right..$

###### Proof..

Applying Lemma 3.5, with $f=\chi_{Q}\,\omega$, we obtain

(6)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}=\|\chi_{Q}\,\omega\|_{L^{p(\cdot)}}\approx\left[\omega^{p_{-}}(Q)\right]^{1/p_{-}},$

for all cube $Q$.

From (6) and since $1<p_{-}\leq p_{-}(Q)$, by Hölder’s inequality, we have

$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}^{p_{-}}\approx\int_{Q}[\omega(x)]^{p_{-}}dx\leq\left[\omega^{p_{-}(Q)}(Q)\right]^{\frac{p_{-}}{p_{-}(Q)}}\|\chi_{Q}\|_{\left(\frac{p_{-}(Q)}{p_{-}}\right)^{\prime}}\leq\left[\omega^{p_{-}(Q)}(Q)\right]^{\frac{p_{-}}{p_{-}(Q)}},$

if $\ell(Q)\leq 1$. So,

(7)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\lesssim\left[\omega^{p_{-}(Q)}(Q)\right]^{\frac{1}{p_{-}(Q)}},$

holds for every cube $Q$ with $\ell(Q)\leq 1$.

On the other hand, $\omega\in RH_{p_{+}}$ and $p_{-}(Q)\leq p_{+}$, then
$\omega\in RH_{p_{-}(Q)}$ and

$\left(\frac{1}{|Q|}\int_{Q}[\omega(x)]^{p_{-}(Q)}dx\right)^{1/p_{-}(Q)}\leq
C|Q|^{-1}\int_{Q}\omega(x)dx\leq
C|Q|^{-1}\|\chi_{Q}\,\omega\|_{L^{p(\cdot)}}\|\chi_{Q}\|_{L^{p^{\prime}(\cdot)}},$

[15, Lemma 2.2] gives
$\|\chi_{Q}\|_{L^{p^{\prime}(\cdot)}}\approx|Q|^{1/p^{\prime}_{+}(Q)}$ if
$\ell(Q)\leq 1$, being $p^{\prime}_{+}(Q)=(p_{-}(Q))^{\prime}$, we obtain

(8)
$\left[\omega^{p_{-}(Q)}(Q)\right]^{1/p_{-}(Q)}\lesssim\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}},\,\,\,\,\text{if}\,\,\,\ell(Q)\leq
1.$

Finally, (6), (7) and (8) give (5). ∎

######  Corollary 3.7.

Let $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ with $0<p_{-}\leq
p_{+}<\infty$ and $p_{\infty}=p_{-}$, and let $s>1/p_{-}$. If
$\omega(x)\lesssim(1+|x|)^{\gamma}$ for some $\gamma\in\mathbb{R}$ and
$\omega^{1/s}\in RH_{sp_{+}}$, then (5) holds.

###### Proof..

Since $(sp(\cdot))_{+}=sp_{+}$, $(sp(\cdot))_{-}=sp_{-}$ and
$(sp(\cdot))_{-}(Q)=sp_{-}(Q)$, Lemma 3.6, applied to $sp(\cdot)$ and
$\omega^{1/s}$, gives

$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}^{1/s}=\|\chi_{Q}\|_{L^{sp(\cdot)}_{\omega^{1/s}}}\approx\left\\{\begin{array}[]{c}\,\,\,\left[\omega^{p_{-}}(Q)\right]^{1/sp_{-}},\hskip
28.45274pt\text{if}\,\,\ell(Q)>1\\\ \\\
\left[\omega^{p_{-}(Q)}(Q)\right]^{1/sp_{-}(Q)},\,\,\text{if}\,\,\ell(Q)\leq
1\end{array}\right..$

So, the lemma follows. ∎

######  Lemma 3.8.

Let $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ with $1<p_{-}\leq
p_{+}<\infty$ and $p_{\infty}=p_{+}$. If $\omega\in RH_{p_{+}}$, then for
every cube $Q$ we have

(9)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\approx\left\\{\begin{array}[]{c}\,\,\,\left[\omega^{p_{+}}(Q)\right]^{1/p_{+}},\hskip
28.45274pt\text{if}\,\,\ell(Q)>1\\\ \\\
\left[\omega^{p_{+}(Q)}(Q)\right]^{1/p_{+}(Q)},\,\,\text{if}\,\,\ell(Q)\leq
1\end{array}\right..$

###### Proof..

By hypothesis $p_{\infty}=p_{+}$, thus there exists $C_{\infty}>0$ such that

$0\leq p_{+}-p(x)\leq\frac{C_{\infty}}{\log(e+|x|)},\,\,\,\text{for
all}\,\,x\in\mathbb{R}^{n}.$

Let $\lambda=[\omega^{p_{+}}(Q)]^{1/p_{+}}$, by applying [3, Lemma 2.8] with
$t=2/p_{-}$, we have

$\int_{Q}\left(\frac{\omega(x)}{\lambda}\right)^{p(x)}dx\leq
C_{2/p_{-}}\int_{Q}\left(\frac{\omega(x)}{\lambda}\right)^{p_{+}}dx+\int_{Q}\frac{dx}{(e+|x|)^{2n}}$
$\leq C_{2/p_{-}}+\int_{\mathbb{R}^{n}}\frac{dx}{(e+|x|)^{2n}}=:M<\infty.$

So, from the definition of the ”norm” $\|\cdot\|_{L^{p(\cdot)}_{\omega}}$ we
have

(10)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\leq\max\\{M^{1/p_{-}},M^{1/p_{+}}\\}[\omega^{p_{+}}(Q)]^{1/p_{+}},\,\,\,\text{for
all cube}\,\,Q.$

Now, if $\ell(Q)>1$, the condition $\omega\in RH_{p_{+}}$, Hölder’s inequality
and [15, Lemma 2.2] give

$\left(\frac{1}{|Q|}\int_{Q}[\omega(x)]^{p_{+}}dx\right)^{1/p_{+}}\leq\frac{C}{|Q|}\int_{Q}\omega(x)dx$
$\leq\frac{C}{|Q|}\|\chi_{Q}\,\omega\|_{L^{p(\cdot)}}\|\chi_{Q}\|_{L^{p^{\prime}(\cdot)}}\leq\frac{C}{|Q|}\|\chi_{Q}\,\omega\|_{L^{p(\cdot)}}|Q|^{1-\frac{1}{p_{+}}}.$

Consequently,

(11) $[\omega^{p_{+}}(Q)]^{1/p_{+}}\leq
C\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}},\,\,\,\text{if}\,\,\ell(Q)>1.$

Then, (10) and (11) give

(12)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\approx[\omega^{p_{+}}(Q)]^{1/p_{+}},\,\,\,\text{if}\,\,\ell(Q)>1.$

Now, we study the case $\ell(Q)\leq 1$. It is easy to check that

$0\leq p_{+}(Q)-p(x)\leq\frac{C}{\log(e+|x|)},\,\,\,\text{for all}\,\,x\in Q.$

Since $1<p_{+}(Q)\leq p_{+}$, we have that $\omega\in RH_{p_{+}(Q)}$. Then, by
reasoning as above, we get

(13)
$\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}\approx\left[\omega^{p_{+}(Q)}(Q)\right]^{1/p_{+}(Q)},\,\,\,\text{if}\,\,\ell(Q)\leq
1.$

Finally, (12) and (13) give (9). ∎

######  Corollary 3.9.

Let $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ with $0<p_{-}\leq
p_{+}<\infty$ and $p_{\infty}=p_{+}$, and let $s>1/p_{-}$. If $\omega^{1/s}\in
RH_{sp_{+}}$, then (9) holds.

###### Proof..

The proof follows from Lemmas 2.1 and 3.8. ∎

##  4 Off-diagonal Fefferman-Stein inequality

We apply extrapolation techniques to obtain an ”off-diagonal” version of the
Fefferman-Stein vector-valued maximal inequality on $L^{p(\cdot)}_{\omega}$.
The following result generalizes Theorem 3.1 obtained in [12].

######  Theorem 4.1.

Let $0\leq\alpha<n$, $1<u<\infty$ and let
$q(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a measurable function with
$0<q_{-}\leq q_{+}<\infty$. If $\omega\in\mathcal{W}_{q(\cdot)}$, then for
$\frac{1}{p(\cdot)}:=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$ and any
$r>s_{\omega,\,q(\cdot)}+\frac{\alpha}{n}$,

(14)
$\left\|\left(\sum_{j\in\mathbb{N}}(M_{\frac{\alpha}{r}}f_{j})^{u}\right)^{1/u}\right\|_{L^{rq(\cdot)}_{\omega^{1/r}}}\lesssim\left\|\left(\sum_{j\in\mathbb{N}}|f_{j}|^{u}\right)^{1/u}\right\|_{L^{rp(\cdot)}_{\omega^{1/r}}},$

holds for all sequences of bounded measurable functions with compact support
$\\{f_{j}\\}_{j=1}^{\infty}$.

###### Proof..

Given $r>s_{\omega,\,q(\cdot)}+\frac{\alpha}{n}$, from the definition of
$s_{\omega,\,q(\cdot)}$, we have $s>s_{\omega,\,q(\cdot)}$ such that
$s+\frac{\alpha}{n}<r$ and $M$ is bounded on
$L^{(sq(\cdot))^{\prime}}_{w^{-1/s}}(\mathbb{R}^{n})$. Define

$\mathcal{F}=\left\\{\left(\left(\sum_{j=1}^{K}(M_{\frac{\alpha}{r}}f_{j})^{u}\right)^{1/u},\left(\sum_{j=1}^{K}|f_{j}|^{u}\right)^{1/u}\right):K\in\mathbb{N},\\{f_{j}\\}_{j=1}^{K}\subset
L^{\infty}_{comp}\right\\},$

where $L^{\infty}_{comp}$ denotes the set of bounded functions with compact
support.

Let $q_{0}=\frac{r}{s}$ and let $p_{0}$ be defined by
$\frac{1}{p_{0}}:=\frac{1}{q_{0}}+\frac{\alpha}{nr}$. Since
$r>s+\frac{\alpha}{n}$ we have that $1<p_{0}<\frac{nr}{\alpha}$. From [14,
Theorem 3] follows that there exists an universal constant $C>0$ such that for
any $(F,G)\in\mathcal{F}$ and any $v\in\mathcal{A}_{1}$ (for the definition of
the $\mathcal{A}_{1}$ class, the reader may refer to [9, Chapter 7])

(15) $\int[F(x)]^{q_{0}}v(x)\,dx\leq
C\left(\int[G(x)]^{p_{0}}[v(x)]^{p_{0}/q_{0}}\,dx\right)^{q_{0}/p_{0}},$

since $v\in\mathcal{A}_{1}$ implies that
$v^{1/q_{0}}\in\mathcal{A}_{p_{0},q_{0}}$ (for the definition of the
$\mathcal{A}_{p,q}$ class, see [14, inequality (1.1)]). On the other hand, by
Proposition 2.3, we have

(16)
$\|F\|^{q_{0}}_{L^{rq(\cdot)}_{\omega^{1/r}}}=\|F^{q_{0}}\|_{L^{sq(\cdot)}_{\omega^{1/s}}}\leq
C\sup\left\\{\int_{\mathbb{R}^{n}}\left|[F(x)]^{q_{0}}g(x)\right|dx:\|g\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}\leq
1\right\\}$

for some constant $C>0$.

Let $\mathcal{R}$ be the operator defined on
$L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}$ by

$\mathcal{R}g(x)=\sum_{k=0}^{\infty}\frac{M^{k}g(x)}{2^{k}\|M\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}^{k}},$

where, for $k\geq 1$, $M^{k}$ denotes $k$ iterations of the Hardy-Littlewood
maximal operator $M$, $M^{0}=M$, and
$\|M\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}$ is the operator norm of the
maximal operator $M$ on $L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}$. It follows
immediately from this definition that:

$(i)$ if $g$ is non-negative, $g(x)\leq\mathcal{R}g(x)$ a.e.
$x\in\mathbb{R}^{n}$;

$(ii)$ $\|\mathcal{R}g\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}\leq
2\|g\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}$;

$(iii)$ $\mathcal{R}g\in\mathcal{A}_{1}$ with
$[\mathcal{R}g]_{\mathcal{A}_{1}}\leq
2\|M\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}$.
Since $F$ is non-negative, we can take the supremum in (16) over those non-
negative $g$ only. For any fixed non-negative $g\in
L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}$, by $(i)$ above we have that

(17) $\int[F(x)]^{q_{0}}g(x)dx\leq\int[F(x)]^{q_{0}}(\mathcal{R}g)(x)dx.$

Then $(iii)$ and (15), and Hölder’s inequality yield

(18) $\int[F(x)]^{q_{0}}(\mathcal{R}g)(x)dx\leq
C\left(\int[G(x)]^{p_{0}}[(\mathcal{R}g)(x)]^{p_{0}/q_{0}}dx\right)^{q_{0}/p_{0}}$
$\leq
C\|G^{p_{0}}\|_{L^{rp(\cdot)/p_{0}}_{\omega^{p_{0}/r}}}^{q_{0}/p_{0}}\|(\mathcal{R}g)^{p_{0}/q_{0}}\|_{L^{(rp(\cdot)/p_{0})^{\prime}}_{\omega^{-p_{0}/r}}}^{q_{0}/p_{0}}$
$=C\|G\|^{q_{0}}_{L^{rp(\cdot)}_{\omega^{1/r}}}\|\mathcal{R}g\|_{L^{\frac{p_{0}}{q_{0}}\left(\frac{rp(\cdot)}{p_{0}}\right)^{\prime}}_{\omega^{-q_{0}/r}}}$

being
$\frac{1}{rp(\cdot)}-\frac{1}{rq(\cdot)}=\frac{1}{p_{0}}-\frac{1}{q_{0}}$ and
$q_{0}=\frac{r}{s}$, we have
$\frac{p_{0}}{q_{0}}\left(\frac{r}{p_{0}}p(\cdot)\right)^{\prime}=\left(\frac{r}{q_{0}}q(\cdot)\right)^{\prime}=(sq(\cdot))^{\prime}$,
so

$=C\|G\|^{q_{0}}_{L^{rp(\cdot)}_{\omega^{1/r}}}\|\mathcal{R}g\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}$

now, $(ii)$ gives

$\leq
C\|G\|^{q_{0}}_{L^{rp(\cdot)}_{\omega^{1/r}}}\|g\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}.$

Thus, (17) and (18) lead to

(19) $\int[F(x)]^{q_{0}}g(x)dx\leq
C\|G\|^{q_{0}}_{L^{rp(\cdot)}_{\omega^{1/r}}},$

for all non-negative $g$ with
$\|g\|_{L^{(sq(\cdot))^{\prime}}_{\omega^{-1/s}}}\leq 1$. Then, (16) and (19)
give (14) for all finite sequences $\\{f_{j}\\}_{j=1}^{K}\subset
L^{\infty}_{comp}$. Finally, by passing to the limit, we obtain (14) for all
infinite sequences $\\{f_{j}\\}_{j=1}^{\infty}\subset L^{\infty}_{comp}$. ∎

######  Corollary 4.2.

Let $0<\alpha<n$, $q(\cdot):\mathbb{R}^{n}\to(0,\infty)$ be a measurable
function with $0<q_{-}\leq q_{+}<\infty$ and
$\omega\in\mathcal{W}_{q(\cdot)}$. If
$\frac{1}{p(\cdot)}:=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$ and
$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega}}\approx|Q|^{-\alpha/n}\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}$
for every cube $Q$, then for any sequence of scalars
$\\{\lambda_{j}\\}_{j\in\mathbb{N}}$, any family of cubes
$\\{Q_{j}\\}_{j\in\mathbb{N}}$, and any $\theta\in(0,\infty)$ fixed we have

$\left\|\sum_{j\in\mathbb{N}}\left(\frac{|\lambda_{j}|\chi_{Q_{j}}}{\|\chi_{Q_{j}}\|_{L^{q(\cdot)}_{\omega}}}\right)^{\theta}\right\|^{1/\theta}_{L^{q(\cdot)/\theta}_{\omega^{\theta}}}\lesssim\left\|\sum_{j\in\mathbb{N}}\left(\frac{|\lambda_{j}|\chi_{Q_{j}}}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\right\|^{1/\theta}_{L^{p(\cdot)/\theta}_{\omega^{\theta}}}.$

###### Proof..

Since
$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega}}\approx|Q|^{-\alpha/n}\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}$
for every cube $Q$ we obtain

$\left\|\sum_{j}\left(\frac{|\lambda_{j}|\chi_{Q_{j}}}{\|\chi_{Q_{j}}\|_{L^{q(\cdot)}_{\omega}}}\right)^{\theta}\right\|^{1/\theta}_{L^{q(\cdot)/\theta}_{\omega^{\theta}}}=\left\|\left\\{\sum_{j}\left(\frac{|\lambda_{j}|\chi_{Q_{j}}}{\|\chi_{Q_{j}}\|_{L^{q(\cdot)}_{\omega}}}\right)^{\theta}\right\\}^{1/\theta}\right\|_{L^{q(\cdot)}_{\omega}}$
$\lesssim\left\|\left\\{\sum_{j}\left(\frac{|\lambda_{j}||Q_{j}|^{\alpha/n}\chi_{Q_{j}}}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\right\\}^{1/\theta}\right\|_{L^{q(\cdot)}_{\omega}},$

it is easy to check that $|Q_{j}|^{\alpha/n}\chi_{Q_{j}}(x)\leq
M_{\frac{\alpha\theta}{N}}(\chi_{Q_{j}})^{\frac{N}{\theta}}(x)$ for all $j$
and all $N\in\mathbb{N}$, so

$\lesssim\left\|\left\\{\sum_{j}\left(\frac{|\lambda_{j}|M_{\frac{\alpha\theta}{N}}(\chi_{Q_{j}})^{\frac{N}{\theta}}}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\right\\}^{1/\theta}\right\|_{L^{q(\cdot)}_{\omega}}=\left\|\left\\{\sum_{j}\left(\frac{|\lambda_{j}|^{\theta}M_{\frac{\alpha\theta}{N}}(\chi_{Q_{j}})^{N}}{\|\chi_{Q_{j}}\|^{\theta}_{L^{p(\cdot)}_{\omega}}}\right)\right\\}^{1/\theta}\right\|_{L^{q(\cdot)}_{\omega}}$
$=\left\|\left\\{\sum_{j}\left(\frac{|\lambda_{j}|^{\theta}M_{\frac{\alpha\theta}{N}}(\chi_{Q_{j}})^{N}}{\|\chi_{Q_{j}}\|^{\theta}_{L^{p(\cdot)}_{\omega}}}\right)\right\\}^{1/N}\right\|^{N/\theta}_{L^{Nq(\cdot)/\theta}_{\omega^{\theta/N}}},$

taking $N$ such that $N/\theta>s_{\omega,\,q(\cdot)}+\frac{\alpha}{n}$, by
Theorem 4.1, we get

$\lesssim\left\|\left\\{\sum_{j}\left(\frac{|\lambda_{j}|^{\theta}\chi_{Q_{j}}}{\|\chi_{Q_{j}}\|^{\theta}_{L^{p(\cdot)}_{\omega}}}\right)\right\\}^{1/N}\right\|^{N/\theta}_{L^{Np(\cdot)/\theta}_{\omega^{\theta/N}}}=\left\|\sum_{j}\left(\frac{|\lambda_{j}|\chi_{Q_{j}}}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\right\|^{1/\theta}_{L^{p(\cdot)/\theta}_{\omega^{\theta}}}.$

This completes the proof. ∎

##  5 Weighted variable estimates for Riesz potential

Let $0<\alpha<n$. The Riesz potential of order $\alpha$ is the fractional
operator $I_{\alpha}$ defined by

(20)
$I_{\alpha}f(x)=\int_{\mathbb{R}^{n}}f(y)|x-y|^{\alpha-n}dy,\,\,\,x\in\mathbb{R}^{n},$

$f\in\mathcal{S}(\mathbb{R}^{n})$. A well known result of Sobolev gives the
boundedness of $I_{\alpha}$ from $L^{p}(\mathbb{R}^{n})$ into
$L^{q}(\mathbb{R}^{n})$ for $1<p<\frac{n}{\alpha}$ and
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}.$ In [2] C. Capone, D. Cruz Uribe
and A. Fiorenza extend this result to the case of Lebesgue spaces with
variable exponents $L^{p(\cdot)}.$ In [20] E. Stein and G. Weiss used the
theory of harmonic functions of several variables to prove that these
operators are bounded from $H^{1}(\mathbb{R}^{n})$ into
$L^{\frac{n}{n-\alpha}}(\mathbb{R}^{n})$. In [21], M. Taibleson and G. Weiss
obtained the boundedness of the Riesz potential $I_{\alpha}$ from the Hardy
spaces $H^{p}(\mathbb{R}^{n})$ into $H^{q}(\mathbb{R}^{n}),$ for $0<p<1$ and
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. These results were extended to the
context of variable Hardy spaces by M. Urciuolo and the author in [18, 17].

In this section we will prove that the Riesz potential $I_{\alpha}$ is bounded
from weighted variable Hardy spaces into weighted variable Lebesgue spaces.
The main tools that we will use are Theorem 4.1, Corollary 4.2 and Theorem
2.9.

######  Theorem 5.1.

Let $0<\alpha<n$, $q(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ with
$0<q_{-}\leq q_{+}<\infty$, and $\omega\in\mathcal{W}_{q(\cdot)}$. If
$\frac{1}{p(\cdot)}:=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$ and
$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega}}\approx|Q|^{-\alpha/n}\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}$
for every cube $Q$, then the Riesz potential $I_{\alpha}$ given by (20) can be
extended to a bounded operator $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})\to
L^{q(\cdot)}_{\omega}(\mathbb{R}^{n})$.

###### Proof..

Let $\omega\in\mathcal{W}_{q(\cdot)}$, by Definition 2.4, there exists
$0<\theta<1$ such that $\frac{1}{\theta}\in\mathbb{S}_{\omega,\,q(\cdot)}$.
Now, we take $q_{0}>\frac{n}{n-\alpha}$ such that
$q_{0}>\theta\left(\kappa_{\omega,\,q(\cdot)}^{1/\theta}\right)^{\prime}$, and
define $\frac{1}{p_{0}}:=\frac{1}{q_{0}}+\frac{\alpha}{n}$. By Proposition
3.1, for $\frac{1}{p(\cdot)}=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$, we have
that $\mathcal{W}_{q(\cdot)}\subset\mathcal{W}_{p(\cdot)}$ and
$s_{\omega,\,p(\cdot)}\leq s_{\omega,\,q(\cdot)}+\frac{\alpha}{n}$. So, given
$f\in S_{0}(\mathbb{R}^{n})$, from Theorem 2.9 (taking $s=p_{0}$) and since
one can always choose atoms with additional vanishing moment, we have that
there exist a sequence of real numbers $\\{\lambda_{j}\\}_{j=1}^{\infty}$, a
sequence of cubes $Q_{j}=Q(z_{j},r_{j})$ centered at $z_{j}$ with side length
$r_{j}$ and $\omega-(p(\cdot),p_{0},\lfloor
ns_{\omega,\,q(\cdot)}+\alpha-n\rfloor)$ atoms $a_{j}$ supported on $Q_{j}$,
satisfying

(21)
$\left\|\sum_{j}\left(\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\chi_{Q_{j}}\right\|_{L^{p(\cdot)/\theta}_{\omega^{\theta}}}^{1/\theta}\lesssim\|f\|_{H^{p(\cdot)}_{\omega}},$

and $f=\sum_{j}\lambda_{j}a_{j}$ converges in $L^{p_{0}}(\mathbb{R}^{n})$. By
Sobolev’s Theorem we have that $I_{\alpha}$ is bounded from
$L^{p_{0}}\left(\mathbb{R}^{n}\right)$ into
$L^{q_{0}}\left(\mathbb{R}^{n}\right)$, so

$|I_{\alpha}f(x)|\leq\sum_{j}|\lambda_{j}||I_{\alpha}a_{j}(x)|,\,\,\,\,\,\textit{a.e.}\,x\in\mathbb{R}^{n}.$

Then

(22)
$\|I_{\alpha}f\|_{L^{q(\cdot)}_{\omega}}\lesssim\left\|\sum_{j}|\lambda_{j}|\,\chi_{2Q_{j}}\cdot
I_{\alpha}a_{j}\right\|_{L^{q(\cdot)}_{\omega}}+\left\|\sum_{j}|\lambda_{j}|\,\chi_{\mathbb{R}^{n}\setminus
2Q_{j}}\cdot I_{\alpha}a_{j}\right\|_{L^{q(\cdot)}_{\omega}}=J_{1}+J_{2},$

where $2Q_{j}=Q(z_{j},2r_{j})$. To estimate $J_{1}$, we again apply Sobolev’s
Theorem and obtain

$\left\|(I_{\alpha}a_{j})^{\theta}\right\|_{L^{q_{0}/\theta}(2Q_{j})}=\left\|I_{\alpha}a_{j}\right\|_{L^{q_{0}}(2Q_{j})}^{\theta}\lesssim\left\|a_{j}\right\|_{L^{p_{0}}}^{\theta}\lesssim\frac{|Q_{j}|^{\frac{\theta}{p_{0}}}}{\left\|\chi_{Q_{j}}\right\|_{L^{p(\cdot)/\theta}_{\omega^{\theta}}}}\lesssim\frac{\left|2Q_{j}\right|^{\frac{\theta}{q_{0}}}}{\left\|\chi_{2Q_{j}}\right\|_{L^{q(\cdot)/\theta}_{\omega^{\theta}}}},$

where $\theta$, $q_{0}$ and $p_{0}$ are given above. The last inequality
follows from the condition
$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega}}\approx|Q|^{-\alpha/n}\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega}}$
assumed for every cube $Q$ and Lemma 3.4 applied to the exponent $q(\cdot)$.
Since $\frac{1}{\theta}\in\mathbb{S}_{\omega,\,q(\cdot)}$ and
$q_{0}>\theta\left(\kappa_{\omega,\,q(\cdot)}^{1/\theta}\right)^{\prime}$, we
apply the $\theta$-inequality and Proposition 2.8 with
$b_{j}=\left(\chi_{2Q_{j}}\cdot I_{\alpha}(a_{j})\right)^{\theta}$ and
$A_{j}=\left\|\chi_{2Q_{j}}\right\|_{L^{q(\cdot)/\theta}_{\omega^{\theta}}}^{-1}$
to obtain

$J_{1}\lesssim\left\|\sum_{j}\left(|\lambda_{j}|\,\chi_{2Q_{j}}\cdot
I_{\alpha}a_{j}\right)^{\theta}\right\|^{1/\theta}_{L^{q(\cdot)/\theta}_{\omega^{\theta}}}\lesssim\left\|\sum_{j}\left(\frac{|\lambda_{j}|}{\left\|\chi_{2Q_{j}}\right\|_{L^{q(\cdot)}_{\omega}}}\right)^{\theta}\chi_{2Q_{j}}\right\|^{1/\theta}_{L^{q(\cdot)/\theta}_{\omega^{\theta}}}.$

Being $\chi_{2Q_{j}}\leq M(\chi_{Q_{j}})^{2}$, by Theorem 4.1, we have

$J_{1}\lesssim\left\|\left\\{\sum_{j}\left(\frac{|\lambda_{j}|^{\theta/2}}{\left\|\chi_{2Q_{j}}\right\|^{\theta/2}_{L^{q(\cdot)}_{\omega}}}M(\chi_{Q_{j}})\right)^{2}\right\\}^{1/2}\right\|^{2/\theta}_{L^{2q(\cdot)/\theta}_{\omega^{\theta/2}}}\lesssim\left\|\sum_{j}\left(\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{q(\cdot)}_{\omega}}}\right)^{\theta}\chi_{Q_{j}}\right\|_{L^{q(\cdot)/\theta}_{\omega^{\theta}}}^{1/\theta}.$

Corollary 4.2 and (21) give

(23)
$J_{1}\lesssim\left\|\sum_{j}\left(\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\chi_{Q_{j}}\right\|_{L^{p(\cdot)/\theta}_{\omega^{\theta}}}^{1/\theta}\lesssim\|f\|_{H^{p(\cdot)}_{\omega}}.$

Now, we estimate $J_{2}$. Let $d=\lfloor
ns_{\omega,\,q(\cdot)}+\alpha-n\rfloor$, and let $a_{j}(\cdot)$ be a
$\omega-(p(\cdot),p_{0},d)$ atom supported on the cube $Q_{j}=Q(z_{j},r_{j})$.
In view of the moment condition of $a_{j}(\cdot)$ we obtain

$I_{\alpha}a_{j}(x)=\int_{Q_{j}}\left(|x-y|^{\alpha-n}-q_{d}(x,y)\right)a(y)dy,\,\,\,\,\,\,\textit{for
all}\,\,\,x\notin 2Q_{j},$

where $q_{d}$ is the degree $d$ Taylor polynomial of the function
$y\rightarrow|x-y|^{\alpha-n}$ expanded around $z_{j}$. By the standard
estimate of the remainder term of the Taylor expansion, there exists $\xi$
between $y$ and $z_{j}$ such that

$\left||x-y|^{\alpha-n}-q_{d}(x,y)\right|\leq
C|y-z_{j}|^{d+1}|x-\xi|^{-n+\alpha-d-1},$

for any $y\in Q_{j}$ and any $x\notin 2Q_{j}$, since
$|x-\xi|\geq\displaystyle{\frac{|x-z_{j}|}{1+\sqrt{n}}}$, we get

$\left||x-y|^{\alpha-n}-q_{d}(x,y)\right|\leq
Cr^{d+1}|x-z_{j}|^{-n+\alpha-d-1},$

this inequality and the condition $(a2)$ allow us to conclude that

(24)
$|I_{\alpha}a_{j}(x)|\lesssim\|a_{j}\|_{1}\,r^{d+1}|x-z_{j}|^{-n+\alpha-d-1}$
$\lesssim|Q_{j}|^{1-\frac{1}{p_{0}}}\|a_{j}\|_{p_{0}}r^{d+1}|x-z_{j}|^{-n+\alpha-d-1}\lesssim\frac{r^{n+d+1}}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}|x-z_{j}|^{-n+\alpha-d-1}$
$\lesssim\frac{\left[M_{\frac{\alpha
n}{n+d+1}}(\chi_{Q_{j}})(x)\right]^{\frac{n+d+1}{n}}}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}},\,\,\,\,\,\,for\,\,all\,\,\,\,x\notin
2Q_{j}.$

We put $r=\frac{n+d+1}{n}$, thus (24) leads to

$J_{2}\lesssim\left\|\left\\{\sum_{j}\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\left[M_{\frac{\alpha}{r}}(\chi_{Q_{j}})\right]^{r}\right\\}^{1/r}\right\|^{r}_{L^{rq(\cdot)}_{\omega^{1/r}}}.$

Since

$r=\frac{n+\lfloor
ns_{\omega,\,q(\cdot)}+\alpha-n\rfloor+1}{n}>s_{\omega,\,q(\cdot)}+\frac{\alpha}{n},$

to apply Theorem 4.1, with $u=r$, we obtain

$J_{2}\lesssim\left\|\left\\{\sum_{j}\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\chi_{Q_{j}}\right\\}^{1/r}\right\|^{r}_{L^{rp(\cdot)}_{\omega^{1/r}}}=\left\|\sum_{j}\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\chi_{Q_{j}}\right\|_{L^{p(\cdot)}_{\omega}}.$

Being $0<\theta<1$, the $\theta$-inequality and (21) give

(25)
$J_{2}\lesssim\left\|\sum_{j}\left(\frac{|\lambda_{j}|}{\|\chi_{Q_{j}}\|_{L^{p(\cdot)}_{\omega}}}\right)^{\theta}\chi_{Q_{j}}\right\|^{1/\theta}_{L^{p(\cdot)/\theta}_{\omega^{\theta}}}\lesssim\|f\|_{H^{p(\cdot)}_{\omega}}.$

Hence, (22), (23) and (25) yield

$\|I_{\alpha}f\|_{L^{q(\cdot)}_{\omega}}\lesssim\|f\|_{H^{p(\cdot)}_{\omega}},\,\,\,\text{for
all}\,\,f\in\mathcal{S}_{0}(\mathbb{R}^{n}).$

Finally, by taking into account that
$p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$, $0<p_{-}\leq p_{+}<\infty$
and $\omega\in\mathcal{W}_{p(\cdot)}$, the theorem follows from Theorem 2.10.
∎

##  6 Example: power weights

For $\gamma>-n$, let $w_{\gamma}(x)=|x|^{\gamma}$ with
$x\in\mathbb{R}^{n}\setminus\\{0\\}$. From the estimates obtained in [9,
Example 7.1.6, pp. 505], we observe that

(26) $\omega_{\gamma}\in\bigcap_{t>1}RH_{t},\,\,\,\text{if}\,\,\gamma\geq 0$

and

(27)
$\omega_{\gamma}\in\bigcap_{1<t<n/|\gamma|}RH_{t},\,\,\,\text{if}\,\,-n<\gamma<0.$

Moreover, for $0\leq\alpha<n$, $0<p<\frac{n}{\alpha}$,
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and each cube $Q$ we have that

(28)
$[\omega_{\gamma}^{p}(Q)]^{-1/p}[\omega_{\gamma}^{q}(Q)]^{1/q}=[\omega_{\gamma
p}(Q)]^{-1/p}[\omega_{\gamma
q}(Q)]^{1/q}\approx|Q|^{-\alpha/n},\,\,\,\text{for every}\,\,\gamma\geq 0$

and

$[\omega_{\gamma}^{p}(Q)]^{-1/p}[\omega_{\gamma}^{q}(Q)]^{1/q}\approx|Q|^{-\alpha/n},\,\,\,\text{for
every}\,\,-n<\gamma<0\,\,\text{with}\,\,q<\frac{n}{|\gamma|}.$

Let $q(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ with
$0<q_{\infty}=q_{-}\leq q_{+}<\infty$, and let $p(\cdot)$ be defined by
$\frac{1}{p(\cdot)}:=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$. If $\gamma\geq 0$,
then, by (28) and Corollary 3.7 (first applied to $q(\cdot)$ and then to
$p(\cdot)$), we get

$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega_{\gamma}}}\approx|Q|^{-\alpha/n}\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega_{\gamma}}}.$

It is clear that $q^{\prime}(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$,
$0<q^{\prime}_{-}\leq q^{\prime}_{+}=q^{\prime}_{\infty}<\infty$,
$q^{\prime}_{+}=(q_{-})^{\prime}$ and $q^{\prime}_{+}(Q)=(q_{-}(Q))^{\prime}$.
Then for every $0\leq\gamma<\frac{n}{q^{\prime}_{+}}$, by (26) and Corollary
3.7; and (27) and Corollary 3.9 (applied to $\omega_{\gamma}$ and $q(\cdot)$;
and $\omega_{-\gamma}$ and $q^{\prime}(\cdot)$ respectively), we obtain that

$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega_{\gamma}}}\|\chi_{Q}\|_{L^{(q(\cdot))^{\prime}}_{\omega_{-\gamma}}}\approx\left\\{\begin{array}[]{c}\,\,\,\left[\omega_{\gamma}^{q_{-}}(Q)\right]^{1/q_{-}}[\omega_{-\gamma}^{q^{\prime}_{+}}(Q)]^{1/q^{\prime}_{+}},\hskip
28.45274pt\text{if}\,\,\ell(Q)>1\\\ \\\
\left[\omega_{\gamma}^{q_{-}(Q)}(Q)\right]^{1/q_{-}(Q)}\left[\omega_{-\gamma}^{q^{\prime}_{+}(Q)}(Q)\right]^{1/q^{\prime}_{+}(Q)},\,\,\text{if}\,\,\ell(Q)\leq
1\end{array}\right\\}\approx|Q|,$

for all cube $Q\subset\mathbb{R}^{n}$, where the last estimate follows from
[9, Example 7.1.6, pp. 505]. In particular, for $s>1/q_{-}$ and
$0\leq\gamma<\frac{sn}{(sq(\cdot))^{\prime}_{+}}$, we have

$\|\chi_{Q}\|_{L^{sq(\cdot)}_{\omega_{\gamma/s}}}\|\chi_{Q}\|_{L^{(sq(\cdot))^{\prime}}_{\omega_{-\gamma/s}}}\approx|Q|.$

Since $\omega^{1/s}_{\pm\gamma}=\omega_{\pm\gamma/s}$, by [3, Theorem 1.5] we
have that the maximal operator $M$ is bounded on
$L^{(sq(\cdot))^{\prime}}_{\omega_{\gamma}^{-1/s}}$. By the left-openness
property of the set $\mathcal{A}_{(sq(\cdot))^{\prime}}$ (see definition 3.3
and [11, Theorem 3.3]), it follows that there exists a constant $\kappa>1$
such that the maximal operator $M$ is bounded on
$L^{(sq(\cdot))^{\prime}/\kappa}_{\omega_{\gamma}^{-\kappa/s}}$. So,
$\omega_{\gamma}\in\mathcal{W}_{q(\cdot)}$ and
$\|\chi_{Q}\|_{L^{q(\cdot)}_{\omega_{\gamma}}}\approx|Q|^{-\alpha/n}\|\chi_{Q}\|_{L^{p(\cdot)}_{\omega_{\gamma}}}$
for every cube $Q\subset\mathbb{R}^{n}$.

## References

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Departamento de Matemática
Universidad Nacional del Sur
Bahía Blanca, 8000 Buenos Aires
Argentina

E-mail address<EMAIL_ADDRESS>


11institutetext: Paper ID 609311institutetext:  Department of Computer
Science, Durham University 22institutetext: Department of Computer Vision
Technology (VIS), Baidu Inc. 33institutetext: Shanghai AI Laboratory, 4
University of Warwick
33email: {yang.bai,yang.long}@durham.ac.uk,33email:
<EMAIL_ADDRESS>
# Action Quality Assessment with Temporal Parsing Transformer

Anonymous ECCV submission  Yang Bai 112*†2*† Desen Zhou 2*2* Songyang Zhang 33
Jian Wang 22 Errui Ding 22
Yu Guan 44 Yang Long 11 Jingdong Wang 2‡2‡

###### Abstract

Action Quality Assessment(AQA) is important for action understanding and
resolving the task poses unique challenges due to subtle visual differences.
Existing state-of-the-art methods typically rely on the holistic video
representations for score regression or ranking, which limits the
generalization to capture fine-grained intra-class variation. To overcome the
above limitation, we propose a temporal parsing transformer to decompose the
holistic feature into temporal part-level representations. Specifically, we
utilize a set of learnable queries to represent the atomic temporal patterns
for a specific action. Our decoding process converts the frame representations
to a fixed number of temporally ordered part representations. To obtain the
quality score, we adopt the state-of-the-art contrastive regression based on
the part representations. Since existing AQA datasets do not provide temporal
part-level labels or partitions, we propose two novel loss functions on the
cross attention responses of the decoder: a ranking loss to ensure the
learnable queries to satisfy the temporal order in cross attention and a
sparsity loss to encourage the part representations to be more discriminative.
Extensive experiments show that our proposed method outperforms prior work on
three public AQA benchmarks by a considerable margin.

###### Keywords:

action quality assessment, temporal parsing transformer, temporal patterns,
contrastive regression

††footnotetext:  * Equal contribution. $\dagger$ Work done when Yang Bai was a
research intern at VIS, Baidu.
$\ddagger$ Corresponding Author.

##  1 Introduction

Action quality assessment(AQA), which aims to evaluate how well a specific
action is performed, has attracted increasing attention in research community
recently [18, 17]. In particular, assessing the action quality accurately has
great potential in a wide range of applications such as health care [34] and
sports analysis [2, 20, 19, 18].

In contrast to the conventional action recognition tasks [25, 4], AQA poses
unique challenges due to the subtle visual differences. Previous works on AQA
either use ranking-based pairwise comparison between test videos[5] or
estimate the quality score with regression-based methods[19, 29]. However,
these methods typically represent a video with its holistic representation,
via the global pooling operation over the output of the backbone network(e.g.,
I3D[4]). Since the videos to be evaluated usually are from the same coarse
action category (e.g., diving) in AQA, it’s crucial to capture fine-grained
intra-class variation to estimate more accurate quality scores. Thus, we
propose to decompose the holistic feature into more fine-grained temporal
part-level representations for AQA.

Figure 1: An action consists of multiple temporally ordered key phases.

To achieve this, a promising strategy is to represent the video by using a set
of atomic action patterns. For example, a diving action consists of several
key phases, such as approach, take off, flight, etc., as illustrated in Fig.1.
The fine-grained patterns enable the model to describe the subtle differences,
which is expected to improve the assessment of action quality effectively.
Nevertheless, it remains challenging to learn such atomic patterns as the
existing AQA datasets do not provide temporal part-level labels or partitions.

In this work, we aim to tackle the aforementioned limitations by developing a
regression-based action quality assessment strategy, which enables us to
leverage the fine-grained atomic action patterns without any explicit part-
level supervision. Our key idea is to model the shared atomic temporal
patterns, with a set of learnable queries for a specific action category.
Similar to the decoding process of transformer applied in natural language
modeling[24], we propose a temporal parsing transformer to decode each video
into a fixed number of part representations. To obtain quality scores, we
adopt the recent state-of-the-art contrastive regression framework[32]. Our
decoding mechanism allows the part representations between test video and
exemplar video to be implicitly aligned via a shared learnable query. Then, we
generate a relative pairwise representation per part and fuse different parts
together to perform the final relative score regression.

To learn the atomic action patterns without the part-level labels, we propose
two novel loss functions on the cross attention responses of the decoder.
Specifically, to ensure the learnable queries satisfy the temporal order in
cross attention, we calculate an attention center for each query by weighted
summation of the attention responses with their temporal clip orders. Then we
adopt a marginal ranking loss on the attention centers to guide the temporal
order. Moreover, we propose a sparsity loss for each query’s attention
distribution to guide the part representations to be more discriminative.

We evaluate our method, named as temporal parsing transformer(TPT), on three
public AQA benchmarks: MTL-AQA[18], AQA-7[17] and JIGSAWS[7]. As a result, our
method outperforms previous state-of-the-art methods by a considerable margin.
The visualization results show that our method is able to extract part-level
representations with interpretable semantic meanings. We also provide abundant
ablation studies for better understanding.

The main contributions of this paper are three folds:

  * •

We propose a novel temporal parsing transformer to extract fine-grained
temporal part-level representations with interpretable semantic meanings,
which are optimized with the contrastive regression framework.

  * •

We propose two novel loss functions on the transformer cross attentions to
learn the part representations without the part-level labels.

  * •

We achieve the new state-of-the-art on three public AQA benchmarks, namely
MTL-AQA, AQA-7 and JIGSAWS.

##  2 Related Work

###  2.1 Action quality assessment

In the past years, the field of action quality assessment (AQA) has been
repaid developed with a broad range of applications such as health care[34],
instructional video analysis[5, 6], sports video analysis[17, 18], and many
others[8, 9]. Existing AQA methods can be categorized into two types:
regression based methods and ranking based methods.

##### Regression based methods

Mainstream AQA methods formulate the AQA task as a regression task based on
reliable score labels, such as scores given by expert judges of sports events.
For example, Pirsiavash et al. [20] took the first steps towards applying the
learning method to the AQA task and trained a linear SVR model to regress the
scores of videos based on handcrafted features. Gordan et al. [8] proposed in
their pioneer work the use of skeleton trajectories to solve the problem of
quality assessment of gymnastic vaulting movements. Parmar et al. [19] showed
that spatiotemporal features from C3D [23] can better encode the temporal
representation of videos and significantly improve AQA performance. They also
propose a large-scale AQA dataset and explore all-action models to better
evaluate the effectiveness of models proposed by the AQA community. Xu et al.
[29] proposed learning multi-scale video features by stacked LSTMs followed
[19]. Pan et al. [16] proposed using spatial and temporal graphs to model the
interactions between joints. Furthermore, they also propose to use I3D [4] as
a stronger backbone network to extract spatiotemporal features. Parmar et al.
[18] introduced the idea of multi-task learning to improve the model capacity
of AQA, and collected AQA datasets with more annotations to support multi-task
learning. To diminish the subjectiveness of the action score from human
judges, Tang et al. [22] proposed an uncertainty-aware score distribution
learning (USDL) framework Recently. However, the video’s final score can only
provide weak supervision concerning action quality. Because two videos with
different low-quality parts are likely to share similar final scores, which
means the score couldn’t provide discriminative information.

##### Ranking based methods

Another branch formulates AQA task as a ranking problem. Doughty et al. [5]
proposed a novel loss function that learns discriminative features when a pair
of videos exhibit variance in skill and learns shared features when a pair of
videos show comparable skill levels. Doughty et al. [6] used a novel rank-
aware loss function to attend to skill-relevant parts of a given video.
However, they mainly focus on longer, more ambiguous tasks and only predict
overall rankings, limiting AQA to applications requiring some quantitative
comparisons. Recently, Yu et al. [32] proposed the Contrastive Regression
(CoRe) framework to learn the relative scores by pair-wise comparison,
highlighting the differences between videos and guiding the models to learn
the key hints for assessment.

###  2.2 Temporal action parsing

Fine-grained action parsing is also studied in the field of action
segmentation or temporal parsing [11, 10, 1, 12, 13, 31]. For example, Zhang
et al.[33] proposed Temporal Query Network adopted query-response
functionality that allows the query to attend to relevant segments. Dian et
al.[21] proposed a temporal parsing method called TransParser that is capable
of mining sub-actions from training data without knowing their labels.
However, different from the above fields, part-level labels are not available
in AQA task. Furthermore, most of the above methods focus more on frame-level
feature enhancement, whereas our proposed method extracts part representations
with interpretable semantic meanings.

##  3 Method

In this section, we introduce our temporal parsing transformer with the
contrastive regression framework in detail.

Figure 2: Overview of our framework. Our temporal parsing transformer converts
the clip-level representations into temporal part-level representations. Then
the part-aware contrastive regressor first computes part-wise relative
representations and then fuses them to estimate the relative score. We adopt
the group-aware regression strategy, following[32]. During training, we adopt
the ranking loss and sparsity loss on the decoder cross attention maps to
guide the part representation learning.

###  3.1 Overview

The input of our network is an action video. We adopt the Inflated 3D
ConvNets(I3D)[4] as our backbone, which first applies a sliding window to
split the video into $T$ overlapping clips, where each clip contains $M$
consecutive frames. Then, each clip goes through the I3D network, resulting in
time series clip level representations
$\boldsymbol{V}=\\{\boldsymbol{v}_{t}\in\mathbb{R}^{D}\\}^{T}_{t=1}$, where
$D$ is feature dimension and $T$ is the total number of clips. In our work, we
do not explore spatial patterns, hence each clip representation
$\boldsymbol{v}_{t}$ is obtained by average pooling across spatial dimensions.
The goal of AQA is to estimate a quality score $\mathbf{s}$ based on the
resulting clips representation $\boldsymbol{V}$. In contrastive regression
framework, instead of designing a network to directly estimate raw score
$\mathbf{s}$, it estimates a relative score between the test video and an
exemplar video $\boldsymbol{V}_{0}$ with known quality score $\mathbf{s}_{0}$,
which is usually sampled from training set. Then, contrastive regression aims
to design a network $\mathcal{F}$ that estimates the relative score
$\Delta\boldsymbol{s}$:

$\Delta\mathbf{s}=\mathcal{F}(\boldsymbol{V},\boldsymbol{V}_{0}),$ (1)

then final score can be obtained by

$\mathbf{s}=\mathbf{s}_{0}+\Delta\mathbf{s}.$ (2)

In our framework, we first adopt a temporal parsing transformer $\mathcal{G}$
to convert the clip level representations $\boldsymbol{V}$ into temporal part
level representations, denoted by
$\boldsymbol{P}=\\{\boldsymbol{p}_{k}\in\mathbb{R}^{d}\\}_{k=1}^{K}$, where
$d$ is the part feature dimension and $K$ is the number of queries, i.e.
temporal atomic patterns. Then for test video and exemplar video, we can have
two set of aligned part representations $\boldsymbol{P}$ and
$\boldsymbol{P}_{0}=\\{\boldsymbol{p}_{k}^{0}\in\mathbb{R}^{d}\\}_{k=1}^{K}$.
Our new formulation can be expressed as:

$\Delta\mathbf{s}=\mathcal{R}(\boldsymbol{P},\boldsymbol{P}_{0}).$ (3)

where $\mathcal{R}$ is the relative score regressor, and

$\boldsymbol{P}=\mathcal{G}(\boldsymbol{V}),\boldsymbol{P}_{0}=\mathcal{G}(\boldsymbol{V}_{0}).$
(4)

An overview of our framework is illustrated in Fig.2. Below we describe the
detailed structure of temporal parsing transformer $\mathcal{G}$ and part-
aware contrastive regressor $\mathcal{R}$.

###  3.2 Temporal parsing transformer

Our temporal parsing transformer takes the clip representations as memory and
exploits a set of learnable queries to decode part representations. Different
from prevalent DETR architecture[3], our transformer only consists of a
decoder module. We found that the encoder module does not provide improvements
in our framework; it even hurts the performance. We guess it might because
that clip-level self-attention smooths the temporal representations, and our
learning strategy cannot decode part presentations in this way without part
labels.

We perform slight modifications to the standard DETR decoder. That is, the
cross attention block in our decoder has a learnable parameter, temperature,
to control the amplification of the inner product. Formally, in the $i$-th
decoder layer, the decoder part feature $\\{p_{k}^{(i)}\in\mathbb{R}^{d}\\}$
and learnable atomic patterns(i.e. query set) $\\{q_{k}\in\mathbb{R}^{d}\\}$
are first summed as a query and then perform cross attention on the embedded
clip representation $\\{\mathbf{v}_{t}\in\mathbb{R}^{d}\\}$:

$\alpha_{k,t}=\frac{\exp^{(\boldsymbol{p}_{k}^{(i)}+q_{k})^{T}\cdot\boldsymbol{v}_{t}/\tau}}{\sum\limits_{j=1}^{T}\exp^{(\boldsymbol{p}_{k}^{(i)}+q_{k})^{T}\cdot\boldsymbol{v}_{j}/\tau}},$
(5)

where $\alpha_{k,t}$ indicates the attention value for query $k$ to clip $t$,
$\tau\in\mathbb{R}$ indicates the learnable temperature to enhance the inner
product to make the attentions more discriminative. Unlike DETR[3], in our
decoder, we do not utilize position embedding of clip id to the memory
$\\{\boldsymbol{v}_{t}\\}$. We expect our query to represent atomic patterns,
instead of spatial anchors, as in the detection task[28, 14]. We found that
adding position encoding significantly drops the performance and makes our
learning strategy fail, which will be shown in the experiment section.

In our experiments, we only utilize one-head attention in our cross attention
blocks. The attention values are normalized across different clips, since our
goal is to aggregate clip representations into our part representation. Then
the updated part representation $\boldsymbol{p}_{k}^{(i)^{\prime}}$ has the
following form:

$\boldsymbol{p}_{k}^{(i)^{\prime}}=\sum\limits_{j=1}^{T}\alpha_{k,j}\boldsymbol{v}_{j}+\boldsymbol{p}_{k}^{(i)}.$
(6)

We then perform standard FFN and multi-head self-attention on decoder part
representations. Similar to DETR[3], our decoder also has a multi-layer
structure.

###  3.3 Part-aware contrastive regression

Our temporal parsing transformer converts the clip representations
$\\{\boldsymbol{v}_{t}\\}$ into part representations
$\\{\boldsymbol{p}_{k}\\}$. Given a test video and exemplar video, we can
obtain two part representation sets $\\{\boldsymbol{p}_{k}\\}$ and
$\\{\boldsymbol{p}_{k}^{0}\\}$. One possible way to estimate the relative
quality score is to fuse each video’s part representations and estimate the
relative score. However, since our temporal parsing transformer allows the
extracted part representations to be semantically aligned with the query set,
we can compute the relative pairwise representation per part and then fuse
them together. Formally, we utilize a multi-layer perceptron(MLP) $f_{r}$ to
generate the relative pairwise representation
$\boldsymbol{r}_{k}\in\mathbb{R}^{d}$ for $k$-th part:

$\boldsymbol{r}_{k}=f_{r}(Concat([\boldsymbol{p}_{k};\boldsymbol{p}_{k}^{0}])).$
(7)

The MLP $f_{r}$ is shared across different parts. To balance the score
distributions across the whole score range, we adopt the group-aware
regression strategy to perform relative score estimation[32]. Specifically, it
first calculates $B$ relative score intervals based on all possible pairs in
training set, where each interval has equal number of pair-samples. Then it
generates a one-hot classification label $\\{l_{n}\\}$, where $l_{n}$
indicates whether the ground truth score $\Delta\mathbf{s}$ lies in $n$-th
interval, and a regression target
$\gamma_{n}=\frac{\Delta\mathbf{s}-x_{left}^{n}}{x_{right}^{n}-x_{left}^{n}}$,
where $x_{left}^{n},x_{right}^{n}$ denote the left and right boundary of
$n$-th interval. Readers can refer to [32] for more details.

We adopt average pooling111 We note that it might be better to weight each
part. However, part weighting does not provide improvements during our
practice. We guess that it may be during the self-attention process in the
decoder, the relations between parts have already been taken into account.  on
the relative part representations $\\{\boldsymbol{r}_{k}\\}$ and then utilize
two two-layer MLPs to estimate the classification label $\\{l_{n}\\}$ and
regression target $\\{\gamma_{n}\\}$. Different from [32], we do not utilize
tree structure. Since we have obtained fine-grained part-level representations
and hence the regression becomes simpler, we found that two-layer MLP works
fine.

###  3.4 Optimization

Since we do not have any part-level labels at hand, it’s crucial to design
proper loss functions to guide the part representation learning. We have
assumed that each coarse action has a set of temporally ordered atomic
patterns, which are encoded in our transformer queries. To ensure that our
query extracts different part representations, we constrain the attention
responses in cross attention blocks for different queries. Specifically, in
each cross attention process, we have calculated the normalized attention
responses $\\{\alpha_{k,t}\\}$ by Eq.5, then we compute an attention center
$\bar{\alpha}_{k}$ for $k$-th query:

$\bar{\alpha}_{k}=\sum\limits_{t=1}^{T}t\cdot\alpha_{k,t},$ (8)

where $T$ is the number of clips and $\sum\limits_{t=1}^{T}\alpha_{k,t}=1$.
Then we adopt two loss functions on the attention centers: ranking loss and
sparsity loss.

##### Ranking loss

To encourage that each query attends to different temporal regions, we adopt a
ranking loss on the attention centers. We wish our part representations have a
consistent temporal order across different videos. To this end, we define an
order on the query index and apply ranking losses to the corresponding
attention centers. We exploit the margin ranking loss, which results in the
following form:

$L_{rank}=\sum_{k=1}^{K-1}\max(0,\bar{\alpha}_{k}-\bar{\alpha}_{k+1}+m)+\max(0,1-\bar{\alpha}_{1}+m)+\max(0,\bar{\alpha}_{K}-T+m),$
(9)

where $m$ is the hyper-parameter margin controlling the penalty, the first
term guides the attention centers of part $k$ and $k+1$ to keep order:
$\bar{\alpha}_{k}<\bar{\alpha}_{k+1}$. From Eq. 8, we have the range of
attention centers: $1\leq\bar{\alpha}_{k}\leq T$. To constrain the first and
last part where $k=1$ and $k=K$, we assume there is two virtual centers at
boundaries: $\bar{\alpha}_{0}=1$ and $\bar{\alpha}_{K+1}=T$. The last two
terms in Eq. 9 constrain the first and last attention centers not collapsed to
boundaries.

##### Sparsity loss

To encourage the part representations to be more discriminative, we further
propose a sparsity loss on the attention responses. Specifically, for each
query, we encourage the attention responses to focus on those clips around the
center $\mu_{k}$, resulting in the following form:

$L_{sparsity}=\sum\limits_{k=1}^{K}\sum\limits_{t=1}^{T}|t-\bar{\alpha}_{k}|\cdot\alpha_{k,t}$
(10)

During training, our ranking loss and sparsity loss are applied to the cross
attention block in each decoder layer.

##### Overall training loss

In addition to the above auxiliary losses for cross attention, our contrastive
regressor $\mathcal{R}$ generates two predictions for the group classification
label $\\{l_{n}\\}$ and regression target $\\{\gamma_{n}\\}$, we follow [32]
to utilize the BCE loss on each group and square error on the ground truth
regression interval:

$L_{cls}=-\sum\limits_{n=1}^{N}l_{n}\log(\tilde{l}_{n})+(1-l_{n})\log(1-\tilde{l}_{n})$
(11)
$L_{reg}=\sum\limits_{n=1}^{N}\mathbbm{1}(l_{n}=1)(\gamma_{n}-\tilde{\gamma}_{n})^{2}$
(12)

where $L_{reg}$ only supervises on the ground truth interval, $\tilde{l}_{n}$
and $\tilde{\gamma}_{n}$ are predicted classification probability and
regression value. The overall training loss is given by:

$L_{all}=\lambda_{cls}L_{cls}+\lambda_{reg}L_{reg}+\lambda_{rank}\sum\limits_{i=1}^{L}L_{rank}^{i}+\lambda_{sparsity}\sum\limits_{i=1}^{L}L_{sparsity}^{i},$
(13)

where $i$ indicates layer id and $L$ is the number of decoder layers,
$\lambda_{cls}$, $\lambda_{reg}$, $\lambda_{rank}$, $\lambda_{sparsity}$ are
hyper-parameter loss weights.

##  4 Experiment

###  4.1 Experimental Settup

##### Datasets

We perform experiments on three public benchmarks: MTL-AQA[18], AQA-7[17], and
JIGSAWS[7]. See Supplement for more details on datasets.

##### Evaluation Metrics

Following prior work[32], we utilize two metrics in our experiments, the
Spearman’s rank correlation and relative L2 distance($R\mbox{-}\ell_{2}$).
Spearman’s rank correlation was adopted as our main evaluation metric to
measure the difference between true and predicted scores. The Spearman’s rank
correlation is deﬁned as follows:

$\rho=\frac{\sum_{i}(p_{i}-\bar{p})(q_{i}-\bar{q})}{\sqrt{\sum_{i}(p_{i}-\bar{p})^{2}\sum_{i}(q_{i}-\bar{q})^{2}}}$
(14)

It focuses on the ranking of test samples. In contrast, relative L2 distance
measures the numerical precision of each sample compared with ground truth.
Formally, it’s defined as:

$R\mbox{-}\ell_{2}=\frac{1}{N}\sum_{n=1}^{N}(\frac{|s_{n}-\hat{s}_{n}|}{s_{max}-s_{min}})^{2}$
(15)

####  4.1.1 Implementation Details

We adopt the I3D backbone pretrained on Kinetics[4] as our local spatial-
temporal feature extractor. The Adam optimizer is applied with a learning rate
$1\times 10^{-4}$ for the backbone and transformer module. The learning rate
for the regression head is set to $1\times 10^{-3}$. The feature dimension is
set to 512 for the transformer block. We select 10 exemplars for each test
sample during the inference stage to align with previous work[32] for fair
comparisons. As for the data-preprocessing on AQA-7 and MTL-AQA datasets, we
sample 103 frames following previous works for all videos. Since our proposed
method requires more fine-grained temporal information, unlike previous work
that segmented the sample frames into 10 clips, we segment the frames into 20
overlapping clips each containing 8 continuous frames. As for the JIGSAWS
dataset, we uniformly sample 160 frames following [22] and divide them into 20
non-overlapping clips as input of the I3D backbone. We select exemplars from
the same difficulty degree on MTL-dataset during the training stage. For AQA-7
and JIGSAWS datasets, all exemplars come from the same coarse classes.

Table 1: Performance comparison on MTL-AQA dataset. ‘w/o DD’ means that
training and test processes do not utilize difficulty degree labels, ‘w/ DD’
means experiments utilizing difficulty degree labels.

Method (w/o DD) | Sp. Corr. | $R\mbox{-}\ell_{2}(\times 100)$
---|---|---
Pose+DCT[20] | 0.2682 | -
C3D-SVR[19] | 0.7716 | -
C3D-LSTM[19] | 0.8489 | -
MSCADC-STL[18] | 0.8472 | -
C3D-AVG-STL[18] | 0.8960 | -
MSCADC-MTL[18] | 0.8612 | -
C3D-AVG-MTL[18] | 0.9044 | -
USDL[22] | 0.9066 | 0.654
CoRe[32] | 0.9341 | 0.365
TSA-Net[27] | 0.9422 | -
Ours | 0.9451 | 0.3222
Method (w/ DD) | Sp. Corr |  $R\mbox{-}\ell_{2}$($\times$ 100)
USDL[22] | 0.9231 | 0.468
MUSDL[22] | 0.9273 | 0.451
CoRe[32] | 0.9512 | 0.260
Ours | 0.9607 | 0.2378

Table 2: Performance comparison on AQA-7 dataset.

Sp. Corr | Diving | Gym Vault | BigSki. | BigSnow. | Sync. 3m | Sync. 10m | Avg. Corr.
---|---|---|---|---|---|---|---
Pose+DCT[20] | 0.5300 | 0.1000 | - | - | - | - | -
ST-GCN[30] | 0.3286 | 0.5770 | 0.1681 | 0.1234 | 0.6600 | 0.6483 | 0.4433
C3D-LSTM[19] | 0.6047 | 0.5636 | 0.4593 | 0.5029 | 0.7912 | 0.6927 | 0.6165
C3D-SVR[19] | 0.7902 | 0.6824 | 0.5209 | 0.4006 | 0.5937 | 0.9120 | 0.6937
JRG[16] | 0.7630 | 0.7358 | 0.6006 | 0.5405 | 0.9013 | 0.9254 | 0.7849
USDL[22] | 0.8099 | 0.7570 | 0.6538 | 0.7109 | 0.9166 | 0.8878 | 0.8102
CoRe[32] | 0.8824 | 0.7746 | 0.7115 | 0.6624 | 0.9442 | 0.9078 | 0.8401
TSA-Net[27] | 0.8379 | 0.8004 | 0.6657 | 0.6962 | 0.9493 | 0.9334 | 0.8476
Ours | 0.8969 | 0.8043 | 0.7336 | 0.6965 | 0.9456 | 0.9545 | 0.8715
$R\mbox{-}\ell_{2}(\times 100)$ | Diving | Gym Vault | BigSki. | BigSnow. | Sync. 3m | Sync. 10m | Avg. $R\mbox{-}\ell_{2}$
C3D-SVR[19] | 1.53 | 3.12 | 6.79 | 7.03 | 17.84 | 4.83 | 6.86
USDL[22] | 0.79 | 2.09 | 4.82 | 4.94 | 0.65 | 2.14 | 2.57
CoRe[32] | 0.64 | 1.78 | 3.67 | 3.87 | 0.41 | 2.35 | 2.12
Ours | 0.53 | 1.69 | 2.89 | 3.30 | 0.33 | 1.33 | 1.68

Table 3: Performance comparison on JIGSAW dataset.

Sp. Corr. | S | NP | KT | Avg.
---|---|---|---|---
ST-GCN[30] | 0.31 | 0.39 | 0.58 | 0.43
TSN[26] | 0.34 | 0.23 | 0.72 | 0.46
JRG[16] | 0.36 | 0.54 | 0.75 | 0.57
USDL[22] | 0.64 | 0.63 | 0.61 | 0.63
MUSDL[22] | 0.71 | 0.69 | 0.71 | 0.70
CoRe[32] | 0.84 | 0.86 | 0.86 | 0.85
Ours | 0.88 | 0.88 | 0.91 | 0.89
$R\mbox{-}\ell_{2}$ | S | NP | KT | Avg.
CoRe[32] | 5.055 | 5.688 | 2.927 | 4.556
Ours | 2.722 | 5.259 | 3.022 | 3.668

###  4.2 Comparison to state-of-the-art

We compare our results with state-of-the-art methods on three benchmarks in
Tab.1, Tab.2 and Tab.3. Our method outperforms priors works on all three
benchmarks under all settings.

On MTL-AQA dataset, we evaluated our experiments with two different settings,
following prior work[32]. Specifically, the MTL-AQA dataset contains the label
of difficult degree, and each video’s quality score is calculated by the
multiplication of the raw score with its difficulty. In the experiment setting
‘w/o DD’, the training and test processes do not utilize difficulty degree
labels. In setting ‘w/ DD’, we exploit the difficulty label by comparing the
test video to the exemplar videos with the same difficulty, and we estimate
the raw score, which is multiplied by the difficulty to get the final quality.
Our method outperforms existing works under both settings. As shown in Tab. 1,
under ‘w/ DD’, our method achieves a Sp. Corr. of 0.9607, and
$R\mbox{-}\ell_{2}$ of 0.2378, outperforms the tree-based CoRe[32]. Note that
our model simply utilizes two shallow MLPs to perform contrastive regression
instead of the tree structure as in [32]. Our transformer extracts fine-
grained part representations, hence the regression becomes easier. Under ‘w/o
DD’, out method achieves 0.9451(Sp. Corr) and 0.3222($R\mbox{-}\ell_{2}$),
outperforms the CoRe and recent TSA-Net[27]. It’s worth noting that TSA-Net
utilizes an external VOT tracker[35] to extract human locations and then
enhance backbone features, which is orthogonal to the main issue of temporal
parsing addressed in our work. Consequently, we expect that our method can be
further improved by incorporating the attention module as in [27].

On AQA-7 dataset, our method achieves state-of-the-art on 5 categories and
comparable performance on the rest category, shown in Tab. 2. In particular,
on average, our method outperforms CoRe by 3.14 Corr.($\times 100$) and TSA-
Net by 2.39 Corr.($\times 100$), and obtains a very small $R\mbox{-}\ell_{2}$
of 1.68($\times 100$), demonstrating the effectiveness of our temporal parsing
transformer.

On the smallest JIGSAW dataset, we perform 4-fold cross validation for each
category, following prior work[32, 22]. Our method achieves an average of 0.89
Corr. and 3.668 $R\mbox{-}\ell_{2}$, achieves new state-of-the-art.

Table 4: Ablation study of different components on MTL-AQA dataset.

Method | TPT | $L_{rank}$ | $L_{sparsity}$ | Sp. Corr. | $R\mbox{-}\ell_{2}$
---|---|---|---|---|---
Baseline | $\times$ | $\times$ | $\times$ | 0.9498 | 0.2893
| ✓ | $\times$ | $\times$ | 0.9522 | 0.2742
| ✓ | ✓ | $\times$ | 0.9583 | 0.2444
Ours | ✓ | ✓ | ✓ | 0.9607 | 0.2378

###  4.3 Ablation Study

In this subsection, we perform ablation studies to evaluate the effectiveness
of our proposed model components and designs. All of our ablation studies are
performed on MTL-AQA dataset under ‘w/ DD’ setting. We build a baseline
network that directly pool the clip features without transformer, and utilize
the resulting holistic representation to perform contrastive regression.

##### Different model components

In this work, we propose a novel temporal parsing transformer(TPT), and
exploit the ranking loss($L_{rank}$) and sparsity loss($L_{sparsity}$) on
cross attention responses to guide the part representation learning. We first
perform experiments to show the effectiveness of each design, the results are
shown in Tab.4. We can observe that with only TPT, the performance only
improves marginally from 0.9498 Corr. to 0.9522 Corr.. With the ranking loss,
the performance is significantly improved, demonstrating the importance of
temporally ordered supervision strategy. The sparsity loss further improves
the performance, showing that the discrimination of parts is also important.

Table 5: More ablation studies on MTL-AQA dataset.

(a) Different part generation strategies.

Method | Sp. Corr. | $R\mbox{-}\ell_{2}$
---|---|---
Baseline | 0.9498 | 0.2893
Adaptive pooling | 0.9509 | 0.2757
Temporal conv | 0.9526 | 0.2758
TPT(ours) | 0.9607 | 0.2378

(b) Effect of order guided supervision.

Method | Sp. Corr. | $R\mbox{-}\ell_{2}$
---|---|---
Baseline | 0.9498 | 0.2893
Diversity loss | 0.9538 | 0.2655
Ranking loss(ours) | 0.9607 | 0.2378

(c) Effect of positional encoding.

Pos. Encode | Memory(clip) | Query(part) | Sp. Corr. | $R\mbox{-}\ell_{2}$
---|---|---|---|---
| ✓ | ✓ | 0.9526 | 0.2741
| ✓ | $\times$ | 0.9532 | 0.2651
Proposed | $\times$ | $\times$ | 0.9607 | 0.2378

(d) Different relative representation generation. Method | Sp. Corr. | $R\mbox{-}\ell_{2}$
---|---|---
Baseline | 0.9498 | 0.2893
Part-enhanced holistic | 0.9578 | 0.2391
Part-wise relative + AvgPool(ours) | 0.9607 | 0.2378

##### Different relative representation generation

Since we have obtained part representations from TPT for each video, we may
have two options to generate relative representation for contrastive
regression. For the first option, we can first fuse the part representations
with a pooling operation for each video, then each video takes the part-
enhanced holistic representation to estimate the relative score. For the
second option, which is our proposed strategy, we first compute a part-wise
relative representation and then apply the AvgPool operation over the parts.
We compare the results of above options in Tab.5(d). We can see that the part-
wise strategy outperforms part-enhanced strategy. It’s worth noting that the
part-enhanced approach also outperforms our baseline network, which implies
that each part indeed encodes fine-grained temporal patterns.

##### Different part generation strategies

Our method utilizes the temporal parsing transformer to extract part
representations. In this ablation study, we compare our method with the other
two baseline part generation strategy, shown in Tab. 5(a). The first strategy
utilizes the adaptive pooling operation cross temporal frames to down-sample
the origin $T$ clip representation into $K$ part representations. The second
strategy replaces the above adaptive pooling with a temporal convolution with
stride $\lfloor T/K\rfloor$, resulting in a representation with $K$ size. We
found that both strategies introduce minor improvements as they can not
capture fine-grained temporal patterns.

##### Effect of position encoding

Different from conventional transformer[3, 24], our transformer decoding
process does not rely on the temporal position encoding. We compare the
results of different position encoding strategies on the memory(clip) and
query(part) in Tab.5(c). To embed the position encoding on queries, we add the
cosine series embedding of $\lfloor T/K\rfloor\times i$ to $i$-th learnable
query, making the queries have positional guidance uniformly distributed
across temporal clips. We keep the ranking loss and sparsity for fair
comparisons. From Tab.5(c), we can observe that adding position encoding hurts
the learning of temporal patterns.

##### Effect of order guided training strategy

Our ranking loss on the attention centers consistently encourages the temporal
order of atomic patterns. To verify the importance of such order guided
supervision, we replace the ranking loss to a diversity loss following the
Associative Embedding[15] to push attention centers:
$L_{div}=\sum\limits_{i=1}^{K}\sum\limits_{j=i+1}^{K}\exp^{-\frac{1}{2\sigma^{2}}(\bar{\alpha}_{i}-\bar{\alpha}_{j})^{2}}$.
Compared with $L_{rank}$, $L_{div}$ does not encourage the order of queries,
but keeps diversity of part representations. As shown in Tab. 5(b), the
performance significantly drops from 0.9607 Corr. to 0.9538 Corr.,
demonstrating the effectiveness of our order guided training strategy.

Figure 3: Visualization of the frames with the highest attention responses in
decoder cross attention maps on MTL-AQA and AQA-7 datasets. Each row
represents a test video from different representative categories (diving from
MTL-AQA, gymnastic vault from AQA-7), whose ID is shown in the left first
frame. Different columns correspond to temporally ordered queries. The above
results show that our transformer is able to capture semantic temporal
patterns with learned queries.  Figure 4: Visualization of cross attention
maps on three video samples from MTL-AQA dataset, where video IDs are shown on
the top. In each subfigure, each row indicates one query, and each column
indicates one clip. We can observe that the bright grids(with high attention
responses) have a consistent temporal order due to ranking loss, and the
attention maps are sparse due to our sparsity loss.

###  4.4 Visualization results

We provide some visualization results in Fig.3 and Fig.4. Samples are from
MTL-AQA dataset trained under ‘w/ DD’ setting and AQA-7 dataset. In Fig.3, we
visualize the clip frames with the highest attention responses in cross
attention maps of the last decoder layer. Since each clip consists of multiple
frames, we select the middle frame of a clip as representative. We can observe
that our transformer can capture semantic temporal patterns with learned
queries. In Fig.4, we visualize the cross attention maps. We can observe that
the attention responses have a consistent temporal order due to our designed
ranking loss, and they are also sparse due to our sparsity loss.

##  5 Conclusion

In this paper, we propose a novel temporal parsing transformer for action
quality assessment. We utilize a set of learnable queries to represent the
atomic temporal patterns, and exploit the transformer decoder to convert clip-
level representations to part-level representations. To perform quality score
regression, we exploit the contrastive regression framework that first
computes the relative pairwise representation per part and then fuses them to
estimate the relative score. To learn the atomic patterns without part-level
labels, we propose two novel loss functions on cross attention responses to
guide the queries to attend to temporally ordered clips. As a result, our
method is able to outperform existing state-of-the-art methods by a
considerable margin on three public benchmarks. The visualization results show
that the learned part representations are semantic meaningful.

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