# Off–Diagonal Two Weight Bumps for Fractional Sparse Operators

Rob Rahm  Texas A&M Mathematics Department<EMAIL_ADDRESS>
###### Abstract.

In this paper, we continue some recent work on two weight boundedness of
sparse operators to the "off–diagonal" setting. We use the new "entropy bumps"
introduced in by Treil–Volberg ([TreVol2016]) and improved by Lacey–Spencer
([LacSpe2015]) and the "direct comparison bumps" introduced by Rahm–Spencer
([RahSpe2018]) and improved by Lerner ([Ler2020]). Our results are "sharp" in
the sense that they are sharp in various particular cases. A feature is that
given the current machinery, the proofs are almost trivial.

###### Key words and phrases:

sparse operators, separated bumps, entropy bumps, direct comparison bumps

###### 2010 Mathematics Subject Classification:

42B20, 42B25

##  1\. Introduction

The topic of this article is two–weight bump conditions for sparse operators
in the “off–diagonal” setting (i.e $q>p$). We continue the line of
investigation concerning entropy bumps that began with Treil–Volberg in
[TreVol2016] and was continued in [LacSpe2015, RahSpe2016, RahSpe2018] and the
line of investigation concerning "direct comparison bumps" introduced in
[RahSpe2018] and continued and improved in [Ler2020]. We will be concerned
with sparse operators of the form ( $0\leq\alpha<d$):

$\displaystyle
T_{\alpha,\mathcal{S}}f:=\sum_{Q\in\mathcal{S}}(\left|Q\right|^{\frac{\alpha}{d}}\langle\left|f\right|\rangle_{Q})1\\!\\!1_{Q},\hskip
36.135pt\textnormal{ where }\hskip 36.135pt\langle
f\rangle_{Q}:=\frac{1}{\left|Q\right|}\int_{Q}f.$

A more general class of these operators are studied in [FacHyt2018]. Unlike
that paper, we make no assumption about the weights being in $A_{\infty}$ and
we only concentrate on the off–diagonal case.

The literature is saturated with results of this type. The main contributions
of this paper are (1) the results are new; (2) the proofs are succinct and
highlight the development of this area and (3) indicate the challenges in
proving various "separated bump theorems".

The first deals with entropy bumps. It was proven in the $p=q=2$ case in
[TreVol2016] and in the general $p=q$ case in [LacSpe2015]; both with
$\alpha=0$.

######  Theorem 1.1.

Let $T_{\alpha,\mathcal{S}}$ be a sparse operator and $1<p<q<\infty$ then:

$\displaystyle\left\|T_{\alpha,\mathcal{S}}\sigma\cdot:L^{p}(w)\to
L^{p}(\sigma)\right\|\lesssim\mathcal{E}+\mathcal{E}^{\ast},$

where:

$\displaystyle\mathcal{E}=\sup_{Q}\frac{w(Q)^{\frac{1}{q}}\sigma(Q)^{\frac{1}{p^{\prime}}}}{\left|Q\right|^{1-\frac{\alpha}{d}}}\rho(Q;\sigma)^{\frac{1}{q}}\varepsilon(\rho(Q;\sigma))^{\frac{1}{q}}$

and

$\displaystyle\mathcal{E}_{\ast}=\sup_{Q}\frac{w(Q)^{\frac{1}{q}}\sigma(Q)^{\frac{1}{p^{\prime}}}}{\left|Q\right|^{1-\frac{\alpha}{d}}}\rho(Q;\sigma)^{\frac{1}{p^{\prime}}}\varepsilon(\rho(Q;\sigma))^{\frac{1}{p^{\prime}}}$

and $\rho(Q;\sigma)$ is the "local $A_{\infty}$ characteristic":

$\displaystyle\rho(Q;\sigma):=\frac{1}{\sigma(Q)}\int_{P}M(\sigma
1\\!\\!1_{P})\hskip 36.135pt\textnormal{and}\hskip
36.135pt\rho(Q;w):=\frac{1}{w(Q)}\int_{P}M(w1\\!\\!1_{P}),$

and $\varepsilon$ is a monotonic increasing function that satisfies
$\sum_{r=0}^{\infty}\varepsilon(2^{r})^{-1}<\infty.$

The next theorem was introduced in [RahSpe2018] and improved in [Ler2020] (for
$p=q$ and $\alpha=0$):

######  Theorem 1.2.

Let $T_{\alpha,\mathcal{S}}$ be a sparse operator and $1<p<q<\infty$ and $p>1$
then:

$\displaystyle\left\|T\sigma\cdot:L^{p}(w)\to
L^{p}(\sigma)\right\|\lesssim\mathcal{D}+\mathcal{D}_{\ast},$

where:

$\displaystyle\mathcal{D}=\sup\frac{{w(Q)^{\frac{1}{q}}}\sigma(Q)^{\frac{1}{p^{\prime}}}}{\left|Q\right|^{1-\frac{\alpha}{d}}}\varepsilon(\langle\sigma\rangle_{Q})^{\frac{1}{q}}$

and

$\displaystyle\mathcal{D}_{\ast}=\sup\frac{{w(Q)^{\frac{1}{q}}}\sigma(Q)^{\frac{1}{p^{\prime}}}}{\left|Q\right|^{1-\frac{\alpha}{d}}}\varepsilon(\langle
w\rangle_{Q})^{\frac{1}{p^{\prime}}}$

and $\varepsilon$ is a function that is decreasing on $(0,1)$ increasing on
$(1,\infty)$ and satisfies $\sum_{r=-\infty}^{\infty}\varepsilon(2^{r})^{-1}$.

##  2\. Background and Discussion

It was noted in [Muc1972, HunMucWhe1973, MucWhe1974] that the standard
Muckenhoupt condition was necessary but not sufficient for the two–weight
boundedness of operators of interest. The purpose of the various "bumps" are
to replace the standard $A_{p}$ (or $A_{p,q}$ in our case) condition with a
slightly bigger condition that is sufficient to give two–weight boundedness.
This is a well–developed area, see [Neu1983, Per1994, CruRezVol2014, Lac2016,
NazRezTreVol2013, Ler2013, CruMoe2013a, CruMoe2013b, Cru2017] and the
references therein for more information.

The "entropy bumps" in Theorem 1.1 were introduced by Treil–Volberg in
[TreVol2016] in the $p=q=2$ case. This was extended to the general $p=q$ case
by Lacey–Spencer in [LacSpe2015]. These operators were also studied in the
off–diagonal $p\leq q$, $\alpha>0$ setting in [RahSpe2016, PanSun2019]. Our
theorem here is "sharp" in the sense that when $\sigma$ and $w$ are
$A_{\infty}$, we recover the sharp results of, for example [CruMoe2013a,
CruMoe2013b, LacMoePerTor2010]. The "direct comparison bumps" were introduced
in [RahSpe2018] and improved in [Ler2020].

The off–diagonal setting allows us to replace the normal norm in the testing
inequalities with $L^{1}$ norms (this is Proposition 3.1 below; see also
[CruMoe2013b]*Theorem1.1). This is why off–diagonal results are sharper than
on–diagonal results. Based on comparisons with Orlicz conjectures, the
following conjecture is made (Lerner almost proves this in [Ler2020]):

######  Conjecture 2.1.

$\left\|T\sigma\cdot:L^{p}(\sigma)\to
L^{p}(w)\right\|\lesssim\mathcal{D}_{p,p}+\mathcal{D}^{\ast}_{p,p}$.

The Orlicz, entropy, and direct comparison bumps are not strictly comparable.
The Orlicz bumps are the most established but require the most in terms of
local integrability. The entropy bumps are guaranteed to be bounded in the one
weight setting (and record important $A_{\infty}$ information about the
operator norms). The direct comparison bumps require the least in terms of
local integrability but are not as popular. They are also easier to verify
than Orlicz bumps or entropy bumps. The weights

$\displaystyle\sigma(x)=\frac{1}{x(1-\log x)^{2}},\hskip 36.135ptw(x)=x^{2}.$

satisfy the two weight conditions in [Ler2020] or even in [RahSpe2018]. Yet
$\sigma$ is not in $L\log L$ and so neither the Orlicz nor the entropy bumps
will detect the boundedness of sparse operators with these weights.

##  3\. Preliminaries and Notation

A collection of cubes, $\mathcal{S}$, is called $\lambda$–Sparse
($0<\lambda<1$) if for every $Q_{0}\in\mathcal{S}$, there holds:

$\displaystyle\sum_{\begin{subarray}{c}Q\in\mathcal{S}:\\\ Q\subset
Q_{0},Q\textnormal{ is
maximal}\end{subarray}}\left|Q\right|\leq\lambda\left|Q_{0}\right|.$

If $\mathcal{S}$ is sparse, the sets
$E_{Q}=Q\setminus\cup_{Q^{\prime}\in\mathcal{S}:Q^{\prime}\subset
Q}Q^{\prime}$ are pairwise disjoint and
$\left|E_{Q}\right|\simeq\left|Q\right|$. If $Q_{0}\in\mathcal{S}$ then using
the pairwise disjointness of the $\\{E_{Q}\\}$ we have the following well–know
estimate:

(3.1) $\displaystyle\sum_{Q\in\mathcal{Q}:Q\subset
Q_{0}}\sigma(Q)\simeq\int_{Q_{0}}\sum_{Q\in\mathcal{Q}:Q\subset
Q_{0}}\langle\sigma\rangle_{Q}1\\!\\!1_{E_{Q}}\leq\int_{Q_{0}}M(\sigma
1\\!\\!1_{Q_{0}})=\rho(Q_{0};\sigma)\sigma(Q_{0}).$

A consequence of [CruMoe2013b]*Theorem 1.1 is (note the $L^{1}$ norms):

######  Proposition 3.1.

For $1<p<q$ there holds

$\displaystyle\left\|T_{\alpha,\mathcal{S}}(\sigma\cdot):L^{p}(\sigma)\to
L^{q}(w)\right\|\lesssim\mathcal{T}+\mathcal{T}^{\ast}$

where:

$\displaystyle\mathcal{T}$ $\displaystyle:=\sup_{\mathcal{S}\textnormal{ is
sparse}}\sup_{R\in\mathcal{S}}\sigma(R)^{-\frac{1}{p}}\left\|\sum_{Q\in\mathcal{S}:R\subset
Q}(\left|Q\right|^{q\frac{\alpha}{d}}\langle\sigma
1\\!\\!1_{R}\rangle_{Q})^{q}1\\!\\!1_{E_{Q}}\right\|_{L^{1}(w)}^{\frac{1}{q}}$
$\displaystyle\mathcal{T}^{\ast}$ $\displaystyle=\sup_{\mathcal{S}\textnormal{
is
sparse}}\sup_{R\in\mathcal{S}}w(R)^{-1/q^{\prime}}\left\|\sum_{Q\in\mathcal{S}:R\subset
Q}\left|Q\right|^{(\frac{\alpha}{d}-1)p^{\prime}}w(Q)^{p^{\prime}}1\\!\\!1_{E_{Q}}\right\|_{L^{1}(\sigma)}^{\frac{1}{p^{\prime}}}.$

Indeed, [CruMoe2013b]*Theorem 1.1 says that if $p<q$ then:

$\displaystyle\left\|T_{{\alpha,\mathcal{S}}}\sigma\cdot:L^{p}(\sigma)\to
L^{q}(w)\right\|\simeq\left\|M_{{\alpha}}\sigma\cdot:L^{p}(\sigma)\to
L^{q}(w)\right\|+\left\|M_{{\alpha}}\sigma\cdot:L^{q^{\prime}}(w)\to
L^{p^{\prime}}(\sigma)\right\|,$

where
$M_{\alpha}f(x):=\sup_{Q}\left|Q\right|^{\frac{\alpha}{d}}\langle\left|f\right|\rangle_{Q}1\\!\\!1_{Q}(x)$
is the fractional maximal operator. The well–known ([Saw1982]) testing
conditions for this operator reduce to the ones in Proposition 3.1. This is
because the linearization of $M_{\alpha}$ is the function inside the norm in
this proposition and the pairwise disjointness of the $\\{E_{Q}\\}$ allows us
to pass the exponent $q$ (and $p^{\prime}$) under the sum.

##  4\. Proof of Theorem 1.1

We will estimate $\mathcal{T}$ from Proposition 3.1 (the estimate for
$\mathcal{T}^{\ast}$ is dual).

Let $\mathcal{S}_{a}$ be those cubes with $\rho(Q;\sigma)\simeq 2^{a}$
(observe that $a\geq 0$) and let $\mathcal{S}_{a}^{\ast}$ be the maximal cubes
in $\mathcal{S}_{a}$. The integral in the definition of $\mathcal{T}$ can be
organized as follows:

(4.1) $\displaystyle\sum_{Q\in\mathcal{S}:Q\subset
R}\left|Q\right|^{q\frac{\alpha}{d}}\langle\sigma\rangle_{Q}^{q}w(Q)=\sum_{a\geq
0}\sum_{Q^{\ast}\in\mathcal{S}_{a}^{\ast}}\sum_{Q\in\mathcal{S}_{a}:Q\subset
Q^{\ast}}(\frac{w(Q)\sigma(Q)^{\frac{q}{p^{\prime}}}}{\left|Q\right|^{q-q\frac{\alpha}{d}}})\sigma(Q)^{\frac{q}{p}}.$

Concerning the inner sum, we bound this by:

$\displaystyle\mathcal{E}^{q}\frac{1}{2^{a}\varepsilon(2^{a})}\sum_{Q\in\mathcal{S}_{a}:Q\subset
Q^{\ast}}\sigma(Q)^{\frac{q}{p}}$
$\displaystyle\mathcal{E}^{q}\frac{1}{2^{a}\varepsilon(2^{a})}\sigma(Q^{\ast})^{\frac{q}{p}-1}\sum_{Q\in\mathcal{S}_{a}:Q\subset
Q^{\ast}}\sigma(Q)\leq\mathcal{E}^{q}\frac{\sigma(Q^{\ast})^{\frac{q}{p}}}{\varepsilon(2^{a})}.$

where we multiplied and divided by
$(\rho(Q;\sigma)\varepsilon(\rho(Q;\sigma))$ and used the fact that
$\rho(Q;\sigma)\simeq 2^{a}$. The "$\leq$" uses (3.1). Now we insert this
estimate into (4.1) and using again that $\frac{q}{p}\geq 1$, the maximality
of the cubes $Q^{\ast}$ and the summability of $\varepsilon$ we get:

$\displaystyle\sum_{a\geq
0}\sum_{Q^{\ast}\in\mathcal{S}_{a}^{\ast}}\sum_{Q\in\mathcal{S}_{a}:Q\subset
Q^{\ast}}\left|Q\right|^{q\frac{\alpha}{d}}\langle\sigma\rangle_{Q}^{q}w(Q)\lesssim\mathcal{E}^{q}\sum_{a\geq
0}\frac{1}{\varepsilon(2^{a})}\left(\sum_{Q^{\ast}\in\mathcal{S}_{a}^{\ast}}\sigma(Q^{\ast})\right)^{\frac{q}{p}}\leq\mathcal{E}^{q}\sigma(R)^{\frac{q}{p}}.$

Taking $q^{\textnormal{th}}$ roots we conclude that
$\mathcal{T}\lesssim\mathcal{E}$.

##  5\. Proof of Theorem 1.2

We will estimate $\mathcal{T}$ from Proposition 3.1 (the estimate for
$\mathcal{T}^{\ast}$ is dual).

Let $\mathcal{S}_{a}$ be those cubes with $\langle\sigma\rangle_{Q}\simeq
2^{a}$. (observe that $-\infty<r<\infty$) and let $\mathcal{S}_{a}^{\ast}$ be
the maximal cubes in $\mathcal{S}_{a}$. The integral in the definition of
$\mathcal{T}$ can be organized as follows:

(5.1) $\displaystyle\sum_{Q\in\mathcal{S}:Q\subset
R}\left|Q\right|^{q\frac{\alpha}{d}}\langle\sigma\rangle_{Q}^{q}w(Q)=\sum_{a\in\mathbb{Z}}\sum_{Q^{\ast}\in\mathcal{S}_{a}^{\ast}}\sum_{Q\in\mathcal{S}_{a}:Q\subset
Q^{\ast}}\left|Q\right|^{q\frac{\alpha}{d}}\langle\sigma\rangle_{Q}^{q}w(Q).$

Concerning the inner sum, this is (similar to the above):

$\displaystyle\sum_{Q\in\mathcal{S}_{a}:Q\subset
Q^{\ast}}(\frac{w(Q)\sigma(Q)^{\frac{q}{p^{\prime}}}}{\left|Q\right|^{q-q\frac{\alpha}{d}}})\sigma(Q)^{\frac{q}{p}}\lesssim\mathcal{D}^{q}\frac{1}{\varepsilon(2^{a})}\sigma(Q^{\ast})^{\frac{q}{p}-1}\sum_{Q\in\mathcal{S}_{r}:Q\subset
Q^{\ast}}\sigma(Q)\leq\mathcal{D}^{q}\frac{\sigma(Q^{\ast})^{\frac{q}{p}}}{\varepsilon(2^{a})}.$

where we multiplied and divided by $\varepsilon(\langle\sigma\rangle_{Q})$ and
used the fact that $\langle\sigma\rangle_{Q}\simeq 2^{a}$. The “$\leq$” is
obtained using sparseness combined with $\sigma(Q)\simeq 2^{a}\left|Q\right|$
for the cubes in question.

Similar to the above proof, the summability condition on $\varepsilon$ implies
that $\mathcal{T}\lesssim\mathcal{D}$.

## References