# Self-improving Poincaré-Sobolev type functionals in product spaces

María Eugenia Cejas  Departamento de Matemática, Facultad de Ciencias Exactas,
Universidad Nacional de La Plata, Calle 50 y 115, (1900) La Plata, Prov. de
Buenos Aires, Argentina<EMAIL_ADDRESS>,  Carolina Mosquera
Department of Mathematics, Facultad de Ciencias Exactas y Naturales,
University of Buenos Aires, Ciudad Universitaria Pabellón I, Buenos Aires 1428
Capital Federal Argentina<EMAIL_ADDRESS>,  Carlos Pérez  Department of
Mathematics, University of the Basque Country, IKERBASQUE (Basque Foundation
for Science) and BCAM –Basque Center for Applied Mathematics, Bilbao, Spain
<EMAIL_ADDRESS>and  Ezequiel Rela  Department of Mathematics, Facultad de
Ciencias Exactas y Naturales, University of Buenos Aires, Ciudad Universitaria
Pabellón I, Buenos Aires 1428 Capital Federal Argentina<EMAIL_ADDRESS>
###### Abstract.

In this paper we give a geometric condition which ensures that
$(q,p)$-Poincaré-Sobolev inequalities are implied from generalized
$(1,1)$-Poincaré inequalities related to $L^{1}$ norms in the context of
product spaces. The concept of eccentricity plays a central role in the paper.
We provide several $(1,1)$-Poincaré type inequalities adapted to different
geometries and then show that our selfimproving method can be applied to
obtain special interesting Poincaré-Sobolev estimates. Among other results, we
prove that for each rectangle $R$ of the form $R=I_{1}\times
I_{2}\subset\mathbb{R}^{n}$ where $I_{1}\subset\mathbb{R}^{n_{1}}$ and
$I_{2}\subset\mathbb{R}^{n_{2}}$ are cubes with sides parallel to the
coordinate axes, we have that

$\left(\frac{1}{w(R)}\int_{R}|f-f_{R}|^{p_{\delta,w}^{*}}\,wdx\right)^{\frac{1}{p_{\delta,w}^{*}}}\leq
c\,(1-\delta)^{\frac{1}{p}}\,[w]_{A_{1,\mathfrak{R}}}^{\frac{1}{p}}\,\Big{(}a_{1}(R)+a_{2}(R)\Big{)},$

where $\delta\in(0,1)$, $w\in A_{1,\mathfrak{R}}$,
$\frac{1}{p}-\frac{1}{p_{\delta,w}^{*}}=\frac{\delta}{n}\,\frac{1}{1+\log[w]_{A_{1,\mathfrak{R}}}}$
and $a_{i}(R)$ are bilinear analog of the fractional Sobolev seminorms
$[u]_{W^{\delta,p}(Q)}$ (See Theorem 2.18). This is a biparameter weighted
version of the celebrated fractional Poincaré-Sobolev estimates with the gain
$(1-\delta)^{\frac{1}{p}}$ due to Bourgain-Brezis-Minorescu.

###### Key words and phrases:

Poincaré - Sobolev inequalities. Muckenhoupt weights.

###### 1991 Mathematics Subject Classification:

Primary: 42B25. Secondary: 42B20.

M.E.C. is partially supported by grant PICT-2018-03017 (ANPCYT)

C.M. is partially supported by grants UBACyT 20020170100430BA, PICT 2018–03399
and PICT 2018–04027.

C. P. is supported by grant MTM2017-82160-C2-1-P of the Ministerio de Economía
y Competitividad (Spain), grant IT-641-13 of the Basque Government, and
IKERBASQUE

E.R. is partially supported by grants UBACyT 20020170200057BA, PIP (CONICET)
11220110101018, by the Basque Government through the BERC 2014-2017 program,
and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo
Ochoa accreditation SEV-2013-0323. This project has received funding from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 777822.

##  1\. Introduction and Background

Poincaré and Poincaré-Sobolev inequalities have been studied extensively in a
wide variety of scenarios, including the general theory for convex sets. The
particular case of cubes is specially interesting, once we have a good result
on cubes there are mechanisms (sometimes chaining methods) to pass to more
general domains, see for instance Chapter 9 from the very recent manuscript
[KLV] (also [HKST15]). See also the last part of Section 5 in [FPW98] to see
the connection with weights. The classical technique for getting this kind of
inequalities is through the use of a representation formula in terms of a
fractional integral. An approach which avoids the use of any representation
formula to obtain Poincaré-Sobolev inequalities on cubes (or balls) was
introduced in [FPW98] and then sharpened in [MP98]. This approach merges with
the John-Nirenberg theory of the $\operatorname{BMO}$ spaces and others under
the phenomenon of the so called self-imroving property. However this
“abstract” general theory has not been developed in the context of rectangles
or, more generally, product spaces as was considered for first time in [ST93]
and later in [LW98], which will be the main purpose of this article. In this
different scenario it is reasonable to expect some sensitivity with respect to
very thin rectangles which is reflected in the dependence on the notion of
eccentricity: roughly, a quantity that indicates how far the given object is
from being a cube. This notion will be essential to our results and we will
properly define it later.

We will study generalized weighted Poincaré and Poincaré-Sobolev type
inequalities related to two product space settings. The most general possible
situation is the case of the family of rectangles in $\mathbb{R}^{n}$ seen as
$n$-fold product of intervals on $\mathbb{R}$, denoted by $\mathcal{R}$. We
will study a variant of this case by considering rectangles $R$ of the form
$R=I_{1}\times I_{2}\subset\mathbb{R}^{n}$ where
$I_{1}\subset\mathbb{R}^{n_{1}}$ and $I_{2}\subset\mathbb{R}^{n_{2}}$ are
cubes with sides parallel to the coordinate axes, denoted by $\mathfrak{R}$.
We will deal with these two situations separately to simplify the
presentation. Our approach consists in starting with some generalized
unweighted Poincaré inequality with respect to an $L^{1}$ mean oscillation and
from there we will improve it in order to gain higher integrability with the
minimum loss in the involved constants.

The motivation to study this situation comes from the following very
interesting $(1,1)$-Poincaré inequality valid for any bounded convex domain
$\Omega\subset\mathbb{R}^{n}$,

(1.1)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Omega}|f-f_{\Omega}|dx\leq\frac{1}{2}\,d(\Omega)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Omega}|\nabla
f|dx,$

where we used $f_{\Omega}$ and
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Omega}f\
dx$ to denote the average of the function $f$ over the set $\Omega$ and
$d(\Omega)$ stands for the diameter of the set $\Omega$. This result is due to
G. Acosta and R. Durán in [AD04] where the constant $\frac{1}{2}$ in front is
best possible. Of course, it is a classical well-known fact that this estimate
holds for any convex set $\Omega$ in $\mathbb{R}^{n}$ with a larger constant
in front. In particular, (1.1) holds in the case of $\Omega$ being a rectangle
of $\mathbb{R}^{n}$ with sides parallel to the axes. The key idea here is that
inequality (1.1) enjoys a sort of self-improving property, meaning that we can
reach higher integrability and therefore obtain $(p,p)$-Poincaré and
$(p^{*},p)$-Poincaré-Sobolev type inequalities, with $p^{*}>p$, both in the
unweighted and the weighted settings. This self-improving phenomena is not new
and can be traced back to [SC92, HaK95] in a more classical setting, but in a
more general abstract way in [FPW98] improved in [MP98]. A strengthened
version of the results from these papers can be found in [PR19] where much
more precise $L^{q}-L^{p}$ weighted Poincaré-Sobolev inequalities over cubes
in $\mathbb{R}^{n}$ were obtained. To be more precise it is remarkable that
the self-improving property is not so much related to the presence of a
gradient on the right hand side of (1.1), but to a discrete summation
condition associated to the functional that it defines. In fact, we will
consider as a starting point estimates of the form

(1.2)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|dx\leq
a(R)\qquad R\in\mathcal{F}$

where $\mathcal{F}$ is a given family of sets and $a:\mathcal{F}\to(0,\infty)$
is some general functional with no restriction. Typical examples of
$\mathcal{F}$ are the family of cubes $\mathcal{Q}$, the families of
rectangles from $\mathcal{R}$ or $\mathfrak{R}$, or any other collection of
sets. The key difference when moving from the basis $\mathcal{R}$ to the basis
$\mathfrak{R}$ is not on the selfimproving phenomena, as it will be clear from
the proofs. The issue will be to find the appropriate analogue for (1.2),
where the functional $a(R)$ will be replaced by a _sum_ of two functionals
$a_{1}(I)+a_{2}(J)$, with $R=I\times J$. Part of the effort will be devoted to
prove these starting points where the oscillation is controlled by a sum of
functionals.

This kind of generalized Poincaré inequality with a general abstract
functional acting on balls (or cubes) over a metric (or quasi-metric) space
with a doubling measure (spaces of homogeneous type) was considered for first
time in [FPW98]. A special geometric condition, denoted by $D_{p}$, was
introduced with the idea of deriving self-improving properties of the given
mean oscillation of the function on the left hand side. This condition has
been recently refined in [PR19], denoted $SD_{p}$, which produces much more
accurate results. We introduce both conditions $D_{p}$ and $SD_{p}$ for the
families $\mathcal{R}$ and $\mathfrak{R}$ in Definition 1.3 and Definition 1.1
below in the context of rectangles. We will make special emphasis on the
$SD_{p}$ type condition since it is highly efficient allowing to improve the
estimates obtained in [FPW98] in a much more precise way.

To present here a preview of the main contributions of this paper, consider
the particular case of $a(R)$ being the average of the $L^{1}$ norm of the
gradient of $f$, namely
$a(R)=d(R)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|\nabla
f|$. The main goal here would be, starting from an inequality of the form
(1.2) and using certain discrete summability conditions satisfied by the
functional $a$, to obtain an inequality involving $L^{q}-L^{p}$ norms of the
form

$\left(\frac{1}{w(R)}\int_{R}|f-f_{R}|^{q}w\right)^{\frac{1}{q}}\leq
C_{w}\,d(R)\left(\frac{1}{w(R)}\int_{R}|\nabla f|^{p}w\right)^{\frac{1}{p}}$

where $w$ is a certain weight. We will try to reach the best possible $q$ but
keeping controlled the constant $C_{w}$ extending and improving the results in
[FPW98] and [PR19].

The precise definition that we need is the following. As usual, we will say
that a _weight_ is a non negative locally integrable function defined on
$\mathbb{R}^{n}$. Also, the Lebesgue measure for a measurable set $\Omega$ in
$\mathbb{R}^{n}$ is denoted by $|\Omega|$.

######  Definition 1.1.

For a given weight $w$, and $s>0$, we say that a functional $a\in
SD_{p,\mathcal{R}}^{s}(w)$ for $0<p<+\infty$ if there is a constant $c$ such
that for any family of disjoint _dyadic_ subrectangles
$\left\\{R_{i}\right\\}_{i}$ of $R$ the following inequality holds

(1.3)
$\left(\sum_{i}a(R_{i})^{p}\frac{w(R_{i})}{w(R)}\right)^{\frac{1}{p}}\leq
c\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{1}{s}}a(R).$

The best possible constant $c$ (the infimum of the constants in last
inequality) is denoted by $\|a\|_{SD_{p,\mathcal{R}}^{s}}$. Usually, if there
is no chance of confusion, we will drop the subscript for the sake of clarity
in the exposition.

Sometimes we will refer to this condition as the smallness preservation
condition. The idea comes from the main application, namely if $L>1$ and if
$R$ is a rectangle, and a family of dyadic rectangles is $L$-small, namely
that

(1.4) $\frac{|\bigcup_{i}R_{i}|}{|R|}\leq\frac{1}{L}.$

then the functional $a$ preserves the smallness by applying (1.3).

######  Remark 1.2.

As it can be easily verified, the exact same definition applies to the family
$\mathfrak{R}$ using the appropriate notation $SD_{p,\mathfrak{R}}^{s}(w)$ and
$\|a\|_{SD_{p,\mathfrak{R}}^{s}}.$

We also recall the following geometric type condition which is somehow a
“rough” version of (1.3). Again, the same definition applies to $\mathfrak{R}$
with the obvious notational modifications.

######  Definition 1.3.

Let $w$ be any weight. We say that the functional $a$ satisfies the weighted
$D_{p,\mathcal{R}}(w)$ condition for $0<p<\infty$ if there is a constant $c$
such that for any $R\in\mathcal{R}$ and any family of disjoint _dyadic_
subrectangles $\left\\{R_{i}\right\\}_{i}$ of $R$ the following inequality
holds:

(1.5)
$\left(\sum_{i}a(R_{i})^{p}\frac{w(R_{i})}{w(R)}\right)^{\frac{1}{p}}\leq
c\,a(R).$

The best possible constant $c$ above is denoted by
$\|a\|_{D_{p,\mathcal{R}}(w)}$. We will write in this case that $a\in
D_{p,\mathcal{R}}(w)$. Observe that $\|a\|_{D_{p,\mathcal{R}}(w)}\geq 1$.

Condition (1.5) in the context of cubes or balls in $\mathbb{R}^{n}$, or more
generally in the context of Spaces of Homogeneous type was introduced in
[FPW98]. It was also considered in [OP02] in the “non-homogeneous” context
where an abstract self-improving property was obtained.

We conclude this introduction by remarking how condition (1.5) can be used, in
the much easier context of cubes $\mathcal{Q}$ with no weight, to derive very
sharp proofs in the theory of fractional Sobolev spaces. Indeed, in this
paper, Corollary 2.11 and many others, are extensions of the fractional
Poincaré-Sobolev inequality

(1.6)
$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|u-u_{Q}|^{p^{*}_{\delta}}dx\right)^{\frac{1}{p^{*}_{\delta}}}\leq
c_{n,p^{*}_{\delta}}\,[u]_{W^{\delta,p}}(Q),\qquad Q\in\mathcal{Q}.$

We are using here the notation $p_{\delta}^{*}$ for the fractional Sobolev
exponent defined by $\frac{1}{p}-\frac{1}{p_{\delta}^{*}}=\frac{\delta}{n}$
and also the notation

$[u]_{W^{\delta,p}(Q)}:=\ell(Q)^{\delta}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+\delta
p}}\,dy\,dx\right)^{1/p}.$

To illustrate the self-improving philosophy in the case of cubes, we will
discuss the main steps of the proof of (1.6) and a further improvement. This
approach avoids completely the use of classical potential theory (see [KLV] in
the classical case and [NPV12] in the fractional case) and this is the reason
why we can go beyond and consider other geometries and degeneracies. Moreover,
due to the presence of a more complicated geometry and to the presence of
degeneracies produced by weights, the proofs of the results in the present
paper require substantial work and involve serious technical difficulties.

We begin with the following easy to get inequality

(1.7)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\lvert
u(x)-u_{Q}\rvert\,dx\leq c_{n}\,[u]_{W^{\delta,p}(Q)}\qquad
Q\in\mathcal{Q},\quad\delta>0$

which we claim encodes a lot of information, at least in the case
$\delta\in(0,1)$. Indeed, it can be shown using the ideas in [FPW98], with the
recently obtained improvements from [CP21], the following weak type version of
(1.6)

(1.8)
$\|u-u_{Q}\|_{L^{p^{*}_{\delta},\infty}\big{(}Q,\frac{dx}{|Q|}\big{)}}\leq
c_{n}\,p^{*}_{\delta}\,\,[u]_{W^{\delta,p}(Q)},$

which holds whenever $p\in[1,\infty)$ and for
$\frac{1}{p}-\frac{1}{p_{\delta}^{*}}=\frac{\delta}{n}$. Inequality (1.8)
really follows from

$\left(\sum_{i}a(Q_{i})^{p^{*}_{\delta}}\frac{|Q_{i}|}{|Q|}\right)^{\frac{1}{p^{*}_{\delta}}}\leq
a(Q),$

where $a(Q)=[u]_{W^{\delta,p}(Q)}$ for any $\delta\in(0,1)$ and any
$p\in[1,\infty)$ (observe that the constant in front is equal to 1). Now,
since the truncation argument works for the functional $[u]_{W^{\delta,p}(Q)}$
as shown in [DIV16], we can replace the weak norm (1.8) by the strong norm

$\|u-u_{Q}\|_{L^{p^{*}_{\delta}}\big{(}Q,\frac{dx}{|Q|}\big{)}}\leq
c_{n}\,p^{*}_{\delta}\,\,[u]_{W^{\delta,p}(Q)}$

(actually we can put the Lorentz norm $\|\|_{L^{p^{*}_{\delta},p}}$ as well).
On the other hand, it turns out that this result is far from being optimal.
Indeed, motivated by the works [BBM02, MS02], M. Milman obtained in [M05],
using ideas from interpolation theory, a very general way of proving the
following self-improvement of (1.7),

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|u-u_{Q}|\,dx\leq
c_{n}\,(1-\delta)^{\frac{1}{p}}\,\,[u]_{W^{\delta,p}(Q)},$

where the highly interesting extra gain $(1-\delta)^{\frac{1}{p}}$ appears in
front. Then, exactly as above, the results in [CP21] combined with the
truncation from [DIV16] yield the following highly interesting improvement of
(1.6).

######  Theorem 1.4.

Let $0<\delta<1$ and $1\leq p<\infty$. Then,

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|u(x)-u_{Q}|^{p^{*}_{\delta}}dx\right)^{\frac{1}{p^{*}_{\delta}}}\leq
c_{n}\,p^{*}_{\delta}\,(1-\delta)^{\frac{1}{p}}\,[u]_{W^{\delta,p}(Q)}\qquad
Q\in\mathcal{Q}$

######  Remark 1.5.

Standard arguments can be used to obtain the strong global estimate in
$\mathbb{R}^{n}$ with the correction factor in front
$(1-\delta)^{\frac{1}{p}}$.

Some extensions of (1.6) to the case of domains can be found in [HV13]. We
also refer to [CDM19] for extensions to the doubling metric case.

We plan to extend this result into the context of the product space
$\mathfrak{R}$ from which we will derive the corresponding version of (1.6)
and then derive its corresponding Poincaré-Sobolev extension from our general
method (see Theorem 2.20).

##  2\. Main results

For the sake of clarity on the presentation, we will present our results for
the basis $\mathcal{R}$ and $\mathfrak{R}$ separately.

###  2.1. Analysis in $\mathcal{R}$

In this section we will study the self-improving phenomena in the context of
$\mathcal{R}$ wich will be served as a model for the case $\mathfrak{R}$ that
will be considered in Section 2.5. First we will obtain Poincaré-Sobolev type
inequalities involving higher order derivatives for rectangles in
$\mathcal{R}$ (some references about higher order Poincaré-Sobolev type
inequalities can be found in [AH96, T00]). To see where is the difficulty
here, note in Definition 1.1 that the condition on the family of rectangles is
related to their measure. The classical one parameter theory for cubes works
fine here, since the $n$-th power of the diameter (or the sidelength) of a
cube is comparable to its measure. However, in the multiparametric setting of
rectangles, this is no longer true. That is why the notion of eccentricity
comes into play.

Our first main theorem on self-improving functionals is the following. We use
here an oscillation with respect to the polynomial $P_{R}$ defined in Section
3.2, equation (3.4). We anticipate here that this object $P_{R}$ is a
projection to a space of polynomials of certain degree, and is the natural
substitute for the average $f_{R}$ when dealing with functionals involving
higher derivatives. The weighted estimate will be valid for any weight in the
Muckenhoupt class $w\in A_{\infty}:=\bigcup_{p>1}A_{p}$ associated to the
basis $\mathcal{R}$ (this can be defined for any basis, see more details about
this class in Section 3.3).

######  Theorem 2.1.

Let $w$ be any $A_{\infty,\mathcal{R}}$ weight in $\mathbb{R}^{n}$. Consider
also the functional $a$ such that for some $p\geq 1$ it satisfies the weighted
condition $SD^{s}_{p,\mathcal{R}}(w)$ from (1.3) with $s>0$ and constant
$\|a\|$. Let $f$ be a locally integrable function such that

(2.1) $\frac{1}{|R|}\int_{R}|f-P_{R}f|\leq a(R)\qquad R\in\mathcal{R}.$

Then, there exists a dimensional constant $c_{n}$ such that for any
$R\in\mathcal{R}$,

(2.2)
$\left(\frac{1}{w(R)}\int_{R}|f-P_{R}f|^{p}\,wdx\right)^{\frac{1}{p}}\leq
c_{n}(1+s)\max\\{\|a\|^{s},1\\}\,a(R).$

This result generalizes and improves on the corresponding result from [PR19]
Theorem 1.24 where the linear constant $s$ is of exponential type.

As consequences of this sort of general template for a self-improving theorem,
we will obtain Poincaré-Sobolev inequalities for higher order derivatives with
a rather precise control on the $A_{p,\mathcal{R}}$-like quantities involved.

######  Remark 2.2.

We will often (but not always) use a compact notation for the weighted local
$L^{q}$ average over a rectangle $R$ defined as follows:

$\left\|f\right\|_{L^{q}\left(\frac{w\,dx}{w(R)}\right)}:=\left(\frac{1}{w(R)}\int_{R}|f|^{q}wdx\right)^{\frac{1}{q}}.$

Similarly, we will use the standard notation for the normalized weak
$(r,\infty)$ (quasi)norm: for any $0<r<\infty$, measurable $E$ and weight $w$,
we define

$\|f\|_{L^{r,\infty}\big{(}\frac{w\,dx}{w(E)}\big{)}}:=\sup_{t>0}t\,\left(\frac{1}{w(E)}w(\\{x\in
E:|f(x)|>t\\})\right)^{1/r}.$

The general Theorem 2.1 will be used to derive several corollaries. One of the
most important consequences is that we will get Poincaré-Sobolev type
inequalities involving higher derivatives instead of the gradient.

We also present here the following result, which can be seen as a weak norm
version of the main theorem, since we are asking the functional to only
satisfy the “rough” $D_{p}$ condition (1.5) but obtaining a self-improving for
the weak norm. It is a self-improving result as in [FPW98] but combined with
the recent improvement obtained in [CP21] where linear bounds in both
$[w]_{A_{\infty},\mathcal{R}}$ and $p$ are obtained instead of exponential.

######  Theorem 2.3.

Let $w$ be any $A_{\infty,\mathcal{R}}$ weight in $\mathbb{R}^{n}$ and $a\in
D_{p,\mathcal{R}}(w)$. Let $f$ be a locally integrable function such that,

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-P_{R}f|\leq
a(R)\qquad R\in\mathcal{R}.$

Then there exists a dimensional constant $c>0$ such that for every
$R\in\mathcal{R}$,

$\big{\|}f-P_{R}f\big{\|}_{L^{p,\infty}\big{(}R,\frac{w\,dx}{w(R)}\big{)}}\leq
c\,p\,[w]_{A_{\infty,\mathcal{R}}}\,\|a\|_{D_{p,\mathcal{R}}(w)}\,a(R).$

This result is completely new and has a lot of potential applications. We will
show some of them in next sections.

Recall that

$\big{\|}g\big{\|}_{L^{q}(X)}\leq\left(\frac{p}{p-q}\right)^{\frac{{1}}{q}}\,\big{\|}g\big{\|}_{L^{p,\infty}(X)}$

whenever $(X,\mu)$ is a probability space, and $0<q<p<\infty$ (sometimes
called Kolmogorov’s inequality.) Then, under the assumption of Theorem 2.3, if
$q<p$,

$\big{\|}f-P_{R}f\big{\|}_{L^{q}\big{(}R,\frac{w\,dx}{w(R)}\big{)}}\leq
c_{p,q}\,[w]_{A_{\infty,\mathcal{R}}}\,\|a\|_{D_{p,\mathcal{R}}(w)}\,a(R).$

###  2.2. Higher order Poincaré

As a consequence of our general result we will prove, using condition (1.3) as
the key point, inequalities for $R\in\mathcal{R}$ of the form

$\left(\frac{1}{w(R)}\int_{R}|f-P_{R}f|^{q}wdx\right)^{\frac{1}{q}}\leq
C_{w}\,d(R)^{m}\left(\frac{1}{w(R)}\int_{R}|\nabla^{m}f|^{p}wdx\right)^{\frac{1}{p}}$

where $C_{w}$ is a constant and $P_{R}f$ is an appropriate optimal polynomial
of degree less than $m$, defined in Section 3.2. The goal is to find the best
possible improvement in the exponent $q$ on the left hand side and precise
estimates on $C_{w}$. We will refer to the above inequality as a higher order
$(q,p)$-Poincaré-Sobolev weighted inequality.

A $(p,p)$ version of the above inequality in the case of cubes with $p\geq 1$,
was proved in [PR19]. Here we improve on that result achieving a higher
Sobolev-type exponent on the left hand side and also valid for the family of
rectangles $\mathcal{R}$. As usual, when proving weighted inequalities related
to rectangles, the appropriate class of weights is the _strong_ class defined
analogously to the standard $A_{p}$ class as follows. We will say that $w\in
A_{p,\mathcal{R}}$ if

$[w]_{A_{p,\mathcal{R}}}:=\sup_{R\in\mathcal{R}}\left(\frac{1}{|R|}\int_{R}w(x)\,dx\right)\left(\frac{1}{|R|}\int_{R}w(x)^{-\frac{1}{p-1}}\,dx\right)^{p-1}<\infty.$

The strong $A_{\infty,\mathcal{R}}$ class is defined in the same way as in the
cubic case and it enjoys the same geometric conditions (see Section 3.3 for
the details) and the corresponding constant $[w]_{A_{\infty,\mathcal{R}}}$ is
defined in (3.6).

The following corollary of a $(p,p)$-Poincaré inequality (for cubes) was
already obtained as a corollary of [PR19, Theorem 1.24] which is a weaker
version of our Theorem 2.1 for the case of cubes.

######  Corollary 2.4 ([PR19]).

Let $p\geq 1$ and let $w\in A_{p}$ in $\mathbb{R}^{n}$. Then the following
$(p,p)$-Poincaré inequality holds

$\left(\frac{1}{w(Q)}\int_{Q}|f-P_{Q}f|^{p}\,w\ dx\right)^{\frac{1}{p}}\leq
C\,[w]^{\frac{1}{p}}_{A_{p}}\ell(Q)^{m}\left(\frac{1}{w(Q)}\int_{Q}|\nabla^{m}f|^{p}\,w\
dx\right)^{\frac{1}{p}},$

where $C=C_{n,m}$ is a structural constant and $\ell(Q)$ is the sidelength of
the cube $Q\subset\mathbb{R}^{n}$.

The method of proof goes beyond Poincaré to reach Poincaré-Sobolev type
inequalities with a higher exponent on the left hand side. In the following
theorem the method seems to work best when the weight adds some degeneracy,
which is a bit surprising. More precisely, we will say that a weight $w\in
A_{\infty,\mathcal{R}}$ is nontrivial whenever
$[w]_{A_{\infty,\mathcal{R}}}>1$. The same notion for other bases of
rectangles or cubes will be used as well.

######  Theorem 2.5.

Let $1\leq p<n$ and let $w\in A_{q,\mathcal{R}}$ in $\mathbb{R}^{n}$ be a
nontrivial weight with $1\leq q\leq p$. Let also $p_{w}^{*}$ be defined by

(2.3)
$\frac{1}{p}-\frac{1}{p_{w}^{*}}=\frac{\delta}{n}\frac{1}{q+\log[w]_{A_{q,\mathcal{R}}}}.$

For a fixed $\delta>0$, let $a$ be the functional defined by

(2.4) $a(R)=d(R)^{\delta}\left(\frac{1}{w(R)}\mu(R)\right)^{1/p},$

where $\mu$ is any Radon measure in $\mathbb{R}^{n}$. Suppose that $f$
satisfies

(2.5) $\frac{1}{|R|}\int_{R}|f-P_{R}f|\leq a(R)$

for every rectangle $R\subset\mathbb{R}^{n}$. Then, there exists a dimensional
constant $C_{n}$ such that for any rectangle $R\subset\mathbb{R}^{n}$

$\left(\frac{1}{w(R)}\int_{R}|f-P_{R}f|^{p_{w}^{*}}\,wdx\right)^{\frac{1}{p_{w}^{*}}}\leq
B_{w,q}\,C_{n}\frac{1}{\delta}a(R),$

where
$B_{w,q}=\frac{1+\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}{\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}$.

Note that the dependence on the weight can be disregarded if we are looking
for the asymptotic behavior when $[w]_{{A_{q,\mathcal{R}}}}\to\infty$ (we can
bound that expression by 2 by considering weights such that
$[w]_{{A_{q,\mathcal{R}}}}\geq e^{q}$). The real problem is with _flat
weights_. We borrow that terminology from [PR18] to refer to weights with
small $A_{\infty}$ constant, close to 1. Those weights are, in some sense,
getting closer to being just a multiple of the Lebesgue measure, and that is
why this situation is counter intuitive, since one would expect an easier
problem for this latter case. Also note that there is some sort of balance
between the exponent $p^{*}_{w}$ and the constant in the inequality. The
closer the $A_{q}$ constant gets to 1, the closer the value of $p^{*}_{w}$
gets to the optimal Sobolev exponent. But it should also be observed that the
constant in the inequality blows up when $[w]_{A_{q,\mathcal{R}}}\to 1$. And
that is precisely the reason why this result can not be specialized on the
limit case of the constant weight, but we can give a reasonable substitute by
using weak norms.

######  Theorem 2.6.

Let $1\leq p<n$ and let $w\in A_{q,\mathcal{R}}$ in $\mathbb{R}^{n}$ with
$1\leq q\leq p$. If we define the exponent $p^{*}_{w}$ as in (2.3), the
functional $a$ as in (2.4) and consider a function satisfying (2.5), there
exists a dimensional constant $c_{n}$ such that for any rectangle
$R\subset\mathbb{R}^{n}$

(2.6)
$\left\|f-P_{R}f\right\|_{L^{p_{w}^{*},\infty}(R,\frac{wdx}{w(R)})}\leq\,c_{n}\,\frac{1}{\delta}a(R).$

The idea is to use a specific choice of the functional $a$ as the starting
point. Since a $(1,1)$-Poincaré inequality involving higher derivatives is
known to hold on rectangles, we obtain the following result.

######  Corollary 2.7.

Let $1\leq p<n$ and let $w\in A_{q,\mathcal{R}}$ in $\mathbb{R}^{n}$ with
$1\leq q\leq p$. Let $p_{w}^{*}$ as in the previous theorem. Then the
following inequality holds,

$\left\|f-P_{R}f\right\|_{L^{p_{w}^{*}}\left(R,\frac{w\,dx}{w(R)}\right)}\leq
B_{w,q}C_{n}\frac{1}{m}[w]_{A_{q,\mathcal{R}}}^{\frac{1}{p}}\,d(R)^{m}\left\|\nabla^{m}f\right\|_{L^{p}\left(R,\frac{w\,dx}{w(R)}\right)}$

for every rectangle $R\in\mathcal{R}$, where as before,
$B_{w,q}=\frac{1+\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}{\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}$.

We also have the estimate for the weak norm

(2.7)
$\left\|f-P_{R}f\right\|_{L^{p_{w}^{*},\infty}(R,\frac{wdx}{w(R)})}\leq\,C_{n,p}\frac{1}{m}[w]_{A_{q,\mathcal{R}}}^{\frac{1}{p}}\,d(R)^{m}\left\|\nabla^{m}f\right\|_{L^{p}\left(R,\frac{w\,dx}{w(R)}\right)}.$

The proof of this result can be found in Section 4.

######  Remark 2.8.

We remark that for any nontrivial weight separated from the constant weight,
say for instance $[w]_{{A_{q,\mathcal{R}}}}\geq e^{q}$, we can apply Theorem
2.5 to an appropriate starting point. Without this non-degeneracy condition,
and in particular in the case of flat weights, we avoid the logarithmic blowup
but at the cost of obtaining only a weak norm estimate. Of course, when $m=1$
the truncation argument can be carried out to reach the strong norm from
(2.7), to wit

$\left\|f-P_{R}f\right\|_{L^{p_{w}^{*}}(R,\frac{wdx}{w(R)})}\leq\,c_{n}[w]_{A_{q,\mathcal{R}}}^{\frac{1}{p}}\,d(R)\left\|\nabla
f\right\|_{L^{p}\left(R,\frac{w\,dx}{w(R)}\right)}.$

###  2.3. Fractional Poincaré-Sobolev and eccentricity.

This section is motivated by the fractional Poincaré-Sobolev inequalities
(1.6). First we will introduce a fractional type Poincaré estimate adapted to
the product space $\mathcal{R}$ from which we will derive the corresponding
Poincaré-Sobolev type result.

As before, we will introduce functionals satisfying the smallness preservation
condition $SD_{p,\mathcal{R}}^{s}$ for some choice of parameters where the
concept of eccentricity will play a central role in the proofs. It comes from
the natural interplay between the notion of diameter and measure for a given
rectangle.

######  Definition 2.9.

Let $R\in\mathcal{R}$ be any rectangle in $\mathbb{R}^{n}$. We define the
eccentricity as the number

(2.8) $e(R):=\frac{|R|^{\frac{1}{n}}}{d(R)}.$

The main property we use is the following observation: all the involved
families of rectangles will be obtained from dyadic partitions of a fixed
initial rectangle. Therefore, for each instance of this initial choice of a
rectangle, all of them are dyadic children of a given rectangle. The main
feature here is that all of them have the same eccentricity.

######  Lemma 2.10.

Let $\tilde{R}$ be any dyadic descendant of $R\in\mathcal{R}$. Then

$e(\tilde{R})=e(R).$

Indeed, if $j$ be the dyadic level to which $\tilde{R}$ belongs. Then,

(2.9)
$e(\tilde{R})=\frac{|\tilde{R}|^{1/n}}{d(\tilde{R})}=\frac{(|R|2^{-jn})^{1/n}}{d(R)2^{-j}}=e(R).$

By introducing this notion of eccentricity, we now are able to formulate other
corollaries from our general self-improving Theorem 2.5. We have the following
intermediate estimate.

######  Corollary 2.11.

Let $w\in A_{q,\mathcal{R}}$ in $\mathbb{R}^{n}$ with $1\leq q\leq p<n$. Let
$p_{w}^{*}$ as in (2.3), namely

$\frac{1}{p}-\frac{1}{p_{w}^{*}}=\frac{\delta}{n}\frac{1}{q+\log[w]_{A_{q,\mathcal{R}}}}.$

Let $R\in\mathcal{R}$ and $\delta>0$. Consider the function

(2.10) $A(R,x)=\int_{R}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+\delta
p}}\,dy,\qquad\qquad x\in R.$

Then the following inequality holds,

(2.11) $\left\|f-f_{R}\right\|_{L^{p_{w}^{*}}(R,\frac{wdx}{w(R)})}\leq
C_{n,p,q}\frac{1}{\delta}\,[w]^{\frac{1}{p}}_{A_{p,\mathcal{R}}}\,\frac{d(R)^{\delta}}{e(R)^{\frac{n}{p}}}\left(\frac{1}{w(R)}\int_{R}A(R,\cdot)w\,dx\right)^{\frac{1}{p}}.$

######  Remark 2.12.

We will prove this result from an initial inequality (1.2) with a non standard
functional $a(R)$ involving the quantity $A(R,x)$, namely

$\frac{1}{|R|}\int_{R}|f-f_{R}|\leq[w]^{\frac{1}{p}}_{A_{p,\mathcal{R}}}\,\frac{d(R)^{\delta}}{e(R)^{\frac{n}{p}}}\left(\frac{1}{w(R)}\int_{R}A(R,x)\,w(x)dx\right)^{\frac{1}{p}}.$

######  Remark 2.13.

There should be here a result as in Theorem 2.19 with the extra factor
$(1-\delta)^{\frac{1}{p}}$ at least in the case $w\in A_{1,\mathcal{R}}$. We
don’t know how to do it. However, we proved the corresponding variation of
this result in the context of $\mathfrak{R}$ in Theorem 2.20.

Note that, on the one hand the above starting point does not involve any
derivative, but on the other hand it is imposing a somewhat stronger control
over the $L^{1}$ oscillation, since it is known that for any $0<\delta<1$, and
any $p\geq 1$ one has

$\ell(Q)^{\delta}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+\delta
p}}dydx\right)^{1/p}\lesssim
C_{\delta,p,n}\ell(Q)\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\nabla
f|^{p}\right)^{1/p}.$

We will in fact provide the proof of an improved version of this inequality
for the case $p=1$ in Lemma 5.3, inequality (5.4). The proof is included in
the Appendix B.

###  2.4. A weaker starting point

The aim of this section is to show that we can replace the $L^{1}$ mean
oscillation in (2.1) as initial hypothesis by a weaker condition to derive the
same results. Indeed, as shown in Theorems 2.1 and Theorem 2.5, the natural
starting point is given by the expression

(2.12)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x)-f_{R}|\,dx\leq
a(R)$

where $a$ is a functional satisfying some sort of discrete summation
condition. We show in next theorem that we can replace the $L^{1}$ mean
oscillation in (2.12) by $L^{\delta}$, $0<\delta<1$ weakening the initial
assumption. This idea was already considered in [LP02] and [LP05] but the
results and the methods are not so precise as the ones we obtain here within a
different context.

######  Theorem 2.14.

Let $f$ be a measurable function and let $\delta\in(0,1)$. Suppose that there
is a functional $a$ such that,

(2.13)
$\inf_{c\in\mathbb{R}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x)-c|^{\delta}\right)^{\frac{1}{\delta}}dx\leq
a(R)\qquad R\in\mathcal{R}.$

If the functional $a$ satisfies the $D_{p,\mathcal{R}}$ condition for some
$p>1$ or the $SD_{1,\mathcal{R}}^{s}$ condition for some $s>0$, then there is
a self-improving to $L^{1}$, namely

(2.14)
$\inf_{c\in\mathbb{R}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x)-c|dx\leq
C\,a(R),\qquad R\in\mathcal{R},$

where $C$ depends on $p,\delta,s,\|a\|_{D_{p,\mathcal{R}}}$ or
$\|a\|_{SD^{s}_{1,\mathcal{R}}}$.

######  Remark 2.15.

This is also very interesting even in the case of $BMO$. In that case $a(R)=1$
and the question can be phrased in terms of minimal conditions on the starting
estimate (2.13) to conclude the membership to $BMO$. An approach to this
problem (in the context of cubes) using Bellman functions can be found in
[LSSVZ15]. However, this result has been improved in several ways in the more
recent work [CPR]. It may be possible to adapt these ideas to the context of
the current paper.

###  2.5. Analysis in $\mathfrak{R}$

We will show that we can apply our techniques to obtain other type of results
for the class $\mathfrak{R}$ which denotes the family formed by rectangles $R$
of the form $R=I_{1}\times I_{2}\subset\mathbb{R}^{n}$ where
$I_{1}\subset\mathbb{R}^{n_{1}}$ and $I_{2}\subset\mathbb{R}^{n_{2}}$ are
cubes with sides parallel to the coordinate axes. We will discuss different
possibilities for the starting inequality (1.2) which will lead to different
results on Poincaré-Sobolev type inequalities more in the spirit of [ST93,
LW98].

The first results in the product case context were obtained by X. L. Shi and
A. Torchinsky in [ST93]. They obtained $(q,p)$-Poincaré-Sobolev inequalities
with weights satisfying certain strong conditions. Later on, Seng-Kee Chua
[Chu95] obtained some results assuming mixed norm assumptions on the gradient
which we will not consider. Later on, G. Lu and R. Wheeden also derived in
[LW98] Poincaré-Sobolev inequalities in the context of vector fields. We
improve these results in the classical context. This case could be certainly
treated as a particular case of the general theory developed for the family
$\mathcal{R}$, since the selfimproving method works in the same way. But here
we will consider different starting points, better adapted to this particular
geometry.

We will use the following notation: for a given function $f:U\to\mathbb{R}$
defined on the open set $U\subset\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$,
we will write $f(x)=f(x_{1},x_{2})$ where $x_{1}$ stands for the first $n_{1}$
variables and $x_{2}$ stands for the remaining $n_{2}$ variables.
$\nabla_{1}f$ will denote the partial gradient of $f$ containing the
$x_{1}$-derivatives and similarly $\nabla_{2}f$ will denote the partial
gradient of $f$ containing the $x_{2}$-derivatives. Recall that the family
$\mathfrak{R}$ is defined by rectangles of the form $R=I_{1}\times I_{2}$
where in $I_{1}\subset\mathbb{R}^{n_{1}}$ and $I_{2}\subset\mathbb{R}^{n_{2}}$
are cubes with sides parallel to the coordinate axes and $n:=n_{1}+n_{2}$. As
already mentioned, the classical method to derive these type of results is
based on finding a representation formula and this is the way was done in the
first paper in this context [ST93]. We follow a completely different path
using some of the key ideas from [FPW98].

######  Theorem 2.16.

Let $w\in A_{q,\mathfrak{R}}$ in $\mathbb{R}^{n}$ and $1\leq q\leq p.$ Let
also

$\frac{1}{p}-\frac{1}{p^{*}}=\frac{1}{n}\frac{1}{(q+\log[w]_{A_{q,\mathfrak{R}}})}.$

Then, there exists a constant $c=c(n,p,q)>0$ such that for every Lipschitz
function $f$ and any $R=I_{1}\times I_{2}\in\mathfrak{R}$,

$\left\|f-f_{R}\right\|_{L^{p^{*}}(R,\frac{wdx}{w(R)})}\leq
c\,[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p}}\left(a_{1}(R)+a_{2}(R)\right),$

where

$a_{1}(R)=\ell(I_{1})\left\|\nabla_{1}f\right\|_{L^{p}(R,\frac{wdx}{w(R)})}\quad\text{
and }\quad
a_{2}(R)=\ell(I_{2})\left\|\nabla_{2}f\right\|_{L^{p}(R,\frac{wdx}{w(R)})}.$

This result will be proved in Section 6. As far as we know these results are
not known in the literature (for instance [Chu95, LW98]) even for the case
$p>1$ which is usually simpler. We will show that this theorem will follow
essentially from the following $(1,1)$-Poincaré inequality in product spaces
which seems to be unknown.

######  Lemma 2.17.

There exists a dimensional constant $c>0$ such that for every Lipschitz
function $f:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\to\mathbb{R}$ and for
any $R=I_{1}\times I_{2}\in\mathfrak{R}$,

(2.15)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|\leq
c\,\ell(I_{1})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|\nabla_{1}f|+c\,\ell(I_{2})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|\nabla_{2}f|.$

This lemma is inspired by the work of Shi and Torchinsky [ST93] but they do
not consider the case $p=1$ since their method fails. Inequality (2.15) will
play a similar role as (1.1). However, there are two different points which
makes the initial estimate (2.15) different to the one we have been
considering so far. First is that we use rectangles in the family
$\mathfrak{R}$ and second that the right hand side is formed by the sum of two
functionals.

We observe that if we enlarge (2.15) replacing both sidelenghts, $\ell(I_{1})$
and $\ell(I_{2})$, by the larger quantity $d(R)$, then the new functional is
essentially the right hand side of (1.1) with $\Omega=R$ since
$|\nabla_{1}f|+|\nabla_{2}f|\approx|\nabla f|$.

We will prove Lemma 2.17 in Section 5 in two ways. The first proof will follow
from Proposition 5.1 as a consequence of a point-wise estimate obtained by Lu
and Wheeden in [LW98]. Additionally, we will provide a second proof as a
consequence of a “fractional” Poincaré type inequality proven in Proposition
5.2. This proposition will be further improved in the following theorem which
can be seen as a biparametric version of the results from [BBM02] and improved
by M. Milman in [M05] (see also [MS02] and the recent interesting works
[BVY21] [DD] [DM10]).

######  Theorem 2.18.

Let $p_{1},p_{2}\in[1,\infty)$ and let $\delta_{1},\delta_{2}\in(0,1).$ Then
there exist dimensional constants $c_{n_{1}},c_{n_{2}}>0$ such that for any
rectangle $R=I_{1}\times I_{2}\in\mathfrak{R}$

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|\leq
c_{n_{1}}(1-\delta_{1})^{\frac{1}{p_{1}}}a_{1}(R)+\,c_{n_{2}}\,(1-\delta_{2})^{\frac{1}{p_{2}}}a_{2}(R),$

where

$a_{1}(R)=\ell(I_{1})^{\delta_{1}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|^{p_{1}}}{|x_{1}-y_{1}|^{n_{1}+p_{1}\delta_{1}}}\,dy_{1}dx_{1}\
dx_{2}\right)^{\frac{1}{p_{1}}}$

and

$a_{2}(R)=\ell(I_{2})^{\delta_{2}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{2}}\frac{|f(y_{1},x_{2})-f(y_{1},y_{2})|^{p_{2}}}{|x_{2}-y_{2}|^{n_{2}+p_{2}\delta_{2}}}\
dy_{1}dx_{2}dy_{2}.\right)^{\frac{1}{p_{2}}}$

Once we have Theorem 2.18 proved, it is immediate to obtain Lemma 2.17 from
the case $p_{1}=p_{2}=1$.

This result will be the starting point to derive a bi-parametric version of
the following (one-parameter) result which can be found in [HMPV].

######  Theorem 2.19.

([HMPV]) Let $\delta\in(0,1)$ and $w\in A_{1}$. Also let $p_{\delta,w}^{*}$ be
the weighted fractional Sobolev exponent defined by

(2.16)
$\frac{1}{p}-\frac{1}{p_{\delta,w}^{*}}=\frac{\delta}{n}\,\frac{1}{1+\log[w]_{A_{1}}}$

There is a constant $C_{n,p}>0$ such that for every cube $Q$ in
$\mathbb{R}^{n}$,

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|u-u_{Q}|^{p_{\delta,w}^{*}}\,wdx\right)^{\frac{1}{p_{\delta,w}^{*}}}\leq
C_{n,p}\,(1-\delta)^{\frac{1}{p}}\,[w]^{\frac{1}{p}}_{A_{1}}a(Q),$

where

$a(Q)=\ell(Q)^{\delta}\,\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+p\delta}}\,dy\,wdx\right)^{\frac{1}{p}}.$

The bi-parametric counterpart of this result is the following.

######  Theorem 2.20.

Let $\delta\in(0,1)$ and $w\in A_{1,\mathfrak{R}}$. Also, let

(2.17)
$\frac{1}{p}-\frac{1}{p_{\delta,w}^{*}}=\frac{\delta}{n}\,\frac{1}{1+\log[w]_{A_{1,\mathfrak{R}}}}$

Then there exists a constant $c$ such that for each $R=I_{1}\times
I_{2}\in\mathfrak{R}$

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|^{p_{\delta,w}^{*}}\,wdx\right)^{\frac{1}{p_{\delta,w}^{*}}}\leq
c\,[w]_{A_{1,\mathfrak{R}}}^{\frac{1}{p}}\,(1-\delta)^{\frac{1}{p}}\Big{(}a_{1}(R)+a_{2}(R)\Big{)}$

where

$a_{1}(R)=\left(\frac{\ell(I_{1})^{p\delta}}{w(R)}\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|^{p}}{|x_{1}-y_{1}|^{n_{1}+p\delta}}\,w(x_{1},x_{2})dx_{1}dx_{2}\
dy_{1}\right)^{\frac{1}{p}}$

and

$a_{2}(R)=\left(\frac{\ell(I_{2})^{p\delta}}{w(R)}\int_{R}\int_{I_{2}}\frac{|f(y_{1},x_{2})-f(y_{1},y_{2})|^{p}}{|x_{2}-y_{2}|^{n_{2}+p\delta}}\,w(y_{1},y_{2})\,dy_{1}dy_{2}\
dx_{2}\right)^{\frac{1}{p}}.$

This result is completely new and the proof can be found in Section 7. An
interesting question here is to extend this result beyond
$A_{1,\mathfrak{R}}$, for instance to the $A_{p,\mathfrak{R}}$ case which is
not known even in the one parameter situation, Theorem 2.20.

###  2.6. Outline

The paper is organized as follows. In Section 3 we will include some
preliminary results needed for our proofs. In particular, we will show the
geometric summability conditions satisfied by different functionals $a(R)$. In
Section 4 we will provide the proofs of the main results in the case of
multiparameter rectangles in the family $\mathcal{R}$. In Section 5 we obtain
the crucial starting $(1,1)$-Poincaré inequalities adapted to the basis
$\mathfrak{R}$. In Section 6 we use this starting point as input for our self-
improving method to obtain Poincaré-Sobolev type inequalities for the basis
$\mathfrak{R}$ of rectangles given by product of cubes. Following the same
line of ideas, in Section 7 we obtain fractional Sobolev-Poincaré inequality
for the biparameter case as presented in Theorem 2.20 even with the extra gain
$(1-\delta)^{\frac{1}{p}}$. Finally, in the Appendix A we provide the proofs
of the so called truncation method and in Appendix B we give the proof of
Lemma 5.3.

###  2.7. Acknowledgement

The last author is very grateful to Professors Oscar Domínguez and Mario
Milman for enlighteling conversations about the results concerning fractional
(uniparametric) Poincaré inequalities with extra gain like in Theorem 5.4. In
particular we ackonwledge the very interesting work [M05].

##  3\. Preliminaries

We include here some well known preliminaries and some needed lemmas and
further properties. Throughout this section, the ambient space will be
$\mathbb{R}^{n}$ and $R$ will denote an arbitrary rectangle in $\mathcal{R}$,
the family of rectangles defined as $n$-fold product of intervals on
$\mathbb{R}$.

###  3.1. Smallness preserving functionals

We start with some model examples of functionals mentioned in the statements
of our results. As we will see, the constant eccentricity is crucial.

######  Example 3.1.

Consider the functional defined by

(3.1) $a(R)=d(R)^{\delta}\left(\frac{1}{w(R)}\int_{R}A(R,x)\,dx\right)^{1/p}$

where $A(R,x)$ is nonnegative and increasing in $R$. That is, $R_{1}\subset
R_{2}$ implies that $A(R_{1},x)\leq A(R_{2},x)$. We will verify that this
functional satisfies Definition 1.1, more precisely we will check that $a\in
SD^{n/\delta}_{p,\mathcal{R}}(w)$ for a weight $w$ in $\mathbb{R}^{n}$. More
importantly, that $\|a\|_{SD^{n/\delta}_{p,\mathcal{R}}(w)}=1$ . Consider
$q>1$ such that $\frac{\delta pq}{n}>1$. Then by the key eccentricity property
in Lemma 2.10

$\displaystyle\sum_{i}a(R_{i})^{p}w(R_{i})$ $\displaystyle=$
$\displaystyle\sum_{i}d(R_{i})^{\delta p}\int_{R_{i}}A(R_{i},x)\,dx$
$\displaystyle=$ $\displaystyle\frac{1}{e(R)^{\delta
p}}\sum_{i}|R_{i}|^{p\delta/n}\int_{R_{i}}A(R_{i},x)\,dx$ $\displaystyle\leq$
$\displaystyle\frac{1}{e(R)^{\delta p}}\left(\sum_{i}|R_{i}|^{\frac{\delta
pq}{n}}\right)^{1/q}\left(\sum_{i}\left[\int_{R_{i}}A(R_{i},x)\,dx\right]^{q^{\prime}}\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\leq$ $\displaystyle\frac{1}{e(R)^{\delta
p}}\left(\sum_{i}|R_{i}|\right)^{\frac{\delta p}{n}}\int_{R}A(R,x)\,dx.$

Now, the eccentricity definition gives the desired bound,

$\displaystyle\sum_{i}a(R_{i})^{p}w(R_{i})$ $\displaystyle\leq$
$\displaystyle\frac{|R|^{\frac{\delta p}{n}}}{e(R)^{\delta
p}}\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p}{n}}\,\int_{R}A(R,x)\,dx$ $\displaystyle=$ $\displaystyle d(R)^{\delta
p}\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p}{n}}\int_{R}A(R,x)\,dx$ $\displaystyle=$
$\displaystyle\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p}{n}}\,a(R)^{p}w(R).$

######  Example 3.2.

The example above belongs to a rather wide family of functionals given by
averages of general measures:
$a(R)=d(R)^{\delta}\left(\frac{\mu(R)}{w(R)}\right)^{1/p}$, $\delta>0$. The
previous computation shows that $a\in SD_{p,\mathcal{R}}^{n/\delta}(w)$ for
any weight $w$ in $\mathbb{R}^{n}.$

The above estimates are sufficient to obtain $(p,p)$-Poincaré inequalities
since we have that $\|a\|_{SD^{n/\delta}_{p,\mathcal{R}}(w)}\leq 1$. In order
to reach higher exponents of integrability on the left hand side and get a
Poincaré-Sobolev type inequality, we need to pay a price allowing
$\|a\|_{SD_{p,\mathcal{R}}^{s}(w)}$ to be larger than 1. A sort of natural way
to do this is to consider the involved weight to be in a “better”
$A_{q,\mathcal{R}}$ class. We follow the line of ideas from [PR19] adapted to
our setting of rectangles.

######  Lemma 3.3.

Consider two indices $p,q$ such that $1\leq q\leq p<n$ and let $w\in
A_{q,\mathcal{R}}$ in $\mathbb{R}^{n}$. For $\delta>0$, consider the
functional

$a(R)=d(R)^{\delta}\left(\frac{1}{w(R)}\mu(R)\right)^{1/p}\qquad
R\in\mathcal{R}.$

For $M>1$ we define $p_{M}^{*}:=p(n,q,M)$ by the condition

$\frac{1}{p}-\frac{1}{p_{M}^{*}}=\frac{\delta}{nqM}.$

Then $a$ satisfies Definition 1.1 with parameters $p=p^{*}_{M}$ and
$s=\frac{nM^{\prime}}{\delta}$, namely if $R\in\mathcal{R}$ and
$\\{R_{i}\\}_{i}$ is any family of pairwise disjoint dyadic subrectangles of
$R$ the following inequality holds:

(3.2)
$\left(\sum_{i}a(R_{i})^{p^{*}_{M}}\,\frac{w(R_{i})}{w(R)}\right)^{\frac{1}{p^{*}_{M}}}\leq[w]_{A_{q,\mathcal{R}}}^{\frac{\delta}{nqM}}\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta}{nM^{\prime}}}a(R).$

Note that $p_{M}^{*}$ is smaller than the classical Sobolev exponent, namely
the sharp one corresponding to the Lebsegue measure case:

$p<p_{M}^{*}<p^{*}=\frac{pn}{n-p}\qquad 1\leq p<n.$

This condition says that the functional $a$ “preserves smallness” for the
exponent $p_{M}^{*}$ defined in (2.3) with index
$s=\frac{nM^{\prime}}{\delta}$ and constant

$\|a\|_{SD_{p^{*}_{M},\mathcal{R}}^{\frac{nM^{\prime}}{\delta}}(w)}\leq[w]_{A_{q,\mathcal{R}}}^{\frac{\delta}{nqM}}.$

That is, $a\in SD_{p^{*}_{M},\mathcal{R}}^{\frac{nM^{\prime}}{\delta}}(w)$.
The notation here is somewhat involved, but we will use it in a more
simplified manner in our applications. Recall that, as in the case of standard
cubic $A_{q}$ weights, the strong $A_{q,\mathcal{R}}$ weights satisfy the
geometric estimate

$\left(\frac{|E|}{|R|}\right)^{q}\leq[w]_{A_{q,\mathcal{R}}}\frac{w(E)}{w(R)},$

valid for any subset $E\subset R$ and any $R\in\mathcal{R}$. Now we can
proceed to the proof of the main estimate for the functional $a$.

###### Proof of Lemma 3.3.

Let $M>1$. For simplicity in the exposition, we will omit the subindex $M$ and
just use $p^{*}$ instead of $p^{*}_{M}$. To verify the smallness preservation
for the functional $a$, we compute, using again the key eccentricity property
in Lemma 2.10,

$\displaystyle\sum_{i}a(R_{i})^{p^{*}}w(R_{i})$ $\displaystyle=$
$\displaystyle\sum_{i}\mu(R_{i})^{\frac{p^{*}}{p}}\left(\frac{d(R_{i})^{\delta}}{w(R_{i})^{\frac{1}{p}-\frac{1}{p^{*}}}}\right)^{p^{*}}$
$\displaystyle=$ $\displaystyle\frac{1}{e(R)^{\delta
p^{*}}}\sum_{i}\mu(R_{i})^{\frac{p^{*}}{p}}\left(\frac{|R_{i}|^{\frac{\delta}{n}}}{w(R_{i})^{\frac{\delta}{qnM}}}\right)^{p^{*}}$
$\displaystyle=$ $\displaystyle\frac{1}{e(R)^{\delta
p^{*}}}\sum_{i}\mu(R_{i})^{\frac{p^{*}}{p}}\left(\frac{|R_{i}|}{w(R_{i})^{\frac{1}{qM}}}\right)^{\frac{\delta
p^{*}}{n}}$ $\displaystyle\leq$
$\displaystyle\frac{[w]_{A_{q,\mathcal{R}}}^{\frac{\delta
p^{*}}{nqM}}}{e(R)^{\delta
p^{*}}}\left(\frac{|R|^{q}}{w(R)}\right)^{\frac{\delta p^{*}}{nqM}}\
\sum_{i}\mu(R_{i})^{\frac{p^{*}}{p}}|R_{i}|^{\frac{\delta
p^{*}}{nM^{\prime}}}.$

Now consider any $t>1$ such that $t\frac{\delta p^{*}}{nM^{\prime}}\geq 1$ and
apply Hölder’s inequality to the sum above to obtain

$\displaystyle\sum_{i}\mu(R_{i})^{\frac{p^{*}}{p}}|R_{i}|^{\frac{\delta
p^{*}}{nM^{\prime}}}$ $\displaystyle\leq$
$\displaystyle\left(\sum_{i}\mu(R_{i})^{\frac{t^{\prime}p^{*}}{p}}\right)^{\frac{1}{t^{\prime}}}\left(\sum_{i}|R_{i}|\right)^{\frac{\delta
p^{*}}{nM^{\prime}}}$ $\displaystyle\leq$
$\displaystyle\left(\sum_{i}\mu(R_{i})\right)^{\frac{p^{*}}{p}}\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p^{*}}{nM^{\prime}}}|R|^{\frac{\delta p^{*}}{nM^{\prime}}}.$
$\displaystyle\leq$
$\displaystyle\mu(R)^{\frac{p^{*}}{p}}\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p^{*}}{nM^{\prime}}}|R|^{\frac{\delta p^{*}}{nM^{\prime}}}.$

Therefore,

$\displaystyle\sum_{i}a(R_{i})^{p^{*}}w(R_{i})$ $\displaystyle\leq$
$\displaystyle\left(\frac{[w]_{A_{q,\mathcal{R}}}|R|^{q}}{w(R)}\right)^{\frac{\delta
p^{*}}{nqM}}\mu(R)^{\frac{p^{*}}{p}}\,\frac{|R|^{\frac{\delta
p^{*}}{nM^{\prime}}}}{e(R)^{\delta
p^{*}}}\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p^{*}}{nM^{\prime}}}$ $\displaystyle=$
$\displaystyle[w]_{A_{q,\mathcal{R}}}^{\frac{\delta
p^{*}}{nqM}}\frac{|R|^{\frac{\delta p^{*}}{n}}}{w(R)^{\frac{\delta
p^{*}}{nqM}}}\mu(R)^{\frac{p^{*}}{p}}\frac{1}{e(R)^{\delta
p^{*}}}\,\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p^{*}}{nM^{\prime}}}$ $\displaystyle=$
$\displaystyle[w]_{A_{q,\mathcal{R}}}^{\frac{\delta
p^{*}}{nqM}}a(R)^{p^{*}}w(R)\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta
p^{*}}{nM^{\prime}}}.$

∎

###  3.2. Optimal polynomials

We also include here some preliminaries on optimal polynomials. Given a
rectangle $R\in\mathcal{R}$ and an integer $m\geq 0$, we consider the space
$\mathcal{P}_{m}$ of polynomials of degree at most $m$ in $n$ variables
endowed with the inner product given by

$<f,g>_{R}:=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}fgdx.$

There is an orthonormal basis with respect to this inner product that we will
denote by $\\{\phi_{\alpha}\\}$, being $\alpha=(\alpha_{1},\dots,\alpha_{n})$
a multiindex of non negative integers such that
$|\alpha|=\alpha_{1}+\cdots+\alpha_{n}\leq m$. An important feature is that

(3.3) $\|\phi_{r}\|_{L^{\infty}}\leq
C\left(\frac{1}{|R|}\int_{R}|\phi_{r}|^{2}dx\right)^{1/2}=C,$

since the space $\mathcal{P}_{m}$ is finite dimensional and therefore all
norms are equivalent. Let $P_{R}$ the projection defined by the formula

(3.4)
$P_{R}(f)=\sum_{r}\left(\frac{1}{|R|}\int_{R}f\phi_{r}dx\right)\phi_{r}.$

We clearly have from (3.3) that

(3.5) $\|P_{R}f\|_{L^{\infty}}\leq
NC^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f|,$

where $N$ depends on $m$. Moreover, as it is the case when $m=0$ and the
projection is over the constants, we have the following optimality property:

$\inf_{\pi\in\mathcal{P}_{m}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-\pi|^{p}\right)^{1/p}\approx\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-P_{R}f|^{p}\right)^{1/p}.$

###  3.3. $A_{\infty,\mathcal{R}}$ weights

We recall that the “strong” $A_{\infty,\mathcal{R}}$ class of weights is
usually defined as

$A_{\infty,\mathcal{R}}=\bigcup_{p>1}A_{p,\mathcal{R}}.$

There are several different characterizations of the belonging of a weight to
this class. We choose here to work with the so called Fujii-Wilson constant
defined as

(3.6)
$[w]_{A_{\infty,\mathcal{R}}}:=\sup_{R\in\mathcal{R}}\frac{1}{w(R)}\int_{R}M_{s}(w\chi_{R})\
dx<\infty.$

Although implicit in [Fuj78], this definition was initially considered in
[HP13] and used to derive a quite sharp Reverse Hölder Inequality (RHI)
estimate. We cite here an improved version from [HPR12], which is similar to
the one obtained [LPR17] in the context of rectangles. The proof is an easy
adaptation from cubes to rectangles and therefore will be omitted.

######  Theorem 3.4.

Let $w\in A_{\infty,\mathcal{R}}$ in $\mathbb{R}^{n}$. Then for any rectangle
$R\in\mathcal{R}$,

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}w^{1+\varepsilon}\
dx\leq
2\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}w\
dx\right)^{1+\varepsilon},$

for any $\varepsilon>0$ such that
$0<\varepsilon\leq\frac{1}{2^{n+1}[w]_{A_{\infty,\mathcal{R}}}-1}$.

##  4\. Proofs Part I: Analysis in $\mathcal{R}$

In this section we will present the proof of our main contributions within the
family of rectangles $\mathcal{R}$ ($n$-fold product of one dimensional
intervals).

We begin by proving Theorem 2.3. To do this we need some previous results.

Given a rectangle $R\in\mathcal{R}$, $M^{d}_{R}$ will stand for the dyadic
maximal function adapted to the rectangle $R$. That is,

$M^{d}_{R}(f)(x):=\sup_{J\ni
x}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{J}|f(y)|\
dy\qquad J\in\mathcal{D}(R)$

where $\mathcal{D}(R)$ is the family of dyadic subrectangles of $R$.

We will also use the appropriate sharp maximal function defined by

$M_{m}^{\sharp}f(x)=\sup_{R\ni
x}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(y)-P_{R}f|\,dy,$

where $m$ is the degree and $P_{R}f$ is the polynomial approximation of degree
less than $m$.

We will need the following result which is similar to the one obtained in
[CP21] where several extensions of the classical John-Nirenberg’s theorem has
been derived. Our result is the context of the basis $\mathcal{R}$ but the
proof is essentially the same as the one given in [CP21] for cubes. It is also
an extension to $\mathcal{R}$ of one of the main results of Karagulyan in
[Kar02]. The key point is to use a Calderón–Zigmund (C–Z) dyadic decomposition
adapted to a given fixed rectangle $R\in\mathcal{R}$. The validity of such a
C–Z decomposition is immediate from the cubic case: the dyadic structure is
exactly the same in both cases. In addition, we will introduce the following
notation for $w\in A_{\infty,\mathcal{R}}$, $r>1$ and for any rectangle
$R\in\mathcal{R}$:

$w_{r}(R)=|R|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}w^{r}\right)^{\frac{1}{r}}=|R|^{\frac{1}{r^{\prime}}}\left(\int_{R}w^{r}\right)^{\frac{1}{r}}.$

######  Theorem 4.1.

Let $f\in L_{loc}^{1}(\mathbb{R}^{n})$ and let $1\leq p<\infty$ and
$1<r<\infty$. Then there is a dimensional constant $c$ such that for any
rectangle $R\in\mathcal{R}$ the following estimate holds

(4.1)
$\left(\frac{1}{w_{r}(R)}\int_{R}\left(\frac{M^{d}_{R}(f-P_{R}f)(x)}{M_{m}^{\sharp}f(x)}\,\right)^{p}w(x)dx\right)^{\frac{1}{p}}\leq
c\,pr^{\prime}.$

Hence, if further $w\in A_{\infty,\mathcal{R}}$ we have

(4.2)
$\left(\frac{1}{w(R)}\int_{R}\left(\frac{M^{d}_{R}(f-P_{R}f)(x)}{M_{m}^{\sharp}f(x)}\,\right)^{p}w(x)dx\right)^{\frac{1}{p}}\leq
c\,p\,[w]_{A_{\infty,\mathcal{R}}}.$

The proof of (4.1) follows the same ideas as in the proof of [CP21, Theorem
7.1]. If we assume the extra condition of $w$ being an $A_{\infty}$ weight,
then we can apply the reverse Hölder inequality for $\mathcal{R}$ stated in
Theorem 3.4 to get (4.2).

######  Corollary 4.2.

Let $f$ and $w$ as before. Then for any rectangle $R\in\mathcal{R}$

$\mbox{$\left\|\frac{\displaystyle M^{d}_{R}(f-P_{R}f)}{\displaystyle
M_{m}^{\sharp}f}\right\|$}_{\exp L(R,\frac{wdx}{w(R)})}\leq
c\,[w]_{A_{\infty,\mathcal{R}}}$

and hence there exist dimensional constants $c_{1},c_{2}>0$ such that for any
rectangle $R\in\mathcal{R}$ and $\lambda,\gamma>0$

$w\big{(}\\{x\in
R:M^{d}_{R}(f-P_{R}f)>\lambda,M_{m}^{\sharp}f(x)\leq\gamma\lambda\\}\big{)}\leq
c_{1}\,{e^{\frac{-c_{2}}{\gamma[w]_{A_{\infty,\mathcal{R}}}}}}\,w(R).$

###### Proof of Theorem 2.3.

With all these ingredients, the proof is an adaptation of the main result in
[FPW98] combined wth the exponential decay good-$\lambda$ type formula from
Corollary 4.2 very much as in [CP21]. We omit the details.

∎

We will provide here the proof of our main general result Theorem 2.1. Before
to proceed with the proof we recall that what we need to prove is that the
quantity

$X=\sup_{R\in\mathcal{R}}\left(\frac{1}{w(R)}\int_{R}\left|\frac{f-P_{R}f}{a(R)}\right|^{p}\,wdx\right)^{1/p}$

is bounded with a precise control on the bound. A brief comment on a technical
step in the proof is necessary here. Since the argument involves the trick of
bounding $X$ by some variation of itself, the partial step of proving that it
is finite is crucial. In many standard situations, this can be achieved by
assuming that the function $f$ is bounded and also by perturbing the
functional $a$ to avoid any possible degeneracy considering
$a_{\varepsilon}(R):=a(R)+\varepsilon$, $\varepsilon>0$. Once a uniform bound
involving $a_{\varepsilon}$ is obtained, a limiting argument proves that the
original $X$ is also finite. The problem is that the hypothesis on $a$
satisfying a $SD_{p,\mathcal{R}}^{s}$ condition is not shared by the perturbed
new functional $a_{\varepsilon}$. However, some weaker variant is indeed true
and it is sufficient for the proof. Here (and only here!) is where the
$A_{\infty}$ condition pops in, remarkably absent in the quantitative
estimate. We will omit the technical details, we refer the reader to [PR19]
for a detailed discussion.

###### Proof of Theorem 2.1.

For the proof we follow the ideas from [PR19], starting with a standard local
C–Z decomposition in the context of $\mathcal{R}$.

We consider the local C-Z decomposition of $\frac{|f-P_{R}f|}{a(R)}$ relative
to $R$ at level $L$ for a large universal constant $L>1$ to be chosen. Let
$\mathcal{D}(R)$ be the family of dyadic subrectangles of $R$ which also
belong to $\mathcal{R}$. The C–Z decomposition yields a collection
$\\{R_{j}\\}_{j}$ of rectangles such that $R_{j}\in\mathcal{D}(R)$, maximal
with respect to inclusion, satisfying

(4.3) $L<\frac{1}{|R_{j}|}\int_{R_{j}}\frac{|f-P_{R}f|}{{a(R)}}\,dy.$

Then, if $R$ is dyadic with $R\supset R_{j}$

(4.4) $\frac{1}{|R|}\int_{R}\frac{|f-P_{R}f|}{{a(R)}}\,dy\leq L$

and hence

(4.5) $L<\frac{1}{|R_{j}|}\int_{R_{j}}\frac{|f-P_{R}f|}{{a(R)}}\,dy\leq
L\,2^{n}$

for each integer $j$. Also note that

$\left\\{x\in
R:M_{R}^{d}\left(\frac{|f-P_{R}f|}{{a(R)}}\chi_{R}\right)(x)>L\right\\}=\bigcup_{j}R_{j}=:\Omega_{L}$

where $M^{d}_{R}$ stands for the dyadic maximal function adapted to the
rectangle $R$. That is,

$M^{d}_{R}(f)(x):=\sup_{J\ni
x}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{J}|f(y)|\
dy\qquad x\in J,J\in\mathcal{D}(R).$

Then, by the Lebesgue differentiation theorem it follows that

(4.6) $\frac{|f(x)-P_{R}f(x)|}{{a(R)}}\leq L\qquad a.e.\ x\notin\Omega_{L}.$

Also, observe that by (4.3) (or the weak type $(1,1)$ property of $M^{d}_{R}$)
and recalling our starting assumption (2.1), we have that $\\{R_{j}\\}_{j}\in
S(L)$, namely

$|\Omega_{L}|=|\bigcup_{j}R_{j}|<\frac{|R|}{L}.$

Now, given the C-Z decomposition of the cube $R$, we decompose the function
$\dfrac{f-P_{R}f}{{a(R)}}$ as

$\displaystyle f-P_{R}f$ $\displaystyle=$
$\displaystyle\left(f-P_{R}f\right)\chi_{\Omega_{L}^{c}}+\sum_{j}\left(P_{R_{j}}f-P_{R}f\right)\chi_{R_{j}}+\sum_{j}\left(f-P_{R_{j}}f\right)\chi_{R_{j}}$
$\displaystyle=$ $\displaystyle A_{1}+A_{2}+A_{3}.$

For $A_{1}$ and $A_{2}$ we have a pointwise estimate. The bound for $A_{1}$
follows from (4.6). Also note that $P_{R_{j}}P_{R}=P_{R}$ and recall the bound
(3.5). Then we can control $A_{2}$ as

$\displaystyle|A_{2}(x)|$ $\displaystyle\leq$
$\displaystyle\sum_{j}\left|(P_{R_{j}}f(x)-P_{R}f(x))\right|\chi_{R_{j}}(x)$
$\displaystyle=$
$\displaystyle\sum_{j}\left|P_{R_{j}}(f-P_{R}f)\right(x)|\chi_{R_{j}}(x)$
$\displaystyle\leq$
$\displaystyle\sum_{j}NC^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{j}}|f-P_{R}f|\chi_{R_{j}}(x)$
$\displaystyle\leq$ $\displaystyle NC^{2}2^{n}La(R)$

since for any given $x\in\Omega$ there is only one $j$ such that $x\in R_{j}$.
So far we have pointwise bounds for $A_{1}$ and $A_{2}$. Now we move on to the
desired $L^{p}$ norm estimate. Consider the local weighted $L^{p}$ space on
$R$ with the measure $\mu:=\frac{w(x)dx}{w(R)}$. By the triangle inequality,
we have that

$\displaystyle\left\|\frac{f-P_{R}f}{a(R)}\right\|_{L^{p}_{\mu}}$
$\displaystyle\leq$
$\displaystyle\left\|\frac{A_{1}}{a(R)}\right\|_{L^{p}_{\mu}}+\left\|\frac{A_{2}}{a(R)}\right\|_{L^{p}_{\mu}}+\left\|\sum_{j}\frac{f-P_{R_{j}}f}{a(R)}\chi_{R_{j}}\right\|_{L^{p}_{\mu}}$
$\displaystyle\leq$
$\displaystyle\tilde{c}L+\left(\frac{1}{w(R)}\int_{R}\sum_{j}\frac{|f-P_{R_{j}}f|^{p}}{a(R)^{p}}\chi_{R_{j}}wdx\right)^{\frac{1}{p}}.$

Since the $R_{j}$’s are disjoint, we can compute

$\displaystyle\int_{R}\sum_{j}\frac{|f-P_{R_{j}}f|^{p}}{a(R)^{p}}\chi_{R_{j}}wdx$
$\displaystyle=$
$\displaystyle\sum_{j}\frac{a(R_{j})^{p}w(R_{j})}{a(R)^{p}w(R_{j})}\int_{R_{j}}\frac{|f-P_{R_{j}}f|^{p}}{a(R_{j})^{p}}wdx$
$\displaystyle\leq$ $\displaystyle
X^{p}\sum_{j}\frac{a(R_{j})^{p}w(R_{j})}{a(R)^{p}},$

where $X$ is the quantity defined as

$X=\sup_{R\in\mathcal{R}}\left(\frac{1}{w(R)}\int_{R}\frac{|f-P_{R}f|^{p}}{a(R)^{p}}wdx\right)^{1/p}.$

Averaging over $w(R)$ and raising to the $1/p$ power, we obtain

$\displaystyle\left(\frac{1}{w(R)}\int_{R}\left|\frac{f-P_{R}f}{a(R)}\right|^{p}\,wdx\right)^{1/p}$
$\displaystyle\leq$
$\displaystyle\tilde{c}L+X\left(\sum_{j}\frac{a(R_{j})^{p}w(R_{j})}{a(R)^{p}w(R)}\right)^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle\tilde{c}L+X\frac{\|a\|}{L^{1/s}}.$

Taking the supremum over all $R\in\mathcal{R}$, we finally obtain

(4.7) $X\leq\tilde{c}L+X\frac{\|a\|}{L^{1/s}},$

for any $L>1$. Choosing here an appropriate $L$ as $L=e\max\\{\|a\|^{s},1\\}$
and using the elementary fact that $\left(e^{1/s}\right)^{\prime}\leq 1+s$, we
finally get that, for any $R\in\mathcal{R}$,

$\left(\frac{1}{w(R)}\int_{R}\left|f-P_{R}f\right|^{p}\,wdx\right)^{1/p}\leq
c_{n}(1+s)\max\\{\|a\|^{s},1\\}\,a(R),$

which is the desired estimate. ∎

Now we are ready to prove the consequences that can be derived from our
general theorem. We start by proving Theorem 2.5, where we exploit the
membership of the weight $w$ to a better $A_{q,\mathcal{R}}$ class to reach a
higher Poincaré-Sobolev exponent.

###### Proof of Theorem 2.5.

Recall that the functional from (2.4) satisfies a
$SD_{p,\mathcal{R}}^{n/\delta}$ condition with norm 1 (see Example 3.2). This
is sufficient to prove $(p,p)$-Poincaré inequalities by using Theorem 2.1.
Here, in order to achieve a higher integrability, the problem reduces to prove
a better smallness preservation condition that comes with a price: the norm
with respect to the better parameters is no longer uniformly bounded on the
weights. Hence, a careful optimization has to be done. We will use the main
estimate from Lemma 3.3, choosing $M>1$ with $s=\frac{nM^{\prime}}{\delta}$,
namely

(4.8)
$\|a\|:=\|a\|_{SD_{p^{*}_{M},\mathcal{R}}^{\frac{nM^{\prime}}{\delta}}(w)}\leq[w]_{A_{q,\mathcal{R}}}^{\frac{\delta}{nqM}}$

where $\displaystyle\frac{1}{p}-\frac{1}{p_{M}^{*}}=\frac{\delta}{nqM}.$ The
choice will depend on $[w]_{A_{q,\mathcal{R}}}$ so that the expression
$s\|a\|^{s}$ remains bounded. We choose

(4.9) $M=1+\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}.$

Observe that both $M$ and
$M^{\prime}=\frac{1+\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}{\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}>1$
and also that this choice of $M$ implies that the exponent $p^{*}_{M}$ is
exactly the value $p^{*}_{w}$ from (2.3). Hence applying Theorem 2.1 with $p$
replaced by $p^{*}_{w}$ and using estimate (4.8) we obtain

$\displaystyle\left(\frac{1}{w(R)}\int_{R}|f-P_{R}f|^{p_{w}^{*}}\,wdx\right)^{\frac{1}{p_{w}^{*}}}$
$\displaystyle\leq$ $\displaystyle C_{n}s\,\|a\|^{s}\,a(R)$
$\displaystyle\leq$ $\displaystyle
C_{n}\frac{M^{\prime}}{\delta}[w]_{A_{q,\mathcal{R}}}^{\frac{\delta}{nqM}\frac{nM^{\prime}}{\delta}}\,a(R)$
$\displaystyle\leq$ $\displaystyle
C_{n}\frac{M^{\prime}}{\delta}[w]_{A_{q,\mathcal{R}}}^{\frac{1}{\log[w]_{A_{q,\mathcal{R}}}}}\,a(R)$
$\displaystyle=$ $\displaystyle
c_{n}\frac{1+\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}{\log[w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}}\frac{1}{\delta}\,a(R).$

This yields the proof of Theorem 2.5 providing a good estimate for weights
with large nontrivial $A_{q,\mathcal{R}}$ constants. A simple computation
shows that for any weight such that $[w]_{A_{q,\mathcal{R}}}>1$, if we choose
$K$ large such that $[w]_{A_{q,\mathcal{R}}}=e^{\frac{q}{K}}$, then we obtain
a cleaner inequality

$\left(\frac{1}{w(R)}\int_{R}|f-f_{R}|^{p_{w}^{*}}\,wdx\right)^{\frac{1}{p_{w}^{*}}}\leq
c_{n}(K+1)\frac{1}{\delta}\,a(R).$

∎

Now we present the proof of Theorem 2.6 covering, in particular, the case of
flat weights.

###### Proof of Theorem 2.6.

We can assume that the weight $w$ satisfies $[w]_{A_{q},\mathcal{R}}\leq
e^{q}$, since in the other case we simply use the previous Theorem 2.5. We
claim that for any family $\\{R_{i}\\}_{i}$ of pairwise disjoint dyadic
subrectangles of $R$, the following inequality holds:

(4.10)
$\left(\sum_{i}a(R_{i})^{p^{*}_{1}}\,\frac{w(R_{i})}{w(R)}\right)^{\frac{1}{p^{*}_{1}}}\leq
e^{\frac{\delta}{n}}\,a(R),$

recalling the notation
$\displaystyle\frac{1}{p}-\frac{1}{p_{1}^{*}}=\frac{\delta}{n}\frac{1}{q}$.
This means that $a\in D_{p^{*}_{1},\mathcal{R}}(w)$ with
$\|a\|_{D_{p^{*}_{1},\mathcal{R}}(w)}\leq e^{\frac{\delta}{n}}$ according to
Definition 1.3. Let’s postpone the proof of the claim for the time being. Now,
applying Theorem 2.3, we get

$\displaystyle\|f-f_{R}\|_{L^{p^{*}_{1},\infty}\big{(}R,\frac{w\,dx}{w(R)}\big{)}}$
$\displaystyle\leq$ $\displaystyle
c\,p^{*}_{1}\,[w]_{A_{\infty,\mathcal{R}}}\,\|a\|_{D_{p^{*}_{1},\mathcal{R}}(w)}\,a(R)$
$\displaystyle\leq$ $\displaystyle
c\,p^{*}_{1}\,[w]_{A_{q,\mathcal{R}}}\,\|a\|_{D_{p^{*}_{1},\mathcal{R}}(w)}\,a(R)$
$\displaystyle\leq$ $\displaystyle
c\,p^{*}_{1}\,e^{q}\,e^{\frac{\delta}{n}}\,a(R).$

Consider here the same choice of
$M=1+\log[w]^{\frac{1}{q}}_{A_{q,\mathcal{R}}}$ as in the proof of Theorem 2.5
above. Since $p^{*}_{1}>p^{*}_{M}$ (just note that $p^{*}_{M}$ is a decreasing
function on $M$), by Jensen’s inequality which holds at weak level (simply use
that the inner part of what is inside $()^{\frac{1}{p}}$ is less or equal than
one) we have,

$\displaystyle\|f-f_{R}\|_{L^{p^{*}_{M},\infty}\big{(}R,\frac{w\,dx}{w(R)}\big{)}}$
$\displaystyle\leq$
$\displaystyle\|f-f_{R}\|_{L^{p^{*}_{1},\infty}\big{(}R,\frac{w\,dx}{w(R)}\big{)}}$
$\displaystyle\leq$ $\displaystyle
c\,p^{*}_{1}\,e^{q}\,e^{\frac{\delta}{n}}\,a(R).$

This is the desired weak type estimate. To conclude the proof of the theorem
we need to check claim (4.10). But this is a variant of the proof of Lemma
3.3, indeed by the key eccentricity property in Lemma 2.10,

$\displaystyle\sum_{i}a(R_{i})^{p^{*}_{1}}w(R_{i})$ $\displaystyle=$
$\displaystyle\sum_{i}\mu(R_{i})^{\frac{p^{*}_{1}}{p}}\left(\frac{d(R_{i})^{\delta}}{w(R_{i})^{\frac{1}{p}-\frac{1}{p^{*}_{1}}}}\right)^{p^{*}_{1}}$
$\displaystyle=$ $\displaystyle\frac{1}{e(R)^{\delta
p^{*}_{1}}}\sum_{i}\mu(R_{i})^{\frac{p^{*}_{1}}{p}}\left(\frac{|R_{i}|^{\frac{\delta}{n}}}{w(R_{i})^{\frac{\delta}{qn}}}\right)^{p^{*}_{1}}$
$\displaystyle=$ $\displaystyle\frac{1}{e(R)^{\delta
p^{*}_{1}}}\sum_{i}\mu(R_{i})^{\frac{p^{*}_{1}}{p}}\left(\frac{|R_{i}|}{w(R_{i})^{\frac{1}{q}}}\right)^{\frac{\delta
p^{*}_{1}}{n}}$ $\displaystyle\leq$ $\displaystyle\frac{1}{e(R)^{\delta
p^{*}_{1}}}\sum_{i}\mu(R_{i})^{\frac{p^{*}_{1}}{p}}\left([w]_{A_{q,\mathcal{R}}}^{\frac{1}{q}}\frac{|R|}{w(R)^{\frac{1}{q}}}\right)^{\frac{\delta
p^{*}_{1}}{n}}$ $\displaystyle\leq$ $\displaystyle\frac{e^{\frac{\delta
p^{*}_{1}}{n}}}{e(R)^{\delta
p^{*}_{1}}}\left(\frac{|R|^{q}}{w(R)}\right)^{\frac{\delta p^{*}_{1}}{nq}}\
\sum_{i}\mu(R_{i})^{\frac{p^{*}_{1}}{p}}$ $\displaystyle\leq$
$\displaystyle\frac{e^{\frac{\delta p^{*}_{1}}{n}}}{e(R)^{\delta
p^{*}_{1}}}\left(\frac{|R|^{q}}{w(R)}\right)^{\frac{\delta p^{*}_{1}}{nq}}\
\mu(R)^{\frac{p^{*}_{1}}{p}}$ $\displaystyle=$ $\displaystyle e^{\frac{\delta
p^{*}_{1}}{n}}a(R)^{p_{1}^{*}}w(R)$

since we assumed that $[w]_{A_{q},\mathcal{R}}\leq e^{q}$. This gives the
claim (4.10) finishing the proof of the theorem. ∎

###### Proof of Corollary 2.7.

In order to apply our self improving method from Theorem 2.5 or Theorem 2.6,
we need a starting point as (2.5) where the functional $a$ is of the form
(2.4). Regarding the starting point, we know from the work of Chua in [Chu06,
Lemma 2.5] that the following unweighted inequality holds 111The results in
[Chu06] are derived for cubes but the same results hold for rectangles as
well.

$\frac{1}{|R|}\int_{R}\mbox{$|f-P_{R}f|$}\,dx\leq
C\,\frac{d(R)^{m}}{|R|}\int_{R}\mbox{$|\nabla^{m}f|$}\,dx,$

where $P_{R}f$ is the “approximating” polynomial as defined in (3.4) and for a
class of Sobolev functions with appropriate smoothness, namely $f\in
W^{m,1}(R)$. This result follows from the well known first order
$(1,1)$-Poincaré inequality on convex sets. From this inequality and using the
standard argument of introducing an $A_{p,\mathcal{R}}$ weight we derive that

(4.11)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mbox{$|f-P_{R}f|$}\,dx\leq
C\,[w]^{1/p}_{A_{p,\mathcal{R}}}d(R)^{m}\left(\frac{1}{w(R)}\int_{R}\mbox{$|\nabla^{m}f|$}^{p}w\,dx\right)^{\frac{1}{p}}.$

Hence, we have our starting point

$\frac{1}{|R|}\int_{R}\mbox{$|f-P_{R}f|$}\,dx\leq a(R),$

where the functional $a$ is defined by

$a(R)=C\,[w]^{\frac{1}{p}}_{A_{p,\mathcal{R}}}d(R)^{m}\left(\frac{1}{w(R)}\int_{R}\mbox{$|\nabla^{m}f|$}^{p}w\,dx\right)^{\frac{1}{p}}$

which is a particular case of (3.1). An application of Theorem 2.5 gives the
result for nontrivial weights with $A_{q}$ constant bounded from below by
$e^{q}$. For a result valid for flat weights, including the unweighted
classical Poincaré-Sobolev inequality for the Lebesgue measure, we can only
use Theorem 2.6 to get a weak inequality. ∎

Now we move on to prove Corollary 2.11.

###### Proof of Corollary 2.11.

Again, the key resides in proving a suitable starting point. Consider, for
$\delta>0$, the functional defined by the expression

$a(R):=[w]^{\frac{1}{p}}_{A_{p,\mathcal{R}}}\frac{d(R)^{\delta}}{e(R)^{\frac{n}{p}}}\left(\frac{1}{w(R)}\int_{R}A(R,x)\,w(x)dx\right)^{\frac{1}{p}}$

where

(4.12) $A(R,x)=\int_{R}\frac{|f(x)-f(y)|^{p}}{|x-y|^{\delta p+n}}\,w(y)dy.$

Note that by Hölder’s inequality combined with the definition, again, of the
$A_{p,\mathcal{R}}$ condition together with the concept of eccentricity we
obtain

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|$
$\displaystyle\approx$
$\displaystyle\frac{1}{|R|}\int_{R}\frac{1}{|R|}\int_{R}|f(x)-f(y)|\,dy\,dx$
$\displaystyle\leq$
$\displaystyle[w]^{\frac{1}{p}}_{A_{p,\mathcal{R}}}\left(\frac{1}{w(R)|R|}\int_{R}\int_{R}|f(x)-f(y)|^{p}\,dy\,w(x)dx\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle[w]^{\frac{1}{p}}_{A_{p,\mathcal{R}}}\,\left(\frac{d(R)^{\delta
p}}{e(R)^{n}w(R)}\int_{R}\int_{R}\frac{|f(x)-f(y)|^{p}}{|x-y|^{\delta
p+n}}\,dy\,w(x)dx\right)^{\frac{1}{p}}$ $\displaystyle=$
$\displaystyle[w]^{\frac{1}{p}}_{A_{p,\mathcal{R}}}\,\frac{d(R)^{\delta}}{e(R)^{\frac{n}{p}}}\left(\frac{1}{w(R)}\int_{R}\int_{R}\frac{|f(x)-f(y)|^{p}}{|x-y|^{\delta
p+n}}\,dy\,w(x)dx\right)^{\frac{1}{p}}$ $\displaystyle=$ $\displaystyle a(R).$

Note that the functional $a$ here is not _exactly_ the same as in the model
example (2.4) due to the presence of the eccentricity in front. But since we
are testing the smallness preservation on families of dyadic rectangles on a
fixed rectangle $R$, any function on the eccentricity moves out of the sum and
therefore this kind of functional also satisfies Lemma 3.3. Again, the result
follows from Theorem 2.5. ∎

Next we present here the proof of Theorem 2.14. Recall that the idea is that
even only being able to start from a weaker starting point (2.13), in the end
we arrive at the same place, namely to the usual $L^{1}$ oscillation initial
point.

###### Proof of Theorem 2.14.

Note that proving (2.14) is the same as proving that the following quantity

(4.13)
$X:=\sup_{R\in\mathcal{R}}\inf_{c\in\mathbb{R}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\frac{|f(x)-c|}{a(R)}dx$

is finite. There are two reductions that make this trivial: the function $f$
being bounded and the functional $a$ being uniformly away from zero. Both can
be formalized in a standard way. For the functional $a$ it suffices to
consider the perturbed functional $a_{\varepsilon}:=a+\varepsilon$ for
$\varepsilon>0$ since this does not affect the properties assumed on $a$. This
observation may (and should) be compared to the remark made prior to the proof
of Theorem 2.1. The absence of weights makes these reductions much easier.

The argument below shows how to truncate the function $f$. Therefore, we can
always assume that the supremum defining $X$ is finite. Let us start then by
observing that we may assume that the function $f$ is bounded. More precisely,
suppose that (2.13) holds and consider $m\in\mathbb{N}$. Then the truncation
$f_{m}:=\min\\{|f|;m\\}$ also verifies (2.13). To see why, consider
$\lambda\in\mathbb{R}$ and also truncate it by introducing
$\lambda_{m}:=\min\\{|\lambda|;m\\}$. We can compute directly both minima and
obtain:

$\displaystyle f_{m}-\lambda_{m}$ $\displaystyle=$
$\displaystyle\min\\{|f|;m\\}-\min\\{|\lambda|;m\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(|f|+m-||f|-m|-\left(|\lambda|+m-||\lambda|-m|\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(|f|-|\lambda|-||\lambda|-m|-||f|-m|\right).$

Hence, taking absolute values,

$\displaystyle|f_{m}-\lambda_{m}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\left(||f|-|\lambda||+|||\lambda|-m|-||f|-m||\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\left(|f-\lambda|+||\lambda|-|f||\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\left(|f-\lambda|+|f-\lambda|\right)$
$\displaystyle\leq$ $\displaystyle|f-\lambda|$

for all $m\in\mathbb{N}$ and for all $\lambda>0$. This pointwise estimate
yields the following

$\inf_{c\in\mathbb{R}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f_{m}(x)-c|^{\delta}\right)^{\frac{1}{\delta}}\leq\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f_{m}(x)-\lambda_{m}|^{\delta}\right)^{\frac{1}{\delta}}\leq\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x)-\lambda|^{\delta}\right)^{\frac{1}{\delta}},$

also for any $\lambda>0$. Therefore we obtain that

$\inf_{c\in\mathbb{R}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f_{m}(x)-c|^{\delta}\right)^{\frac{1}{\delta}}\leq\inf_{\lambda\in\mathbb{R}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x)-\lambda|^{\delta}\right)^{\frac{1}{\delta}}\leq
a(R).$

This means that the same initial hypothesis also holds for any truncation of
$f$ at height $m$, independently of $m$.

Now we start with the proof of inequality (2.12) assuming that the function
$f$ is bounded. Given a rectangle $R\in\mathcal{R}$, we have by hypothesis
that, for some $c_{R}\in\mathbb{R}$,

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta}dx\right)^{\frac{1}{\delta}}\leq
2.$

We perform the standard dyadic C-Z decomposition of the function
$\left|\frac{f-c_{R}}{a(R)}\right|^{\delta}$ adapted to $R$ at level $L>2$ to
be chosen later. This means that we will have a collection $\\{R_{i}\\}_{i}$
of disjoint maximal dyadic rectangles satisfying

(4.14)
$L<\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta}dx\right)^{\frac{1}{\delta}}\leq
2^{\frac{n}{\delta}}L.$

Lets denote by $\Omega_{L}$ the union of such rectangles. As usual, this set
is the level set of the dyadic maximal function relative to $R$:

(4.15) $\Omega_{L}=\left\\{x\in
R:M^{d}_{R}\left(\left|\frac{f-c_{R}}{a(R)}\right|^{\delta}\right)(x)>L^{\delta}\right\\},$

where

(4.16) $M^{d}_{R}(g)(x)=\sup_{x\in
P\in\mathcal{D}(R)}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}|g(y)|dy.$

Now, consider the following pointwise decomposition.

$\displaystyle\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta}$ $\displaystyle=$
$\displaystyle\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta}\chi_{\Omega}(x)+\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta}\chi_{\Omega^{c}}(x)$
$\displaystyle\leq$
$\displaystyle\sum_{i}\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta}\chi_{R_{i}}(x)+L^{\delta}$

since the Lebesgue differentiation theorem implies that outside $\Omega$ the
function $\frac{f-c_{R}}{a(R)}$ is pointwise bounded by $L$. The usual
argument at this stage would be to find a way to intercalate some average-like
quantity related to $R_{i}$ instead of the original object $R$. We proceed as
follows in the sum above. Denote $g_{R}=\frac{f-c_{R}}{a(R)}$, then

$\displaystyle\sum_{i}|g_{R}(x)|^{\delta}\chi_{R_{i}}(x)$ $\displaystyle=$
$\displaystyle\sum_{i}\left(|g_{R}(x)|^{\delta}-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}|g_{R}(y)|^{\delta}dy\right)\chi_{R_{i}}(x)$
$\displaystyle+\sum_{i}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}|g_{R}(y)|^{\delta}\chi_{R_{i}}(x)$
$\displaystyle\leq$
$\displaystyle\sum_{i}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}\left|\frac{f(x)-f(y)}{a(R)}\right|^{\delta}dy\right)\chi_{R_{i}}(x)+2^{n}L^{\delta},$

by using the elementary bound $||a|^{\delta}-|b|^{\delta}|\leq|a-b|^{\delta}$
for any $a,b\in\mathbb{R}$ and $0<\delta<1$. The second term is obtained using
the maximality from (4.14). Now, by convexity the $\delta$ power can be moved
out from the integral and using that the rectangles are disjoint we obtain

$\sum_{i}|g_{R}(x)|^{\delta}\chi_{R_{i}}(x)\leq\left(\sum_{i}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}\left|\frac{f(x)-f(y)}{a(R)}\right|dy\chi_{R_{i}}(x)\right)^{\delta}+2^{n}L^{\delta}.$

Therefore, collecting all estimates, we get

$\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta}\leq\left(\sum_{i}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}\left|\frac{f(x)-f(y)}{a(R)}\right|dy\chi_{R_{i}}(x)\right)^{\delta}+2^{n}L^{\delta}+L^{\delta}.$

Now we can compute the desired $L^{1}$ norm.

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\left|\frac{f(x)-c_{R}}{a(R)}\right|^{\delta\frac{1}{\delta}}$
$\displaystyle\leq$
$\displaystyle\frac{2^{\frac{1}{\delta}-1}}{|R|}\sum_{i}|R_{i}|\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}\left|\frac{f(x)-f(y)}{a(R)}\right|dydx+2^{\frac{n+2-\delta}{\delta}}L.$

Now, since the double average over $R_{i}$ of $f$ is less or equal than twice
the infimum of the oscillations on $R_{i}$ we obtain, writing
$C_{\delta}=2^{\frac{n+2-\delta}{\delta}}$

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\left|\frac{f(x)-c_{R}}{a(R)}\right|dx$
$\displaystyle\leq$
$\displaystyle\frac{2^{\frac{1}{\delta}}}{a(R)|R|}\sum_{i}a(R_{i})|R_{i}|\inf_{c\in\mathbb{R}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{i}}\left|\frac{f(x)-c}{a(R_{i})}\right|dy+C_{\delta}L$
$\displaystyle\leq$ $\displaystyle
X\frac{2^{\frac{1}{\delta}}}{a(R)|R|}\sum_{i}a(R_{i})|R_{i}|+C_{\delta}L$

where $X$ is defined by (4.13). Here there are two options, as it is presented
in the statement of the theorem. In the case of the functional $a$ satisfying
the $SD_{1,\mathcal{R}}^{s}$ condition for some $s>1$, we use this to obtain

$\displaystyle\inf_{c\in\mathbb{R}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\left|\frac{f(x)-c}{a(R)}\right|dx$
$\displaystyle\leq$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\left|\frac{f(x)-c_{R}}{a(R)}\right|dx$
$\displaystyle\leq$ $\displaystyle
2^{\frac{1}{\delta}}X\frac{\|a\|_{SD_{1,\mathcal{R}}^{s}}}{L^{1/s}}+2^{\frac{n+2-\delta}{\delta}}L.$

Therefore, taking the supremum over all rectangles $R$ on the left hand side
we obtain

$X\leq
2^{\frac{1}{\delta}}X\frac{\|a\|_{SD_{1,\mathcal{R}}^{s}}}{L^{1/s}}+2^{\frac{n+2-\delta}{\delta}}L.$

Now we proceed essentially as in (4.7). Choose $L$ large enough, namely
$L=e\max\\{\max\\{2^{\frac{s}{\delta}}\|a\|_{SD_{1,\mathcal{R}}^{s}}^{s},1\\}$
to note that

$X\leq\frac{1}{e^{\frac{1}{s}}}X+2^{\frac{n+2-\delta}{\delta}}e\max\\{2^{\frac{s}{\delta}}\|a\|_{SD_{1,\mathcal{R}}^{s}}^{s},1\\}.$

As in (4.7), we use that $(e^{\frac{1}{s}})^{\prime}\leq 1+s$ to conclude that

$X\lesssim_{\delta,s}\max\\{\|a\|^{s}_{SD_{1,\mathcal{R}}^{s}},1\\},$

which is the desired estimate.

The other possibility is that the functional $a$ satisfies a standard $D_{p}$
condition for $1<p<\infty$. In that case, using Hölder’s inequality in the sum
involving the $R_{i}$’s, we obtain 222 Actually, we are using that
$D_{p}\subset SD_{1,\mathcal{R}}^{s}$ with
$\|a\|_{SD_{1,\mathcal{R}}^{s}}\leq\frac{\|a\|_{D_{p,\mathcal{R}}}}{L^{\frac{1}{p^{\prime}}}}$

$\displaystyle X$ $\displaystyle\leq$ $\displaystyle
2^{\frac{1}{\delta}}X\frac{1}{a(R)|R|}\sum_{i}a(R_{i})|R_{i}|^{\frac{1}{p}+\frac{1}{p^{\prime}}}+2^{\frac{n+2-\delta}{\delta}}L$
$\displaystyle\leq$ $\displaystyle
2^{\frac{1}{\delta}}X\left(\frac{\sum_{i}a(R_{i})^{p}|R_{i}|}{a(R)^{p}|R|}\right)^{1/p}\left(\frac{\sum_{i}|R_{i}|}{|R|}\right)^{\frac{1}{p^{\prime}}}+2^{\frac{n+2-\delta}{\delta}}L$
$\displaystyle\leq$ $\displaystyle
2^{\frac{1}{\delta}}X\frac{\|a\|_{D_{p,\mathcal{R}}}}{L^{\frac{1}{p^{\prime}}}}+2^{\frac{n+2-\delta}{\delta}}L.$

Once again, we are in the same situation as before and we can argue in the
same way: choosing carefully the size of $L$, we obtain again an inequality of
the form

$X\lesssim_{\delta,p}\|a\|_{D_{p,\mathcal{R}}}^{p^{\prime}}.$

∎

##  5\. Analysis in $\mathfrak{R}$: Bi-paramater (1,1)-Poincaré

We start recalling the following $(1,1)$-Poincaré inequality

(5.1)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|\leq\ell(I_{1})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|\nabla_{1}f|+\ell(I_{2})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|\nabla_{2}f|\qquad
R\in\mathfrak{R}$

which is the statement of Lemma 2.17. As already mentioned the idea of
considering this type of Poincaré inequality follows from the work of Shi and
Torchinsky [ST93].

We will prove inequality (5.1) in two ways, the first one using a point-wise
estimate obtained by Lu and Wheeden in [LW98] and the second proof will follow
as a consequence of a “fractional” Poincaré type inequalities that we will
show in Proposition 5.2 and improve in Theorem 2.18.

Recall that for a function $f$ defined on
$\mathbb{R}^{n}=\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$, $\nabla_{1}f$
will denote the partial gradient of $f$ containing the $x_{1}$-derivatives and
similarly for $\nabla_{2}f$ the partial gradient of $f$ containing the
$x_{2}$-derivatives. From now on, $I_{1}$ will always denote a cube in
$\mathbb{R}^{n_{1}}$ while $I_{2}$ will be a cube in $\mathbb{R}^{n_{2}}$. Let
$\ell_{1}:=\ell(I_{1})$ and $\ell_{2}:=\ell(I_{2})$.

######  Proposition 5.1.

[LW98] Let $R\in\mathfrak{R}$ be of the form $R=I_{1}\times I_{2}$. If
$f\in\text{Lip}(R),$ we have the following pointwise estimate

$|f(x_{1},x_{2})-f_{R}|\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\frac{\ell_{1}|\nabla_{1}f(y_{1},y_{2})|+\ell_{2}|\nabla_{2}f(y_{1},y_{2})|}{\left(\frac{|x_{1}-y_{1}|^{2}}{\ell_{1}^{2}}+\frac{|x_{2}-y_{2}|^{2}}{\ell_{2}^{2}}\right)^{\frac{n-1}{2}}}\
dy_{1}dy_{2}.$

From this we provide the first proof of the $(1,1)$-Poincaré inequality (5.1).

###### First proof of Lemma 2.17.

Let’s put $A:=\displaystyle\int_{R}|f(x_{1},x_{2})-f_{R}|\,dx_{1}dx_{2}$.
Then, using the pointwise estimate from Proposition 5.1 we directly obtain

$\displaystyle A$ $\displaystyle\leq$
$\displaystyle\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\frac{\ell_{1}|\nabla_{1}f(y_{1},y_{2})|+\ell_{2}|\nabla_{2}f(y_{1},y_{2})|}{\left(\frac{|x_{1}-y_{1}|^{2}}{\ell_{1}^{2}}+\frac{|x_{2}-y_{2}|^{2}}{\ell_{2}^{2}}\right)^{\frac{n-1}{2}}}\
dy_{1}dy_{2}dx_{1}dx_{2}$ $\displaystyle\leq$
$\displaystyle\int_{R}\frac{\ell_{1}|\nabla_{1}f(y_{1},y_{2})|}{|I_{1}||I_{2}|}\int_{R}\frac{1}{\left(\frac{|x_{1}-y_{1}|^{2}}{\ell_{1}^{2}}+\frac{|x_{2}-y_{2}|^{2}}{\ell_{2}^{2}}\right)^{\frac{n-1}{2}}}\
dx_{1}dx_{2}dy_{1}dy_{2}$
$\displaystyle+\int_{R}\frac{\ell_{2}|\nabla_{2}f(y_{1},y_{2})|}{|I_{1}||I_{2}|}\int_{R}\frac{1}{\left(\frac{|x_{1}-y_{1}|^{2}}{\ell_{1}^{2}}+\frac{|x_{2}-y_{2}|^{2}}{\ell_{2}^{2}}\right)^{\frac{n-1}{2}}}\
dx_{1}dx_{2}dy_{1}dy_{2}$ $\displaystyle\leq$
$\displaystyle\ell_{1}\int_{R}|\nabla_{1}f(y_{1},y_{2})|\
dy_{1}dy_{2}+\ell_{2}\int_{R}|\nabla_{2}f(y_{1},y_{2})|\ dy_{1}dy_{2}.$

In the above argument we have use the following inequality

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\frac{dx_{1}dx_{2}}{\left(\frac{|x_{1}-y_{1}|^{2}}{\ell_{1}^{2}}+\frac{|x_{2}-y_{2}|^{2}}{\ell_{2}^{2}}\right)^{\frac{n-1}{2}}}$
$\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\frac{dx_{1}dx_{2}}{\left|(\frac{x_{1}}{\ell_{1}},\frac{x_{2}}{\ell_{2}})-(\frac{y_{1}}{\ell_{1}},\frac{y_{2}}{\ell_{2}})\right|^{\frac{n-1}{2}}}$
$\displaystyle=$
$\displaystyle\frac{1}{|I_{1}||I_{2}|}\int_{\frac{I_{1}}{\ell_{1}}\times\frac{I_{2}}{\ell_{2}}}\frac{\ell_{1}^{n_{1}}\ell_{2}^{n_{2}}\
dz_{1}dz_{2}}{\left|(z_{1},z_{2})-(\frac{y_{1}}{\ell_{1}},\frac{y_{2}}{\ell_{2}})\right|^{\frac{n-1}{2}}}$
$\displaystyle\leq$
$\displaystyle\left|\frac{I_{1}}{\ell_{1}}\times\frac{I_{2}}{\ell_{2}}\right|=1.$

∎

We can improve this result using fractional type Poincaré inequalities. The
idea is to “interpolate” between the oscillation

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}\times
I_{2}}|f-f_{I_{1}\times I_{2}}|$

and the right hand side of (5.1) inspired by the following one parameter
$(1,1)$ fractional type Poincaré inequality which is easy to derive:

(5.2)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|f(t)-f_{Q}|dt\leq
c_{n}\,\ell(Q)^{\delta}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|f(t)-f(s)|}{|t-s|^{n+\delta}}\,dt\,ds.$

The result is the following,

######  Proposition 5.2.

Let $R=I_{1}\times I_{2}$ be a rectangle and $\delta_{1},\delta_{2}\in(0,1)$.
Then there exist dimensional constants $c_{n_{1}},c_{n_{2}}>0$ such that

$\displaystyle\qquad\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|$
$\displaystyle\leq$ $\displaystyle
c_{n_{1}}\ell(I_{1})^{\delta_{1}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|}{|x_{1}-y_{1}|^{n_{1}+\delta_{1}}}\
dx_{1}dy_{1}dx_{2}$ $\displaystyle+\
c_{n_{2}}\ell(I_{2})^{\delta_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{2}}\frac{|f(y_{1},x_{2})-f(y_{1},y_{2})|}{|x_{2}-y_{2}|^{n_{2}+\delta_{2}}}\
dy_{1}dx_{2}dy_{2}.$

###### Proof.

By the triangle inequality

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|$
$\displaystyle\approx$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x_{1},x_{2})-f(y_{1},y_{2})|\,dy_{1}dy_{2}\
dx_{1}dx_{2}$ $\displaystyle\leq$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x_{1},x_{2})-f(y_{1},x_{2})|\,dy_{1}dy_{2}\
dx_{1}dx_{2}$ $\displaystyle+\
\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(y_{1},x_{2})-f(y_{1},y_{2})|\,dy_{1}dy_{2}\
dx_{1}dx_{2}$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}|f(x_{1},x_{2})-f(y_{1},x_{2})|\,dy_{1}\
dx_{1}dx_{2}$ $\displaystyle+\
\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{2}}|f(y_{1},x_{2})-f(y_{1},y_{2})|\,dy_{1}dy_{2}\
dx_{2}=A+B.$

We compute $A$ to derive the first term of (5.2). A similar estimate holds for
$B$ to derive the second term of (5.2)

$\displaystyle A$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})||x_{1}-y_{1}|^{\delta_{1}}}{|x_{1}-y_{1}|^{\delta_{1}}}\,dy_{1}dx_{1}\
dx_{2}$ $\displaystyle\leq$
$\displaystyle\ell(I_{1})^{\delta_{1}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|}{|x_{1}-y_{1}|^{\delta_{1}}}\,dy_{1}dx_{1}\
dx_{2}$ $\displaystyle\leq$ $\displaystyle
c_{n_{1}}\,\ell(I_{1})^{\delta_{1}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|}{|x_{1}-y_{1}|^{n_{1}+\delta_{1}}}\,dy_{1}dx_{1}\
dx_{2}.$

∎

We remark that the fractional functional right hand side of (5.2) is smaller
than the gradient functional from (5.1), so we could have avoided the use of
Lu-Wheeden pointwise estimate from Proposition 5.1 and in addition we are
getting a better estimate than we were looking for. Indeed, to verify last
assertion we use the one-parameter fractional estimate from the following
lemma.

######  Lemma 5.3.

There exists a dimensional constant $c_{n}>0$ such that for any
$\delta\in(0,1)$, and any cube $Q$ in $\mathbb{R}^{n}$

(5.4)
$\ell(Q)^{\delta}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|f(t)-f(s)|}{|t-s|^{n+\delta}}\,dt\,ds\leq\frac{c_{n}}{\delta(1-\delta)}\ell(Q)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\nabla
f|\,dx.$

Since we do not find the proof of this in the literature we provide an
argument in Appendix B.

###### Second proof of Lemma 2.17.

Let $\delta_{1}=\delta_{2}=\frac{1}{2}\in(0,1).$ We compare the first term in
the fractional functional (5.2) with the first term of the gradient functional
(5.1). To verify this we use the one-parameter fractional estimate from (5.4):

$\displaystyle\ell(I_{1})^{\delta_{1}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}\times
I_{2}}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|}{|x_{1}-y_{1}|^{n_{1}+\delta_{1}}}\
dx_{1}dy_{1}dx_{2}$ $\displaystyle=$ $\displaystyle=\
\ell(I_{1})^{\delta_{1}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|}{|x_{1}-y_{1}|^{n_{1}+\delta_{1}}}\
dx_{1}dy_{1}dx_{2}$ $\displaystyle\leq\
\frac{c_{n_{1}}}{\delta_{1}(1-\delta_{1})}\ell(I_{1})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}|\nabla_{1}f|\
dx_{1}dx_{2}.$ $\displaystyle=\
4c_{n_{1}}\ell(I_{1})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}|\nabla_{1}f|\
dx_{1}dx_{2}.$

A similar estimate holds for the second direction $I_{2}$ with $\delta_{2}$.
Then applying Theorem 2.18 with $p_{1}=p_{2}=1$ we are done or directly from
“rough” Fractional Poincaré inequality (5.2).

∎

To conclude with the proof of the lemma, it remains to prove Theorem 2.18. We
recall again that there is very interesting improvement of (5.2) obtained by
Bourgain, Brezis and Mironescu [BBM02] The result is the following.

######  Theorem 5.4.

Let $\delta\in(0,1)$. Then there exists a dimensional constant $c_{n}>0$ such
that

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|f(x)-f_{Q}|dt\leq
c_{n}\,(1-\delta)^{\frac{1}{p}}\,\ell(Q)^{\delta}\,\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+\delta
p}}\,dy\,dx\right)^{1/p}.$

for every cube $Q$ in ${\mathbb{R}}^{n}$.

######  Remark 5.5.

We remark that the case $p>1$ does not follow from the case $p=1$ using
Jensen’s Also recall that there is sharper model version in Theorem 1.4.

Our contribution here is the generalization stated in Theorem 2.18.

###### Proof of Theorem 2.18.

The proof follows the same steps as in Proposition 5.2, but using the more
precise estimate from Theorem 5.4.

Let $R=I_{1}\times I_{2}\in\mathfrak{R}$. By the triangle inequality

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|$
$\displaystyle\approx\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x_{1},x_{2})-f(y_{1},y_{2})|\,dy_{1}dy_{2}\
dx_{1}dx_{2}$
$\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(x_{1},x_{2})-f(y_{1},x_{2})|\,dy_{1}dy_{2}\
dx_{1}dx_{2}$
$\displaystyle+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f(y_{1},x_{2})-f(y_{1},y_{2})|\,dy_{1}dy_{2}\
dx_{1}dx_{2}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}|f(x_{1},x_{2})-f(y_{1},x_{2})|\,dy_{1}\
dx_{1}dx_{2}$
$\displaystyle+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{2}}|f(y_{1},x_{2})-f(y_{1},y_{2})|\,dy_{1}dy_{2}\
dx_{2}$ $\displaystyle:=A+B.$

We compute $A.$ A similar estimate holds for $B.$ Now, from Theorem 5.4 we
have,

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|u(x)-u(y)|\,dx\,dy\leq
c_{n}\,(1-\delta_{1})^{\frac{1}{p_{1}}}\,\ell(Q)^{\delta_{1}}\,\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|u(x)-u(y)|^{p_{1}}}{|x-y|^{n+p_{1}\delta_{1}}}\,dy\,dx\right)^{\frac{1}{p_{1}}}$

for every cube $Q$ in $\mathbb{R}^{n}$. Applying this to the cube
$I_{1}\subset\mathbb{R}^{n_{1}}$

$\displaystyle A$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}|f(x_{1},x_{2})-f(y_{1},x_{2})|\,dy_{1}dx_{1}\
dx_{2}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}|f(x_{1},x_{2})-f(y_{1},x_{2})|\,dy_{1}dx_{1}\
dx_{2}$
$\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{2}}c_{n_{1}}\,(1-\delta_{1})^{\frac{1}{p_{1}}}\ell(I_{1})^{\delta_{1}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{1}}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|^{p_{1}}}{|x_{1}-y_{1}|^{n_{1}+p_{1}\delta_{1}}}\,dy_{1}dx_{1}\right)^{\frac{1}{p_{1}}}dx_{2}$
$\displaystyle\leq
c_{n_{1}}\,(1-\delta_{1})^{\frac{1}{p_{1}}}\ell(I_{1})^{\delta_{1}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|^{p_{1}}}{|x_{1}-y_{1}|^{n_{1}+p_{1}\delta_{1}}}\,dy_{1}dx_{1}\
dx_{2}\right)^{\frac{1}{p_{1}}}\,$

by Jensen’s inequality. ∎

The same idea could be used to obtain similar results involving $m$-fold
products of cubes. Let $R$ be a rectangle of the form $R=\prod_{i=1}^{m}I_{i}$
in $\mathbb{R}^{n}$ written as a product of cubes with
$I_{i}\subset{\mathbb{R}}^{n_{i}}$, $n=\sum_{i=1}^{m}n_{i}$ and $m\leq n$. We
fix some notation: the points in $\mathbb{R}^{n}$ are of the form ${\bf
x}=(x_{1},\dots,x_{m})$, where each $x_{i}$ is itself a string of $n_{i}$ real
numbers forming a vector in $\mathbb{R}^{n_{i}}$. For any $0\leq i\leq m$, we
denote

${\bf x}_{i}=\left\\{\begin{array}[]{cc}(x_{1},\dots,x_{m})&i=0\\\
(y_{1},\dots,y_{i},x_{i+1},\dots,x_{m})&1\leq i<m\\\
(y_{1},\dots,y_{m})&i=m.\end{array}\right.$

We also adopt the notation $d{\bf x}_{i}:=dy_{1}\dots dy_{i}dx_{i}\dots
dx_{n}.$

######  Theorem 5.6.

Let $R=\prod_{i=1}^{m}I_{i}$ be a rectangle in $\mathbb{R}^{n}$ written as a
product of cubes with $I_{i}\subset{\mathbb{R}}^{n_{i}},$ and
$n=\sum_{i=1}^{m}n_{i}$ and $m\leq n$. Let $0<\delta_{i}<1\leq p_{i}<\infty$,
$1\leq i\leq m$. Then there are some dimensional constants $c_{n_{i}}$ such
that,

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|\leq\sum_{i=1}^{m}c_{n_{i}}(1-\delta_{i})^{\frac{1}{p_{i}}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{i}}\frac{|f({\bf
x}_{i-1})-f({\bf x}_{i})|^{p_{i}}}{|x_{i}-y_{i}|^{n_{i}+p_{i}\delta_{i}}}\
dy_{1}\dots dy_{i}dx_{i}\dots dx_{m}\right)^{\frac{1}{p_{i}}}.$

Going back to the simplest product case, namely the case of product of two
cubes, in the next section we will prove the self-improving result in Theorem
2.16.

##  6\. Proofs part II: the Biparameter Poincaré-Sobolev inequality

The goal of this section is to prove Theorem 2.16, namely, for any
$R\in\mathfrak{R}$

(6.1) $\left\|f-f_{R}\right\|_{L^{p^{*}}(\frac{w\,dx}{w(R)})}\leq
C[w]_{A_{q,\mathfrak{R}}}^{\frac{1}{p}}\left[\ell(I_{1})\left\|\nabla_{1}f\right\|_{L^{p}(\frac{w\,dx}{w(R)})}+\ell(I_{2})\left\|\nabla_{2}f\right\|_{L^{p}(\frac{w\,dx}{w(R)})}\right],$

where

$\displaystyle\frac{1}{p}-\frac{1}{p^{*}}=\frac{1}{n}\frac{1}{q+\log[w]_{A_{q,\mathfrak{R}}}}.$

Since this result involves a weighted estimate for a weight in the class
$A_{p,\mathfrak{R}}$, let us first introduce its obvious definition adapted to
the geometry of the basis $\mathfrak{R}$. For a weight $w$ in
$\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$, $n=n_{1}+n_{2}$, we will say
that $w\in A_{p,\mathfrak{R}}$ if

(6.2)
$[w]_{A_{p,\mathfrak{R}}}:=\sup_{R\in\mathfrak{R}}\left(\frac{1}{|R|}\int_{R}w(x)\,dx\right)\left(\frac{1}{|R|}\int_{R}w(x)^{-\frac{1}{p-1}}\,dx\right)^{p-1}<\infty.$

and in the case $p=1$, for a finite constant $c$

(6.3) $\frac{1}{|R|}\int_{R}w(x)\,dx\leq c\,\inf_{R}w\qquad R\in\mathfrak{R}$

and the smallest of the constants $c$ is denoted by
$[w]_{A_{1,\mathfrak{R}}}$.

The strong $A_{\infty,\mathfrak{R}}$ class is defined in the same way as in
the cubic or strong case and it enjoys the same geometric conditions,

$A_{\infty,\mathfrak{R}}=\bigcup_{p>1}A_{p,\mathfrak{R}}.$

We emphasize here that in the context of $\mathfrak{R}$, the main difficulty
is not in the self-improving method. The cube-product structure does not get
affected by dyadic decompositions, so the standard procedures can be used to
obtain Calderón–Zygmund coverings and many consequences of C–Z. In fact an
inspection of the proof of Theorem 2.1, Theorem 2.5 and Theorem 2.6 will show
that the dyadic analysis will produce the analogous results for the basis
$\mathfrak{R}$. So we leave to the interested reader to check the details to
prove the following claim.

######  Claim 6.1.

There are analogous versions of Theorem 2.1, Theorem 2.3, Theorem 2.5 and
Theorem 2.6 in the context of $\mathfrak{R}$ with the same hypothesis and
conclusions with the obvious modifications.

Hence, the real problem here is in to find a useful starting point and also to
check appropriate $D_{p}$ or $SD_{p}^{s}$-like conditions for the involved
functionals. The first problem was solved in Section 5, with two proofs for
Lemma 2.17. The second will be studied below, in Lemma 6.2.

Regarding the starting point, we have a sort of unweighted $(1,1)$-Poincaré
inequality proved in Lemma 2.17 that can be used as a starting point for our
self-improving method. In the same way as in (4.11), for a weight $w\in
A_{p,\mathfrak{R}}$ we can apply Hölder’s inequality to (5.1) to obtain, for a
rectangle $R=I_{1}\times I_{2}$, that

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|\leq[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p}}\,\ell(I_{1})\left\|\nabla_{1}f\right\|_{L^{p}\left(\frac{w\,dx}{w(R)}\right)}+[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p}}\,\ell(I_{2})\left\|\nabla_{2}f\right\|_{L^{p}\left(\frac{w\,dx}{w(R)}\right)}.$

We can define the functionals

(6.4)
$a_{i}(R):=[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p}}\,\ell(I_{i})\left\|\nabla_{i}f\right\|_{L^{p}\left(\frac{w\,dx}{w(R)}\right)},\qquad
i=1,2$

to obtain our bi-parameter starting point

(6.5)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|\leq
a_{1}(R)+a_{2}(R).$

The novelty here is that our functional $a$ is a _sum_ of two functionals with
certain structure which is _not_ exactly as in (3.1). The presence of the
sidelenght instead of the diameter makes the situation non standard, so we
need to find what kind of geometric condition is satisfied. We summarize all
those properties in the following lemma.

######  Lemma 6.2.

Let $R\in\mathfrak{R}$ and let $R=I_{1}\times I_{2}$, where
$I_{1}\subset\mathbb{R}^{n_{1}}$ and $I_{2}\subset\mathbb{R}^{n_{2}}$ are
cubes. Also put $n=n_{1}+n_{2}$. For a weight $w\in A_{q}$ with $1\leq q\leq
p<n$, consider the functional

(6.6)
$a(R)=\ell(I_{j})^{\delta}\left(\frac{1}{w(R)}\int_{R}A(R,x)\,dx\right)^{1/p}\qquad
j=1,2$

where $A(R,x)$ is nonnegative and increasing in $R$, that is, $R_{1}\subset
R_{2}$ implies that $A(R_{1},x)\leq A(R_{2},x)$.

Then

  1. (1)

$a\in SD_{p,\mathfrak{R}}^{\frac{n}{\delta}}(w)$ with norm less or equal than
1.

  2. (2)

Consider the Sobolev-type exponent $p^{*}_{w}$ defined by the usual condition
$\frac{1}{p}-\frac{1}{p_{w}^{*}}=\frac{\delta}{nq}$. Then $a\in
D_{p_{w}^{*},\mathfrak{R}}(w)$ with
$\|a\|_{D_{p_{w}^{*},\mathfrak{R}}}\leq[w]^{\frac{\delta}{nq}}_{A_{q,\mathfrak{R}}}.$

  3. (3)

For a given $M>1$, consider now the Sobolev-type exponent $p^{*}_{w}$ defined
by $\frac{1}{p}-\frac{1}{p_{w}^{*}}=\frac{\delta}{nq}\frac{1}{M}$. Then $a\in
SD^{s}_{p_{w}^{*},\mathfrak{R}}(w)$ with $s=\frac{nM^{\prime}}{\delta}$ and
$\|a\|_{SD^{s}_{p_{w}^{*},\mathfrak{R}}(w)}\leq[w]_{A_{q,\mathcal{R}}}^{\frac{\delta}{nqM}}$.

###### Proof.

The arguments are very similar to those given before, so we will omit some of
details of the proof, but mention the key idea behind them. As in all the
previous cases, it is crucial to quantify the eccentricity of the objects that
we are dealing with. Motivated by the fact that
$|R|=\ell(I_{1})^{n_{1}}\ell(I_{2})^{n_{2}}$, we will define an _eccentricity
related_ quantity as

$E(R)=\frac{\ell(I_{2})^{n_{2}}}{\ell(I_{1})^{n_{2}}}.$

This is equivalent to the expression

$\ell(I_{1})^{n}.E(R)=|R|.$

In other words, the quantity $E(R)$ reflects the relation between the volume
of the $n$-dimensional cube built from the side length $\ell(I_{1})$ with
respect to the actual measure of the rectangle $R$. The crucial property here
is that this quantity is invariant with respect to dyadic children from $R$,
namely as in Lemma 2.10, $E(\tilde{R})=E(R)$ for any $\tilde{R}$ dyadic
descendant of $R\in\mathfrak{R}$

With this idea in mind and using the quantity $E(R)$, the proof of (1) is a
variation of the argument given in Example 3.1, the proof of (2) can be
translated from the proof of (4.10) and finally the proof of (3) is an
appropriate variation of the proof of Lemma 3.3.

Indeed, consider a family of disjoint dyadic subrectangles from
$R\in\mathfrak{R}$ denoted by $\\{R_{i}\\}$. Of course, each subrectangle is
of the form $R_{i}=I_{i}\times J_{i}$ where $I_{i}$ and $J_{i}$ are dyadic
subcubes of $I$ and $J$ respectively. ∎

######  Remark 6.3.

To be able to apply our main Theorem, the easiest way to proceed is to note
that any $D_{p}$-like condition enumerated in Lemma 6.2, is preserved when
summing two functionals. This can be verified as follows (for the unweighted
$D_{p,\mathfrak{R}}$ as an example, the other situations are similar). Let
$\\{R_{i}\\}_{i}$ be a family of pairwise disjoint dyadic subrectangles of
$R=I_{1}\times I_{2}.$ Then, using Minkowsky inequality,

$\displaystyle\left(\sum_{i}(a_{1}(R_{i})+a_{2}(R_{i}))^{p}\frac{|R_{i}|}{|R|}\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{i}a_{1}(R_{i})^{p}\frac{|R_{i}|}{|R|}\right)^{\frac{1}{p}}$
$\displaystyle+\left(\sum_{i}a_{2}(R_{i})^{p}\frac{|R_{i}|}{|R|}\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\|a_{1}\|_{D_{p,\mathfrak{R}}}a_{1}(R)+\|a_{2}\|_{D_{p,\mathfrak{R}}}a_{2}(R)$
$\displaystyle\leq$ $\displaystyle C\left(a_{1}(R)+a_{2}(R)\right),$

with
$C=\max\left\\{\|a_{1}\|_{D_{p,\mathfrak{R}}},\|a_{2}\|_{D_{p,\mathfrak{R}}}\right\\}$.

We can now proceed to present the proof of Theorem 2.16.

###### Proof of Theorem 2.16.

Suppose first that $w$ is a weigh such that $[w]_{A_{q,\mathfrak{R}}}\geq
e^{q}$. Since the functionals from (6.4) namely

$a_{i}(R):=[w]^{\frac{1}{p}}_{A_{p,\mathfrak{R}}}\ell(I_{i})\left\|\nabla_{i}f\right\|_{L^{p}\left(\mu\right)},\qquad
i=1,2$

satisfy a $SD_{p_{w}^{*},\mathfrak{R}}^{s}$ condition given in Lemma 6.2 (with
$\delta=1$), (3), the same holds for

$a(R):=a_{1}(R)+a_{2}(R).$

Hence, we can apply the $\mathfrak{R}$-version of Theorem 2.5 to this
functional to obtain the proof with a uniform bound using the assumption on
$w$. More precisely, we obtain that

$\left\|f-f_{R}\right\|_{L^{p^{*}}(\frac{w\,dx}{w(R)})}\leq
c_{p,q}[w]_{A_{p,\mathfrak{R}}}^{\frac{1}{p}}\left[\ell(I_{1})\left\|\nabla_{1}f\right\|_{L^{p}(\frac{w\,dx}{w(R)})}+\ell(I_{2})\left\|\nabla_{2}f\right\|_{L^{p}(\frac{w\,dx}{w(R)})}\right].$

Now suppose that we are dealing with a trivial weight (such as
$[w]_{A_{q,\mathfrak{R}}}=1$) or more generally with flat weights such that
$[w]_{A_{q,\mathfrak{R}}}\leq e^{q}$. To avoid the blowup, we can apply the
$\mathfrak{R}$-version of Theorem 2.3 to get the weak inequality

$\|f-f_{R}\|_{L^{p^{*}_{w},\infty}\big{(}R,\frac{w\,dx}{w(R)}\big{)}}\leq
c\,p^{*}_{1}\,e^{q}\,e^{\frac{1}{n}}\,a(R),$