Maximal Ergodic Theorem On Weighted $L^p_w(X)$ spaces

TL;DR

This paper establishes the boundedness of the maximal ergodic operator on weighted $L^p$ spaces with ergodic $A_p$ weights, extending classical ergodic theorems to weighted settings using transference techniques.

Contribution

It introduces a new analysis of the maximal ergodic operator on weighted $L^p$ spaces with ergodic $A_p$ weights, employing transference methods.

Findings

Boundedness of maximal ergodic operator on weighted spaces.

Extension of ergodic theorems to weighted $L^p$ spaces.

Use of transference method for weighted ergodic analysis.

Abstract

In this paper, we study the maximal ergodic operator on $L^p_w(X, \mathcal{B}, \mu)$ spaces, $1 \leq p < \infty$, where $(X, \mathcal{B}, \mu)$ is a probability space equipped with an invertible measure preserving transformation $U$ and $w$ is an ergodic $A_p$ weight using transference method.

Equations

$${\left\{I_{N,j}\right\}}_{j\in \mathbb{Z}}={\left\{[(j-1)2^N+1,\dots ,j{2}^N]\right\}}_{j\in \mathbb{Z}}$$

$$2mLI$$

$$2mRI$$

$$(2m+1)I$$

$$M^{\prime }a(m)=\underset{r>0}{\sup }\frac{1}{(2r+1)}\sum _{n\in I_r}|a(m-n)|$$

$$(M_J^{\prime }a)(m)=\underset{J>r>0}{\sup }\frac{1}{(2r+1)}\sum _{n\in I_r}|a(m-n)|$$

$$Ma(m)=\underset{m\in I}{\sup }\frac{1}{|I|}\sum _{n\in I}|a(n)|$$

$$M_{J}a(m)=\underset{J\ge m\in I}{\sup }\frac{1}{|I|}\sum _{n\in I}|a(n)|$$

$$M_{d}a(m)=\underset{m\in I}{\sup }\frac{1}{|I|}\sum _{k\in I}|a(k)|$$

$$M^{\#}a(m)=\underset{m\in I}{\sup }\frac{1}{|I|}\sum _{n\in I}|a(n)-a_I|$$

$$\biggl{(}\frac{1}{\lvert I\rvert}\sum_{k\in I}w(k)\biggr{)}\biggl{(}\frac{1}{\lvert I\rvert}\sum_{k\in I}w(k)^{-\frac{1}{p-1}}\biggr{)}^{p-1}\leq C$$

$$\frac{1}{|I|}\sum _{k\in I}w(k)\le Cw(m)$$

$$\sum _{n\in \mathbb{Z}}|a(n){|}^{p}w(n)<\infty$$

$$\left\lVert a\right\rVert_{\ell_{w}^{p}({\mathbb{Z}})}=\biggl{(}\sum_{k\in{\mathbb{Z}}}{\lvert a(k)\rvert}^{p}w(k)\biggr{)}^{\frac{1}{p}}$$

$$w(\left\{m\in \mathbb{Z}:Ma(m)>\lambda \right\})\le \frac{C}{{\lambda }^p}\sum _{m\in \mathbb{Z}}|a(m){|}^{p}w(m)$$

$$esssup_{x\in X}\sup_{N\geq 1}\biggl{(}\frac{1}{2N+1}\sum_{k=-N}^{N}w(U^{k}x)\biggr{)}\biggl{(}\frac{1}{2N+1}\sum_{k=-N}^{N}w(U^{k}x)^{\frac{-1}{p-1}}\biggr{)}^{p-1}\leq C$$

$$esssu{p}_{x\in X}\underset{N\ge 1}{\sup }\frac{1}{2N+1}\sum _{k=-N}^{N}w(U^{k}x)\le Cw(U^{m}x)$$

$${\int }_{x\in X}|f(x){|}^{p}w(x)d\mu (x)<\infty$$

$$\left\lVert f\right\rVert_{L^{p}_{w}(X)}=\biggl{(}\int_{x\in X}{\lvert f(x)\rvert}^{p}w(x)d\mu(x)\biggr{)}^{\frac{1}{p}}$$

$$M^{\prime }a(m)\le Ma(m)\le 3M^{\prime }a(m)$$

$$|\left\{m\in \mathbb{Z}:M^{\prime }a(m)>4\lambda \right\}|\le 3|\left\{m\in \mathbb{Z}:M_{d}a(m)>\lambda \right\}|$$

$$1.$$

$$2.$$

$$w(\left\{m\in \mathbb{Z}:Ma(m)>\lambda \right\})\le \frac{C}{{\lambda }^p}\sum _{m\in \mathbb{Z}}|a(m){|}^{p}w(m)$$

$$\sum _{m\in \mathbb{Z}}Ma(m{)}^{p}w(m)\le C_p\sum _{m\in \mathbb{Z}}|a(m){|}^{p}Mw(m)$$

$$\sum _{\left\{m\in \mathbb{Z}:Ma(m)>\lambda \right\}}w(m)\le \frac{C_1}{\lambda }\sum _{m\in \mathbb{Z}}|a(m)|Mw(m)$$

$$\sum _{\left\{n\in \mathbb{Z}:|a(n)|>C\right\}}Mw(n)=0$$

$$\left\{n\in \mathbb{Z}:|a(n)|>C\right\}=\varnothing$$

$$\left\{n\in \mathbb{Z}:|Ma(n)|>C\right\}=\varnothing$$

$$\sum _{\left\{n\in \mathbb{Z}:|Ma(n)|>C\right\}}w(n)=0$$

$${\Vert Ma\Vert }_{{\ell }_w^{\infty }(\mathbb{Z})}\le {\Vert a\Vert }_{{\ell }_{Mw}^{\infty }(\mathbb{Z})}$$

$$\frac{1}{|I_j|}\sum _{k\in I_j}|a(k)|>\frac{\lambda }{4}$$

Full text

††footnotetext: 2020 Mathematics Subject Classification: Primary 28D05; Secondary 37A46.††footnotetext: Calderon-Zygmund decomposition, Maximal Ergodic Operator, Transference Method, Ergodic Rectangles, Ergodic Weights.

1 Introduction

In this paper, using transference method, we prove strong type, weak type inequalities for maximal ergodic operators on $L^{p}_{w}(X,\mathcal{B},\mu)$ spaces, $1\leq p<\infty$, where $(X,\mathcal{B},\mu)$ is a probability space equipped with an invertible measure preserving transformation $U$ and $w$ is an ergodic $A_{p}$ weight.

For standard results on boundedness of various maximal operators in harmonic analysis we refer to [7]. Boundedness of maximal ergodic operator for $L^{p}$ spaces with weights can be found in [3]. In [3] the characterization of those positive functions $w$ such that the maximal ergodic operator associated with an invertible measure preserving transformation on a probability space is a bounded operator on $L^{p}(wd\mu)$ is given. In their proof the ergodic analogue of Calderon-Zygmund decomposition and the concept of ergodic rectangles are used. In this paper we prove the same result using Calderon-Coifman-Weiss transference principle $\text{\cite[cite]{[\@@bibref{}{Cald1}{}{}]}},\text{\cite[cite]{[\@@bibref{}{Coif2}{}{}]}},\text{\cite[cite]{[\@@bibref{}{Mich2}{}{}]}}$.

2 Notation

Throughout this paper, ${\mathbb{Z}}$ denotes set of all integers and ${\mathbb{Z}}_{+}$ denotes set of all positive integers. For a given interval $I$ in ${\mathbb{Z}}$ (We always mean finite interval of integers) , $\lvert I\rvert$ always denotes the cardinality of $I$. For each positive integer N, consider collection of disjoint intervals of cardinality $2^{N}$,

[TABLE]

The set of intervals which are of the form $I_{N,j}$ where $N\in{\mathbb{Z}}_{+}$ and $j\in{\mathbb{Z}}$ are called dyadic intervals. For fixed $N$, $I_{N,j}$ are disjoint. Given a dyadic interval $I=\left\{[(j-1)2^{N}+1,\dots,j2^{N}]\right\}_{j\in{\mathbb{Z}}}$ and a positive integer $m$, we define

[TABLE]

Note that $2mLI,2mRI$ are dyadic intervals each of length $(2m)2^{N}$. However $(2m+1)I,m\in{\mathbb{Z}}_{+}$ are not dyadic intervals.

3 Definitions

Maximal Operators

Let $\left\{a(n):n\in{\mathbb{Z}}\right\}$ be a sequence. We define the following three types of Hardy-Littlewood maximal operators as follows:

Definition 3.1**.**

If $I_{r}$ is the interval $\left\{-r,-r+1,\dots,0,1,2,\dots,r-1,r\right\}$, define centered Hardy-Littlewood maximal operator

[TABLE]

For any positive integer $J$, define truncated centered Hardy-Littlewood maximal operator as

[TABLE]

Definition 3.2**.**

We define Hardy-Littlewood maximal operator as follows

[TABLE]

where the supremum is taken over all intervals containing $m$. For any positive integer $J$, define truncated Hardy-Littlewood maximal operator as

[TABLE]

Definition 3.3**.**

We define dyadic Hardy-Littlewood maximal operator as follows:

[TABLE]

where supremum is taken over all dyadic intervals containing $m$.

Definition 3.4**.**

Given a sequence $\left\{a(n):n\in{\mathbb{Z}}\right\}$ and an interval $I$, let $a_{I}$ denote average of ${\left\{a(n):n\in{\mathbb{Z}}\right\}}$ on $I$. Let, $a_{I}=\frac{1}{\lvert I\rvert}\sum_{m\in I}a(m)$. Define the sharp maximal operator $M^{\#}$ as follows

[TABLE]

where the supremum is taken over all intervals $I$ containing $m$. We say that sequence $\left\{a(n):n\in{\mathbb{Z}}\right\}$ has bounded mean oscillation if the sequence $M^{\#}a$ is bounded. The space of sequences with this property is denoted by BMO(${\mathbb{Z}}$). We define a norm in BMO(${\mathbb{Z}}$) by $\left\lVert a\right\rVert_{\star}=\left\lVert M^{\#}a\right\rVert_{\infty}$. The space BMO(${\mathbb{Z}}$) is studied in [8].

Weights

Definition 3.5**.**

For a fixed $p$, $1<p<\infty$, we say that a non-negative sequence $\left\{w(n):n\in{\mathbb{Z}}\right\}$ belongs to class $A_{p}$ if there is a constant $C$ such that, for all intervals $I$ in ${\mathbb{Z}}$, we have

[TABLE]

This constant $C$ is called $A_{p}$ constant. We say that $\left\{w(m):m\in{\mathbb{Z}}\right\}$ belongs to class $A_{1}$ if there a constant C such that, for all intervals $I$ in ${\mathbb{Z}}$,

[TABLE]

for all $m\in I$. This constant $C$ is called $A_{1}$ constant.

Let $1\leq p<\infty$ and $\left\{w(n):n\in{\mathbb{Z}}\right\}\in A_{p}({\mathbb{Z}})$. We say that a sequence $\left\{a(n):n\in{\mathbb{Z}}\right\}$ is in $\ell^{p}_{w}({\mathbb{Z}})$ if

[TABLE]

We define norm in $\ell^{p}_{w}({\mathbb{Z}})$ by

[TABLE]

For a given sequence $\left\{a(n):n\in{\mathbb{Z}}\right\}\in\ell_{w}^{p}({\mathbb{Z}})$, the weighted weak(p,p) inequality for a non-negative weight sequence $\left\{w(n):n\in{\mathbb{Z}}\right\}$ is as follows:

[TABLE]

For a subset $A$ of ${\mathbb{Z}}$, $w(A)$ denotes $\sum_{k\in A}w(k)$.

The following definition is from [3]

Definition 3.6**.**

Let $(X,\textbf{B},\mu)$ be a probability space and $U$ an invertible measure preserving transformation on X. Suppose $1<p<\infty$ and $w:X\to{\mathbb{R}}$ be a non-negative integrable function. The function $w$ is said to satisfy ergodic $A_{p}$ condition if

[TABLE]

The function $w$ is said to satisfy ergodic $A_{1}$ condition if

[TABLE]

for $m=-N,-N+1,\dots,N$

Definition 3.7**.**

Let $1\leq p<\infty$. We say that a measurable function $f\in L^{p}_{w}(X)$ if

[TABLE]

We define norm in $L^{p}_{w}(X)$ by

[TABLE]

4 Relations between Maximal operators

In the following lemmas, we give relations between maximal operators. For the proofs of the following lemmas, refer [1]. These relations will be used when we prove the weighted inequalities for maximal ergodic operators.

Lemma 4.1**.**

Given a sequence $\left\{a(m):m\in{\mathbb{Z}}\right\}$, the following relation holds:

[TABLE]

Lemma 4.2**.**

If $\textbf{a}=\left\{a(k):k\in{\mathbb{Z}}\right\}$ is a non-negative sequence with $\textbf{a}\in\ell_{1}$, then

[TABLE]

In the following lemma, we see that in the norm of BMO(${\mathbb{Z}}$) space, we can replace the average $a_{I}$ of $\left\{a(n)\right\}$ by a constant $b$. The proof is similar to the proof in continuous version [7]. The second inequality follows from $\lvert\lvert a\rvert-\lvert b\rvert\rvert\leq\lvert a\rvert-\lvert b\rvert$.

Lemma 4.3**.**

Consider a non-negative sequence $\textbf{a}=\left\{a(k):k\in{\mathbb{Z}}\right\}$. Then the following are valid.

[TABLE]

5 Weighted Classical Results for Maximal Operators

In this section, for a given sequence $\left\{a(n):n\in{\mathbb{Z}}\right\}$ in $\ell^{p}_{w}({\mathbb{Z}})$, we prove weighted weak(p,p) inequality with respect to the weight $\left\{w(n):n\in{\mathbb{Z}}\right\}\in A_{p}$ which is as follows:

[TABLE]

Inequality A4 will be proved via several theorems, Theorem [5.1] to Theorem [5.3].

The proof of Theorem5.1 uses Calderon-Zygmund decomposition. For the proofs of the corresponding results for the continuous version, we refer [7]. The proofs of Theorem [5.1] and Theorem [5.3] are same as the proof for the continuous versions of the corresponding results apart from the fact that the constants obtained here are slightly different from the constants obtained for the continuous version due to the nature of dyadic intervals in ${\mathbb{Z}}$ [See [2]]. So, we give here the proof of Theorem[5.1] and Theorem 5.3 for the sake of completeness.

Theorem 5.1**.**

If $\left\{w(n):n\in{\mathbb{Z}}\right\}\in A_{p}$ , $1\leq p<\infty$ and $\left\{a(n):n\in{\mathbb{Z}}\right\}$ is a sequence in $\ell^{p}({\mathbb{Z}})$, then for $p>1$, there exists a constant $C_{p}$ such that

[TABLE]

Furthermore, for $p=1$, there exists a constant $C_{1}$ such that

[TABLE]

Proof.

We will show that $\left\lVert Ma\right\rVert_{\ell^{\infty}_{w}({\mathbb{Z}})}\leq\left\lVert a\right\rVert_{\ell^{\infty}_{Mw}({\mathbb{Z}})}$ and that weak(1,1) inequality holds; the strong(p,p) inequality then follows from the Marcinkiewicz interpolation theorem.

Take $C>\left\lVert a\right\rVert_{\ell^{\infty}_{Mw}({\mathbb{Z}})}$. Then

[TABLE]

which shows that

[TABLE]

Hence $\lvert a(n)\rvert\leq C,\forall n\in{\mathbb{Z}}$ which implies that $\lvert Ma(n)\rvert\leq C,\forall n\in{\mathbb{Z}}$ Therefore

[TABLE]

So,

[TABLE]

which gives $w(\left\{n\in{\mathbb{Z}}:\lvert Ma(n)\rvert>C\right\})=0$.

Therefore $\left\lVert Ma\right\rVert_{\ell^{\infty}_{w}({\mathbb{Z}})}\leq C$. Taking $\inf\left\{C:\left\lVert a\right\rVert_{\ell^{\infty}_{Mw}({\mathbb{Z}})}<C\right\}$ we get

[TABLE]

To prove the weak(1,1) inequality we may assume that $\left\{a(n):n\in{\mathbb{Z}}\right\}\in\ell^{1}({\mathbb{Z}})$. Form Calderon-Zygmund decomposition of sequence $\left\{a(n):n\in{\mathbb{Z}}\right\}$ at height $\frac{\lambda}{4}>0$. Then we get a sequence $\left\{I_{j}\right\}$ of dyadic intervals in ${\mathbb{Z}}$ such that

[TABLE]

Further as we showed in the proof of Lemma [$\ref{Sak_lem_2.12}](See$ [1]),

[TABLE]

It follows that

[TABLE]

Since by Lemma[4.1], $\left\{m:Ma(m)>\lambda\right\}\subseteq\left\{m:M^{\prime}a(m)>\frac{\lambda}{3}\right\}$, it follows that

[TABLE]

∎

The proof of the following theorem is similar to the proof of corresponding result in continuous version [7]. We state here without proof.

Theorem 5.2**.**

Let ${\left\{a(n):n\in{\mathbb{Z}}\right\}}$ be a non-negative sequence and $\left\{w(n):n\in{\mathbb{Z}}\right\}\in A_{p},1\leq p<\infty$ be a non-negative weight sequence. Let $I$ be an interval such that $a(m)>0$ for some $m\in I$. Then,

[TABLE] 2. 2.

Given a finite set $S\subset I$ ,

[TABLE]

$\ref{Sak_own_7.1}[A]$ follows from Holder’s inequality and the $A_{p}$ condition. $\ref{Sak_own_7.1}[B]$ follows by taking $a=\chi_{S}$ in $\ref{Sak_own_7.1}[A]$.

Theorem 5.3**.**

Assume $\left\{w(n):n\in{\mathbb{Z}}\right\}\in A_{p}$. Given a non-negative sequence $\left\{a(n):n\in{\mathbb{Z}}\right\}\in\ell_{w}^{p}({\mathbb{Z}})$, for $1\leq p<\infty$, the weighted weak(p,p) inequality holds:

[TABLE]

Proof.

Let $\left\{a(n):n\in{\mathbb{Z}}\right\}\in\ell_{w}^{p}({\mathbb{Z}})$. Form the Calderon-Zygmund decomposition of $\left\{a(n):n\in{\mathbb{Z}}\right\}$ at height $\frac{\lambda}{12}$ to get a collection of disjoint intervals $\left\{I_{j}\right\}$ such that $a(I_{j})>\frac{\lambda}{12}\lvert I_{j}\rvert$. By the proof of Lemma$[{\ref{Sak_lem_2.12}}]$ in [1] and Lemma[$\ref{Sak_lem_2.11}]$ , we have

[TABLE]

Therefore, using Theorem[5.2], we have

[TABLE]

∎

Theorem 5.4**.**

If $w\in A_{p},1<p<\infty$, then $M$ is bounded on $\ell_{w}^{p}({\mathbb{Z}})$.

The proof follows from Theorem[5.3] and Marcinkiewicz interpolation theorem.

6 Maximal Ergodic Operator

Let $(X,\textbf{B},\mu)$ be a probability space and $U$ an invertible measure preserving transformation on $X$. We define maximal ergodic operator as

[TABLE]

For any positive integer $J$, we also define truncated maximal ergodic operator as

[TABLE]

In the following theorem using transference, we prove that the maximal ergodic operator is bounded on $L^{p}_{w}(X,\mathcal{B},\mu),1<p<\infty$ where $w$ is ergodic $A_{p}$ weight and the maximal ergodic operator satisfies weak type (1,1) inequality on $L^{1}_{w}(X,\mathbf{B},\mu)$ space.

Theorem 6.1**.**

Let $(X,\textbf{B},\mu)$ be a probability space and $U$ an invertible measure preserving transformation on $X$. satisfies

If $w$ is an ergodic $A_{p}$ weight, $1<p<\infty$ and $f\in L^{p}_{w}(X,\mathcal{B},\mu)$, then the maximal ergodic operator

[TABLE] 2. 2.

If w is an ergodic $A_{1}$ weight and $f\in L^{1}_{w}(X,\mathcal{B},\mu)$ , then

[TABLE]

Proof.

Take $p,1\leq p<\infty$. Fix $J>0$ and take a function $f\in L^{p}_{w}(X)$.

[TABLE]

It is enough to prove that $\tilde{M_{J}}$ satisfies (1) and (2) with constants not depending on $J$. Let $\lambda>0$ and put

[TABLE]

For x lying outside a $\mu$ null set and a positive integer $L$, define sequences

[TABLE]

[TABLE]

Using Lemma[4.1], observe that for an integer $m$ with $\lvert m\rvert\leq L$

[TABLE]

Therefore,

[TABLE]

by choosing $L$ appropriately. Conclusion $(1)$ of the theorem now follows by using the Marcinkiewicz interpolation theorem. ∎

Now, we prove the converse of Theorem[6.1] for $p>1$ with the additional assumptions (1) $(X,\mathcal{B},\mu)$ is a probability space and (2) $U$ is ergodic measure preserving transformation. Using transference method, we prove the converse of Theorem[6.1]. A direct proof can be seen in [3]. For this we require the concept of ergodic rectangles which we define below [3].

Definition 6.2** (Ergodic Rectangle).**

Let $E$ be a subset of $X$ with positive measure and let $K\geq 1$ be such that $U^{i}E\cap U^{j}E=\phi\quad\text{if}\quad i\neq j$ and $-K\leq i,j\leq K$. Then the set $R=\cup_{i=-K}^{K}U^{i}E$ is called ergodic rectangle of length $2K+1$ with base $E$.

For the proof of following lemma[6.3], refer[3].

Lemma 6.3**.**

Let $(X,\textbf{B},\mu)$ be a probability space, $U$ an ergodic invertible measure preserving transformation on $X$ and $K$ a positive integer.

If $F\subseteq X$ is a set of positive measure then there exists a subset $E\subseteq F$ of positive measure such that $E$ is base of an ergodic rectangle of length $2K+1$. 2. 2.

There exists a countable family $\left\{E_{j}\right\}$ of bases of ergodic rectangles of length $2K+1$ such that $X=\cup_{j}E_{j}$.

Theorem 6.4**.**

Let $(X,\mathcal{B},\mu)$ be a probability space, U an invertible ergodic measure preserving transformation on $X$. If $\tilde{M}f$ is bounded on $L^{p}_{w}(X)$ for some $1<p<\infty$, then $w\in A_{P}(X)$.

Proof.

For the given function $w$ on $X$, for a.e $x\in X$ define the sequence $w_{x}(k)=w(U^{-k}x)$. We shall prove that

[TABLE]

This will prove that $w\in A_{p}(X)$. In order to prove this, we shall prove that the Hardy-Littlewood maximal operator $M$ is bounded on $\ell^{p}_{w_{x}}({\mathbb{Z}})$ and

[TABLE]

where $C_{p}$ is independent of $x$. In order to prove the above inequality, take a sequence $a=\left\{a(n):n\in{\mathbb{Z}}\right\}\in\ell^{p}_{w_{x}}({\mathbb{Z}})$.

Let $R=\cup_{k=-2J}^{2J}U^{k}E$ be an ergodic rectangle of length $4J+1$ with base $E$. Let $F$ be any measurable subset of $E$. Then $F$ is also base of an ergodic rectangle of length $4J+1$. Let $R^{\prime}=\cup_{k=-2J}^{2J}U^{k}F$. Define function $f$ and $w$ as follows.

[TABLE]

Then,

[TABLE]

Using Lemma[4.1], it is easy to observe that for $-J\leq m\leq J$ and $x\in F$

[TABLE]

Now,

[TABLE]

So from the above estimates

[TABLE]

Since $F$ was an arbitrary subset of $E$, we get

[TABLE]

a.e $x\in E$. Since $U$ is ergodic, $X$ can be written as countable union of bases of ergodic rectangles of length $4J+1$. Therefore for a.e $x\in X$,

[TABLE]

Since $C$ is independent of $J$, a.e $x\in X$,

[TABLE]

It follows that the sequence $\left\{w_{x}(n):n\in{\mathbb{Z}}\right\}$ as defined by $w_{x}(k)=w(U^{k}x)$ belongs to $A_{p}({\mathbb{Z}})$ a.e $x\in X$ and $A_{p}$ weight constant for $w_{x}$ is independent of $x$ so that $w\in A_{p}(X)$. ∎

7 Conclusion

The study of maximal ergodic operator on $L^{p}_{w}(X,\mathcal{B},\mu)$ spaces paves the way to study this operator on variable $L^{{p(\cdot)}}(X,\mathcal{B},\mu)$ spaces. Using Rubio de Francia extrapolation method [6] and appropriate variable Holder’s inequality, we hope to achieve this result.