Bilinear estimates on Morrey spaces by using average

TL;DR

This paper studies the boundedness of bilinear fractional integral operators on Morrey spaces, focusing on the case where the parameter t is between 0 and 1, extending previous results to this range.

Contribution

It extends the analysis of bilinear fractional integral operators on Morrey spaces to include the case where 0<t<1, filling a gap in existing research.

Findings

Established boundedness results for 0<t<1 on Morrey spaces.

Extended previous work that covered t=1 and t>1.

Provided new estimates for bilinear fractional integral operators.

Abstract

This paper is a follow up of [6]. We investigate the boundedness of the bilinear fractional integral operator introduced by Grafakos in [3]. When the local integrability index $s$ falls 1 with weights and $t$ exceeds 1, He and Yan obtained some results on this operator was worked on Morrey spaces earlier in [7]. Later in the paper [6], we considered the case $t=1$. This paper handles the remaining case $0<t<1$.

Equations

$${\mathcal{J}}_{\alpha }[f_1,f_2](x)\equiv {\int }_{{\mathbb{R}}^n}\frac{f_1(x+y)f_2(x-y)}{|y{|}^{n-\alpha }}\mathrm{d}y,x\in {\mathbb{R}}^n,$$

$${\mathcal{M}}_q^p({\mathbb{R}}^n)\equiv \left\{f\in L^0({\mathbb{R}}^n):\Vert f{\Vert }_{{\mathcal{M}}_q^p}\equiv \underset{Q\in \mathcal{D}}{\sup }|Q{|}^{\frac{1}{p}-\frac{1}{q}}{\left({\int }_Q|f(x){|}^{q}dx\right)}^{\frac{1}{q}}<\infty \right\},$$

$${\mathcal{J}}_{\alpha }[f_1,f_2](x)\equiv {\int }_{{\mathbb{R}}^n}\frac{f_1(x+y)f_2(x-y)}{|y{|}^{n-\alpha }}\mathrm{d}y,x\in {\mathbb{R}}^n,$$

$$0<\alpha <n,1<q_1\le p_1<\infty ,1<q_2\le p_2<\infty ,1\le t\le s<\infty .$$

$$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2},\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2},$$

$$\frac{1}{s}=\frac{1}{p}-\frac{\alpha }{n},\frac{q}{p}=\frac{t}{s},s<\min (q_1,q_2).$$

$$\Vert {\mathcal{J}}_{\alpha }[f_1,f_2]{\Vert }_{{\mathcal{M}}_t^s}\lesssim \Vert f_1{\Vert }_{{\mathcal{M}}_{q_1}^{p_1}}\Vert f_2{\Vert }_{{\mathcal{M}}_{q_2}^{p_2}}.$$

$$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2},\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2},$$

$$\frac{1}{s}=\frac{1}{p}-\frac{\alpha }{n},\frac{q}{p}=\frac{t}{s}.$$

$$\Vert {\mathcal{J}}_{\alpha }[f_1,f_2]{\Vert }_{{\mathcal{M}}_t^s}\lesssim \Vert f_1{\Vert }_{{\mathcal{M}}_{q_1}^{p_1}}\Vert f_2{\Vert }_{{\mathcal{M}}_{q_2}^{p_2}}$$

$$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2},\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2},$$

$$\frac{1}{s}=\frac{1}{p}-\frac{\alpha }{n},\frac{q}{p}=\frac{t}{s},s<\min (q_1,q_2).$$

$$\Vert {\mathcal{J}}_{\alpha }[f_1,f_2]{\Vert }_{{\mathcal{M}}_t^s}\lesssim \Vert f_1{\Vert }_{{\mathcal{M}}_{q_1}^{p_1}}\Vert f_2{\Vert }_{{\mathcal{M}}_{q_2}^{p_2}}$$

$$Mf(x)\equiv \underset{Q}{\sup }\frac{{\chi }_Q(x)}{|Q|}{\int }_Q|f(y)|\mathrm{d}y,x\in {\mathbb{R}}^n,$$

$$\Vert M^{(\eta )}f{\Vert }_{{\mathcal{M}}_q^p}\lesssim \Vert f{\Vert }_{{\mathcal{M}}_q^p},f\in {\mathcal{M}}_q^p({\mathbb{R}}^n)$$

$$m_Q(f)\equiv \frac{1}{|Q|}{\int }_{Q}f(x)\mathrm{d}x,$$

$${\mathcal{D}}_j\equiv \{Q_{jk}:k\in {\mathbb{Z}}^n\},j\in \mathbb{Z},$$

$$\mathcal{D}\equiv \{Q_{jk}:j\in \mathbb{Z},k\in {\mathbb{Z}}^n\}=\bigcup _{j\in \mathbb{Z}}{\mathcal{D}}_j.$$

$${\Vert \sum _{j=1}^{\infty }{f}_j\Vert }_{L^p}\lesssim {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{L^p}$$

$${\Vert \sum _{j=1}^{\infty }{f}_j\Vert }_{{\mathcal{M}}_q^p}\lesssim {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}$$

$$|Q{|}^{\frac{1}{p}-\frac{1}{q}}{\Vert \sum _{j=1}^{\infty }{\chi }_Q{f}_j\Vert }_{L^q}\lesssim {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}.$$

$$|Q{|}^{\frac{1}{p}-\frac{1}{q}}{\Vert \sum _{j\in \mathbb{N},Q_j\subset Q}{\chi }_Q{f}_j\Vert }_{L^q}\lesssim {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}$$

$$|Q{|}^{\frac{1}{p}-\frac{1}{q}}{\Vert \sum _{j\in \mathbb{N},Q_j\supset Q}{\chi }_Q{f}_j\Vert }_{L^q}\lesssim {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}.$$

$$|Q{|}^{\frac{q}{p}-1}{\left({\Vert \sum _{j\in \mathbb{N},Q_j\supset Q}{\chi }_Q{f}_j\Vert }_{L^q}\right)}^q\lesssim {\left({\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}\right)}^q.$$

$$|Q{|}^{\frac{q}{p}-1}{\left({\Vert \sum _{j\in \mathbb{N},Q_j\supset Q}{\chi }_Q{f}_j\Vert }_{L^q}\right)}^q$$

$$\le |Q{|}^{\frac{q}{p}-1}\sum _{j\in \mathbb{N},Q_j\supset Q}{\left(|Q{|}^{\frac{1}{q}-1}{\Vert {\chi }_Q{f}_j\Vert }_{L^1}\right)}^q.$$

$${\Vert m_{Q_J}(f_J){\chi }_{Q_J}\Vert }_{{\mathcal{M}}_q^p}\le {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p},$$

$$\Vert f_J{\Vert }_{L^1}\le |Q_J{|}^{1-\frac{1}{p}}{\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}.$$

$$|Q{|}^{\frac{q}{p}-1}{\left({\Vert \sum _{j\in \mathbb{N},Q_j\supset Q}{\chi }_Q{f}_j\Vert }_{L^q}\right)}^q$$

$$\lesssim {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}.$$

$${\Vert \sum _{j=1}^{\infty }{f}_j\Vert }_{{\mathcal{M}}_q^p}\lesssim {\Vert \sum _{j=1}^{\infty }{m}_{Q_j}^{(u)}(f_j){\chi }_{Q_j}\Vert }_{{\mathcal{M}}_q^p}$$

$${\mathcal{J}}_{\alpha }[f_1,f_2](x)$$

$${\mathcal{I}}_{\alpha }[f_1,f_2](x)\equiv {\int }_{{\mathbb{R}}^n}\frac{f_1(y_1)f_2(y_2)}{(|x-y_1|+|x-y_2|{)}^{2n-\alpha }}\mathrm{d}{y}_1\mathrm{d}{y}_2,x\in {\mathbb{R}}^n,$$

Full text

Keywords Morrey spaces, bilinear fractional integral operators, dyadic cubes, average technique.

Mathematics Subject Classifications (2010) Primary 42B35; Secondary 42B25

1. Introduction

The bilinear fractional integral operator of Grafakos type $\mathcal{J}_{\alpha}$, which is given by

[TABLE]

for non-negative measurable functions or more general complex-valued measurable functions $f_{1},f_{2}$ subject to a certain integrability condition, it is known that operator $\mathcal{J}_{\alpha}$ is bounded bilinear fractional integral operator from product Lebesgue space $L^{p_{1}}(\mathbb{R}^{n})\times L^{p_{2}}(\mathbb{R}^{n})$ to Lebesgue space $L^{s}(\mathbb{R}^{n})$ in [1]. We would like to expand this result and show that operator $\mathcal{J}_{\alpha}$ is bounded bilinear fractional integral operator from product Morrey space $\mathcal{M}^{p_{1}}_{q_{1}}(\mathbb{R}^{n})\times\mathcal{M}^{p_{2}}_{q_{2}}(\mathbb{R}^{n})$ to Morrey space $\mathcal{M}^{s}_{t}$. We discuss it under $s<\min(q_{1},q_{2})$. In the paper [6], when $1\leq t\leq s<\min(q_{1},q_{2})$, we obtained the result. When $0<t\leq s<1$, in the paper [7], He and Yan show bilinear estimates with weights. In this paper, we give an alternative proof of non-weighted type of their results. Furthermore developing this proof, we obtain the case $0<t\leq 1\leq s<\min(q_{1},q_{2})$.

Let $0<q\leq p<\infty$. Define the Morrey space ${\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})$ by

[TABLE]

where $\mathcal{D}$ denotes the family of all dyadic cubes in $\mathbb{R}^{n}$. We recall the definition of the dyadic cubes precisely in Section 2.

Grafakos introduced the bilinear fractional integral operator in [3].

Definition 1.1**.**

Let $0<\alpha<n$. Define the bilinear fractional integral operator by

[TABLE]

for measurable functions $f_{1},f_{2}$ defined in $\mathbb{R}^{n}$.

First, He and Yan, under the condition $\frac{1}{q_{1}}+\frac{1}{q_{2}}<1$ and $1<t\leq s<\infty$, investigated the operator $\mathcal{J}_{\alpha}$ acting on Morrey spaces in [7] earlier. Next, we prove the case $1\leq t\leq s<\min(q_{1},q_{2})$ in [6] as follows:

Theorem 1.2**.**

Let

[TABLE]

Define $p$ and $q$ by

[TABLE]

Assume that

[TABLE]

Then for all $f_{1}\in{\mathcal{M}}^{p_{1}}_{q_{1}}({\mathbb{R}}^{n})$ and $f_{2}\in{\mathcal{M}}^{p_{2}}_{q_{2}}({\mathbb{R}}^{n})$,

[TABLE]

We investigate the boundedness property of ${\mathcal{J}}_{\alpha}$ when $0<t\leq 1$ in this paper. Two cases must be considered according to the value of $s$. In the Theorems 4.2 and 4.6 in [7] which are boundednesses of the bilinear fractional integral operator on Morrey spaces with weights, taking weights to identical equal to 1 implies the following theorem. So their results cover the Theorem 1.3 below.

Theorem 1.3**.**

[7, Theorems 4.2 and 4.6]** Let $0<\alpha<n$, $1<q_{j}\leq p_{j}<\infty$, $0<q\leq p<\infty$ and $0<t\leq s<1$ for $j=1,2$. Assume that

[TABLE]

Then we have

[TABLE]

for any $f_{1}\in\mathcal{M}^{p_{1}}_{q_{1}}(\mathbb{R}^{n})$ and for any $f_{2}\in\mathcal{M}^{p_{2}}_{q_{2}}(\mathbb{R}^{n})$.

Theorem 1.4**.**

Let $0<\alpha<n$, $1<q_{j}\leq p_{j}<\infty$, $0<q\leq p<\infty$ and $0<t\leq 1\leq s$ for $j=1,2$ with

[TABLE]

Then we have

[TABLE]

for any $f_{1}\in\mathcal{M}^{p_{1}}_{q_{1}}(\mathbb{R}^{n})$ and for any $f_{2}\in\mathcal{M}^{p_{2}}_{q_{2}}(\mathbb{R}^{n})$.

First we collect some necessary facts in Section 2, Next we give an alternative proof of Theorem 1.3 in Section 3, Finally Theorem 1.4 is proved in Section 4.

2. Preliminaries

For $f\in L^{1}_{\rm loc}(\mathbb{R}^{n})$, define a function $Mf$ by

[TABLE]

where the supremum is over all cubes in $\mathbb{R}^{n}$. This is called the Hardy-Littlewood maximal function. And the $\eta$-powered Hardy-Littlewood maximal function is defined by $M^{(\eta)}f\equiv\left(M[|f|^{\eta}]\right)^{\frac{1}{\eta}}$, for $f\in L^{\eta}_{\rm loc}(\mathbb{R}^{n})$. In [2] Chiarenza and Frasca showed that

[TABLE]

if $0<\eta<q\leq p<\infty$. For a dyadic cube $Q$ in $\mathbb{R}^{n}$, an average $m_{Q}(f)$ over $Q$ is defined by

[TABLE]

for an integrable function $f$. Moreover, we write $m_{Q}^{(\eta)}(f)\equiv\left(m_{Q}(|f|^{\eta})\right)^{\frac{1}{\eta}}$ for $\eta>0$ and a measurable function $f$. A dyadic cube is a set of the form $Q_{jk}\equiv\displaystyle\prod_{i=1}^{n}[k_{i}2^{-j},(k_{i}+1)2^{-j})$ for some $j\in\mathbb{Z}$, $k=(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}$, and define as follows:

[TABLE]

We show a key estimate which is interesting of its own right by using the following proposition:

Proposition 2.1**.**

[4, Lemma 3.1]* Let $0<p\leq 1$. Then*

[TABLE]

for all non-negative sequences $\{f_{j}\}_{j=1}^{\infty}$ of integrable functions such that each $f_{j}$ is supported on a cube $Q_{j}$.

This proposition implies the following theorem.

Theorem 2.2**.**

Let $0<q\leq p<1$. Then

[TABLE]

for all non-negative sequences $\{f_{j}\}_{j=1}^{\infty}$ of integrable functions such that each $f_{j}$ is supported on a dyadic cube $Q_{j}$.

Proof.

Fix a dyadic cube $Q$. It suffices to show that

[TABLE]

We split this estimate into two parts:

[TABLE]

and

[TABLE]

The first estimate is a consequence of Proposition 2.1. Let us prove the second inequality.

We will show

[TABLE]

By the $q$-triangle inequality

[TABLE]

We fix $J\in{\mathbb{N}}$. Then we have

[TABLE]

or equivalently,

[TABLE]

Consequently,

[TABLE]

This completes the proof. ∎

Similarly, the estimate of $u$-powered type holds, too.

Theorem 2.3**.**

[9]* Let $0<q\leq 1\leq p<u$. Then*

[TABLE]

for all non-negative sequences $\{f_{j}\}_{j=1}^{\infty}$ of integrable functions such that each $f_{j}$ is supported on a dyadic cube $Q_{j}$.

We obtain the following lemma similar to in [6, Lemma 2.1].

Lemma 2.4**.**

Let $f_{1},f_{2}\geq 0$ be measurable functions. Then we have

[TABLE]

In addition, we use [6, Lemma 2.3]. For the bilinear fractional integral operator $\mathcal{I}_{\alpha}$, $0<\alpha<2n$ defined by

[TABLE]

for integrable functions $f_{1},f_{2}$, introduced by Kenig and Stein [1], we obtain the pointwise estimate

[TABLE]

for non-negative measurable functions $f_{1},f_{2}$ in [8, Lemma 4.1]. Therefore using following lemma, we show that $\mathcal{I}_{\alpha}$ is bounded from $\mathcal{M}^{p_{1}}_{q_{1}}\times\mathcal{M}^{p_{2}}_{q_{2}}$ to $\mathcal{M}^{s}_{t}$.

Lemma 2.5**.**

Let

[TABLE]

for $j=1,2$. Assume

[TABLE]

[TABLE]

Then

[TABLE]

for all non-negative measurable functions $f_{1},f_{2}$.

By similar to prove Lemma 2.5, we obtain the result of $u$-powered type.

Lemma 2.6**.**

Let

[TABLE]

for $j=1,2$. Assume

[TABLE]

[TABLE]

Then we have

[TABLE]

for all non-negative measurable functions $f_{1},f_{2}$.

Proof.

Let $L=L(x)$ be a positive number that is specified shortly. Similar to a method by Hedberg in [5], We decompose

[TABLE]

First, we estimate the quantity $S_{1}$.

[TABLE]

Next, we estimate the quantity $S_{2}$. By Hölder’s inequality,

[TABLE]

Hence we obtain

[TABLE]

In particular, choose the constant $L=L(x)$ to optimize the right-hand side:

[TABLE]

Then we have

[TABLE]

Therefore, using H$\ddot{\rm o}$lder’s inequality for Morrey spaces, the ${\mathcal{M}}^{p_{1}}_{q_{1}}({\mathbb{R}}^{n})$-boundedness of $M^{(u)}$ and the ${\mathcal{M}}^{p_{1}}_{q_{1}}({\mathbb{R}}^{n})$-boundedness of $M^{(u)}$, we have

[TABLE]

∎

3. Proof of Theorem 1.3

For each $Q\in\mathcal{D}_{j}$, we abbreviate

[TABLE]

We calculate its average $m_{Q}(F_{j,Q})$ as follows:

[TABLE]

Hence by Theorem 2.2 and Lemma 2.4,

[TABLE]

Finally using Lemma 2.5, we obtain the result.

4. Proof of Theorem 1.4

Define $F_{j,Q}$ as above, and fix a parameter $u$ such that $s<u<\min(q_{1},q_{2})$. Thus we calculate its average $m^{(u)}_{Q}(F_{j,Q})$ as follows:

[TABLE]

Hence by Theorem 2.3 and Lemma 2.4,

[TABLE]

Finally using Lemma 2.6, we obtain the result.

Acknowledgement

The author would like to be thankful to his advisor Professor Kotaro Tsugawa for his guidance and advise, and be grateful to Professor Yoshihiro Sawano, in Tokyo Metropolitan University, for his many kinds of ideas and answering many questions. In particular, Sawano gave him a hint to Theorem 2.2.