A sharp boundedness result for restricted maximal operators of Vilenkin-Fourier series on martingale Hardy spaces

TL;DR

This paper establishes a sharp boundedness result for restricted maximal operators of Vilenkin-Fourier series on martingale Hardy spaces, identifying the precise range of p for which the operators are bounded.

Contribution

It determines the maximal subspace of p-values where the restricted maximal operators are bounded from Hardy spaces to Lebesgue spaces, and proves the result's sharpness.

Findings

Boundedness of the operators for 0<p≤1.

Identification of the maximal subspace of p-values.

Proof of the sharpness of the boundedness result.

Abstract

The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.

Equations

$${\Vert S_{n}f\Vert }_p$$

$$\underset{\lambda >0}{\sup }\lambda \mu \left(|S_{n}f|>\lambda \right)$$

$$\underset{n\in \mathbb{N}}{\sup }{\Vert S_{M_n+1}f\Vert }_{L_{p,\infty }}=\infty .$$

$${\tilde{S}}_p^{\ast }f:=\underset{n\in \mathbb{N}}{\sup }\frac{\left|S_{n}f\right|}{{\left(n+1\right)}^{1/p-1}{\log }^{\left[p\right]}\left(n+1\right)},0<p\le 1$$

$$\overline{\underset{n\to \infty }{\lim }}\frac{{\left(n+1\right)}^{1/p-1}{\log }^{\left[p\right]}\left(n+1\right)}{\phi \left(n\right)}=+\infty ,$$

$$\underset{n\in \mathbb{N}}{\sup }{\Vert \frac{S_{n}f}{\phi \left(n\right)}\Vert }_{L_{p,\infty }\left(G_m\right)}=\infty ,\text{ for }0<p<1,$$

$$\underset{n\in \mathbb{N}}{\sup }{\Vert \frac{S_{n}f}{\phi \left(n\right)}\Vert }_1=\infty .$$

$${\Vert S_{n_k}f\Vert }_1\le c{\Vert f\Vert }_1$$

$$\underset{k\in \mathbb{N}}{\sup }{\Vert D_{n_k}\Vert }_1<c<\infty ,$$

$$S^{\#}f:=\underset{n\in \mathbb{N}}{\sup }\left|S_{M_n}f\right|$$

$$Z_{m_k}:=\{0,1,\dots ,m_k-1\}$$

$${\mu }_k\left(\{j\}\right):=1/m_k(j\in Z_{m_k})$$

$$x:=(x_0,x_1,\dots ,x_k,\dots )\left(x_k\in Z_{m_k}\right).$$

$$I_0\left(x\right):=G_m,$$

$$I_n(x):=\{y\in G_m\mid y_0=x_0,\dots ,y_{n-1}=x_{n-1}\}(x\in G_m,n\in \mathbb{N}).$$

$$\overline{I_N}=\stackrel{N-1}{\bigcup _{s=0}}{I}_s\backslash I_{s+1}.$$

$$M_0:=1,M_{k+1}:=m_k{M}_k(k\in \mathbb{N}),$$

$$n=\sum _{j=0}^{\infty }{n}_j{M}_j,\text{ where }{n}_j\in Z_{m_j}(j\in \mathbb{N})$$

$$⟨n⟩:=\min \{j\in \mathbb{N}:n_j\ne 0\}\text{ and }\left|n\right|:=\max \{j\in \mathbb{N}:n_j\ne 0\},$$

$$\rho \left(n\right):=\left|n\right|-⟨n⟩,\text{ for all }n\in \mathbb{N}.$$

$${\Vert f\Vert }_p:={\left({\int }_{G_m}{\left|f\right|}^{p}d\mu \right)}^{1/p}\left(0<p<\infty \right).$$

$${\Vert f\Vert }_{L_p,\infty }:=\underset{\lambda >0}{\sup }\lambda \mu {\left(f>\lambda \right)}^{1/p}<+\infty .$$

$$r_k\left(x\right):=\exp \left(2\pi ıx_k/m_k\right)\left({ı}^2=-1,x\in G_m,k\in \mathbb{N}\right).$$

$${\psi }_n(x):=\prod _{k=0}^{\infty }{r}_k^{n_k}\left(x\right)\left(n\in \mathbb{N}\right).$$

$$\hat{f}\left(k\right)$$

$$S_{n}f$$

$$D_n$$

$$D_{M_{n}}\left(x\right)=\left\{\begin{array}[]{ll}M_{n},&\text{ if \ }x\in I_{n},\\ 0,&\text{ if \ }x\notin I_{n},\end{array}\right.$$

$$D_n\left(x\right)={\psi }_n(x)\left(\underset{j=0}{\sum ^{\infty }}{D}_{M_j}\left(x\right)\stackrel{m_j-1}{\sum _{u=m_j-n_j}}{r}_j^u\left(x\right)\right).$$

$$f^{\ast }=\underset{n\in \mathbb{N}}{\sup }\left|f^{\left(n\right)}\right|.$$

$$f^{\ast }\left(x\right)=\underset{n\in \mathbb{N}}{\sup }\frac{1}{\mu \left(I_n\left(x\right)\right)}\left|{\int }_{I_n\left(x\right)}f\left(u\right)d\mu \left(u\right)\right|$$

$${\Vert f\Vert }_{H_p}:={\Vert f^{\ast }\Vert }_p<\infty .$$

Full text

Research was supported by project TÁMOP-4.2.2.A-11/1/KONV-2012-0051, by Shota Rustaveli National Science Foundation grants DI/9/5-100/13, DO/24/5-100/14, YS15-2.1.1-47 and by a Swedish Institute scholarship no. 10374-2015.

2010 Mathematics Subject Classification. 42C10, 42B25.

Key words and phrases: Vilenkin system, partial sums, maximal operator, Vilenkin-Fourier series, martingale Hardy space.

1. Introduction

Pointwise convergence problems are of fundamental importance in Fourier analysis, and as it is well known they are closely related to studying boundedness of associated maximal operators. In the present paper we will deal with maximal operators. Let us first recall in brief a historical development of this theory.

It is well-known (for details see e.g. [2] and [11]) that Vilenkin systems do not form bases in the space $L_{1}\left(G_{m}\right).$ Moreover, (for details see e.g. [18, 19]) there is a function in the martingale Hardy space $H_{1}\left(G_{m}\right),$ such that the partial sums of $f$ are not bounded in $L_{1}\left(G_{m}\right)$-norm, but Watari [17] (see also Gosselin [10] and Young [20]) proved that there exist absolute constants $c$ and $c_{p}$ such that, for $n=1,2,...,$

[TABLE]

In [14] it was proved that there exists a martingale $f\in H_{p}\left(G_{m}\right)\left(0<p<1\right),$ such that

[TABLE]

The reason of divergence of $S_{M_{n}+1}f$ is that the Fourier coefficients of $f\in H_{p}\left(G_{m}\right)$ ($0<p<1$) are not bounded (see Tephnadze [13]).

Uniform and point-wise convergence and some approximation properties of the partial sums in $L_{1}\left(G_{m}\right)$ norms were investigated by Goginava [8, 9] and Avdispahić, Memić [1]. Some related results can also be found in the recent PhD thesis by Tephnadze [15]. Moreover, Fine [4] obtained sufficient condition for the uniform convergence it is in complete analogy with the Dini-Lipschitz condition. Guličev [12] estimated the rate of uniform convergence of a Walsh-Fourier series using Lebesgue constants and modulus of continuity. Uniform convergence of subsequences of partial sums was also investigated by Goginava and Tkebuchava [7]. This problem was considered for the Vilenkin group $G_{m}$ by Fridli [5], Blahota [3] and Gát [6].

In [14] the following maximal operator was considered:

[TABLE]

(where $\left[x\right]$ denotes integer part of $x$). It was proved that the maximal operator $\widetilde{S}_{p}^{\ast}$ is bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the space $L_{p}\left(G_{m}\right).$ Moreover, if $0<p\leq 1$ and $\varphi:\mathbb{N}_{+}\rightarrow[1,\infty)$ is a non-decreasing function satisfying the condition

[TABLE]

then

[TABLE]

and

[TABLE]

It is also known (for details see e.g. Weisz [19]) that

[TABLE]

holds if and only if

[TABLE]

where $D_{n_{k}}$ denotes the $n_{k}$th Dirichlet kernel with respect to Vilenkin system. Moreover, the corresponding subsequence $S_{n_{k}}$ of the partial sums $S_{{n}}$ are bounded from the Hardy space $H_{p}\left(G_{m}\right)$ to the Hardy space $H_{p}\left(G_{m}\right),$ for all $p>0.$

It is also well-known (for details see e.g. Weisz [19] and Tephnadze [15]) that the following restricted maximal operator

[TABLE]

is bounded from the martingale Hardy space $H_{p}\left(G_{m}\right)$ to the Lebesgue space $L_{p}\left(G_{m}\right),$ for all $p>0$.

In this paper we find the maximal subspace of positive numbers, for which the restricted maximal operator of partial sums with respect to Vilenkin systems in this subspace is bounded from the Hardy space $H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ As applications, both some well-known and new results are pointed out.

The paper is organized as follows: Some Preliminaries (definitions, notations and basic facts) are presented in Section 2. The main result (Theorem 1) and some of its consequences (Corollaries 1-5) are presented and discussed in Section 3. Theorem 1 is proved in Section 5. For this proof we need some Lemmas, one of them is new and of independent interest (see Section 4).

2. Preliminaries

Let $\mathbb{N}_{+}$ denote the set of the positive integers, $\mathbb{N}:=\mathbb{N}_{+}\cup\{0\}$ and assume that $m:=(m_{0},m_{1},\ldots)$ is a sequence of positive integers not less than 2.

Denote by

[TABLE]

the additive group of integers modulo $m_{k}.$

Define the group $G_{m}$ as the complete direct product of the group $Z_{m_{k}}$ with the product of the discrete topologies of $Z_{m_{k}}$‘s.

The product measure $\mu$ of the measures

[TABLE]

is a Haar measure on $G_{m}$ with $\mu\left(G_{m}\right)=1$.

If the sequence $m:=(m_{0},m_{1},\ldots)$ is bounded, then $G_{m}$ is called a bounded Vilenkin group, otherwise it is called an unbounded one. In the present paper we deal only with bounded Vilenkin groups.

The elements of $G_{m}$ are represented by sequences

[TABLE]

A base for the neighbourhood of $G_{m}$ can be given as follows:

[TABLE]

[TABLE]

Denote $I_{n}:=I_{n}\left(0\right)$ for $n\in\mathbb{N}$ and $\overline{I_{n}}:=G_{m}\backslash I_{n}$.

It is evident that

[TABLE]

The generalized number system based on $m$ is defined in the following way

[TABLE]

Every $n\in\mathbb{N}$ can be uniquely expressed as

[TABLE]

and only a finite number of $n_{j}$‘s differ from zero.

Let

[TABLE]

that is $M_{\left|n\right|}\leq n\leq M_{\left|n\right|+1}.$ Set

[TABLE]

The norm (or quasi-norm) of the space $L_{p}(G_{m})$ is defined by

[TABLE]

The space $L_{p,\infty}\left(G_{m}\right)$ consists of all measurable functions $f$ for which

[TABLE]

Next, we introduce on $G_{m}$ an orthonormal system which is called Vilenkin system.

First, we define the complex valued function $r_{k}\left(x\right)\colon G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher functions as

[TABLE]

Let us define the Vilenkin system $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$ as:

[TABLE]

Specifically, we call this system the Walsh-Paley one if $m\equiv 2$.

The Vilenkin system is orthonormal and complete in $L_{2}\left(G_{m}\right)$ (see e.g. [2, 16]).

Now, we present the usual definitions in Fourier analysis.

If $f\in L_{1}\left(G_{m}\right)$ we can establish Fourier coefficients, the partial sums of Fourier series, Dirichlet kernels with respect to the Vilenkin system in the usual manner:

[TABLE]

Recall that (see [2])

[TABLE]

and

[TABLE]

The $\sigma$-algebra generated by the intervals $\left\{I_{n}\left(x\right):x\in G_{m}\right\}$ will be denoted by $\digamma_{n}$ $\left(n\in\mathbb{N}\right)$. Let us denote a martingale with respect to $\digamma_{n}\left(n\in\mathbb{N}\right)$ by $f=\left(f_{n}:n\in\mathbb{N}\right)$ (for details see e.g. [18]). The maximal function of a martingale $f$ is defined by

[TABLE]

In the case $f\in L_{1}\left(G_{m}\right),$ the maximal function is also given by

[TABLE]

For $0<p<\infty$ the Hardy martingale spaces $H_{p}\left(G_{m}\right)$ consist of all martingales, for which

[TABLE]

A bounded measurable function $a$ is a $p$-atom, if there exists an interval $I$, such that

[TABLE]

The Hardy martingale spaces $H_{p}\left(G_{m}\right)$ have an atomic characterization for $0<p\leq 1$. In fact the following theorem is true (see e.g. Weisz [18, 19]):

Theorem W**.**

A martingale $f=\left(f_{n}:n\in\mathbb{N}\right)\in H_{p}\left(G_{m}\right)$ $\left(0<p\leq 1\right)$ if and only if there exists a sequence $\left(a_{k}:\ k\in\mathbb{N}\right)$ of $p$-atoms and a sequence $\left(\mu_{k}:\ k\in\mathbb{N}\right)$ of real numbers, such that for every $n\in\mathbb{N},$

[TABLE]

and

[TABLE]

Moreover,

[TABLE]

where the infimum is taken over all decomposition of $f$ of the form (4).

If $f\in L_{1}\left(G_{m}\right),$ then it is easily shown that the sequence $\left(S_{M_{n}}f:n\in\mathbb{N}\right)$ is a martingale.

If $f=\left(f_{n},n\in\mathbb{N}\right)$ is a martingale, then Vilenkin-Fourier coefficients are defined in a slightly different manner:

[TABLE]

Vilenkin-Fourier coefficients of $f\in L_{1}\left(G_{m}\right)$ are the same as the martingale $\left(S_{M_{n}}f:n\in\mathbb{N}\right)$ obtained from $f$.

3. The Main Result

Our main theorem reads as follows:

Theorem 1**.**

a) Let $0<p\leq 1$ and $\left\{\alpha_{k}:k\in\mathbb{N}\right\}$ be a subsequence of positive natural numbers, such that

[TABLE]

Then the maximal operator

[TABLE]

is bounded from the Hardy space $H_{p}$ to the Lebesgue space $L_{p}.$

b) Let $0<p<1$ and $\left\{\alpha_{k}:k\in\mathbb{N}\right\}$ be a subsequence of positive natural numbers satisfying the condition

[TABLE]

Then there exists a martingale $f\in H_{p}$ such that

[TABLE]

Remark 1**.**

Since $L_{p}\subset L_{p,\infty}$ part b) means in particular that the statement in part a) is sharp in a special sense for the case $0<p<1.$

We also mention the following well-known consequences (for details see e.g. the books [18, 19] and [14]):

Corollary 1** (Tephnadze [14]).**

Let $0<p\leq 1$ and $f\in H_{p}$. Then the maximal operator

[TABLE]

is not bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

In fact, we only have to notice that

[TABLE]

The second part of Theorem 1 implies our Corollary.

Corollary 2**.**

Let $p>0$ and $f\in H_{p}$. Then the maximal operator

[TABLE]

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

We notice that

[TABLE]

Thus, the second part of Theorem 1 gives again Corollary 2.

Corollary 3**.**

Let $p>0$ and $f\in H_{p}$. Then the maximal operator

[TABLE]

is bounded from the Hardy space $H_{p}$ to the space $L_{p}.$

We find that $\left|M_{n}\right|=\left\langle M_{n}\right\rangle=n,$ $\rho\left(M_{n}\right)=0$. Using part b) of Theorem 1, we immediately get Corollary 3.

Since $S_{n}P=P$ for every $P\in\mathcal{P}$, where $\mathcal{P}$ is the set of all Vilenkin polynomials. The set $\mathcal{P}$ is dense in the space $L_{1}(G_{m})$. Combining Lemma 1 and part a) of Theorem 1, we obtain that under condition (5) the restricted maximal operator of partial sums is bounded from the space $L_{1}(G_{m})$ to the space $weak-L_{1}(G_{m})$ It follows that

Corollary 4**.**

Let $f\in L_{1}$ and $\left\{\alpha_{k}:k\in\mathbb{N}\right\}$ be a subsequence of positive natural numbers, satisfying condition (5). Then

[TABLE]

Corollary 5**.**

Let $f\in L_{1}$. Then

[TABLE]

4. Lemmas

First, we note the following well-known result, which was proved in Weisz [18, 19]:

Lemma 1**.**

Suppose that an operator $T$ is sub-linear and, for some $0<p\leq 1$

[TABLE]

for every $p$-atom $a$, where $I$ denotes the support of the atom $a$. If $T$ is bounded from $L_{\infty}$ to $L_{\infty},$ then

[TABLE]

Moreover, if $p<1,$ then we have weak (1,1) type estimate, i.e. it holds that

[TABLE]

for all $f\in L_{1}.$

The next Lemma can be found in Tephnadze [13]:

Lemma 2**.**

Let $n\in\mathbb{N}$ and $x\in I_{s}\backslash I_{s+1},$ $0\leq s\leq N-1$. Then

[TABLE]

We also need the following estimate of independent interest:

Lemma 3**.**

Let $n\in\mathbb{N}$, $\left|n\right|\neq\left\langle n\right\rangle$ and $x\in I_{\left\langle n\right\rangle+1}\left(e_{\left\langle n\right\rangle}\right)$ where $e_{k}:=(0,\dots,0,1,0,\dots)\in G_{m}$ (only the $k$-th coordinate is one, the others are zero). Then

[TABLE]

Proof.

Let $x\in I_{\left\langle n\right\rangle+1}\left(e_{\left\langle n\right\rangle}\right)$. Since

[TABLE]

and

[TABLE]

Applying (2) and (3) we can conclude that

[TABLE]

Let $x_{n}=1.$ Then we readily get for $s_{n}<m_{n}$ that

[TABLE]

It follows that

[TABLE]

Moreover, by using the same arguments as above it is easily seen that

[TABLE]

The proof is complete. ∎

5. Proof of the main theorem

Proof of Theorem 1.

First, we prove part a). Combining (2) and (3) we easily conclude that if condition (5) holds, then

[TABLE]

It follows that $\widetilde{S}^{\ast,\vartriangle}$ is bounded from $L_{\infty}$ to $L_{\infty}.$ By Lemma 1 we obtain that the proof of part a) will be complete if we show that

[TABLE]

for every $p$-atom $a$ with support $I=I_{N}$. Since $S_{\alpha_{k}}\left(a\right)=0$ when $\alpha_{k}\leq M_{N},$ we can suppose that $\alpha_{k}>M_{N}.$ (That is, $|\alpha_{k}|\geq N$.)

Let $t\in I_{N}$ and $x\in I_{s}\backslash I_{s+1},$ $1\leq s\leq N-1.$ If $\left\langle\alpha_{k}\right\rangle\geq N,$ we get that $s<N\leq\left\langle\alpha_{k}\right\rangle$ and since $x-t\in I_{s}\backslash I_{s+1},$ by combining (2) and (3) we obtain that

[TABLE]

Analogously, by combining again (2) and (3) we can conlude that (7) holds, for $s<\left\langle\alpha_{k}\right\rangle\leq N-1$.

It follows that

[TABLE]

Let $0<p\leq 1$, $t\in I_{N}$ and $x\in I_{s}\backslash I_{s+1}$, $\left\langle\alpha_{k}\right\rangle\leq s\leq N-1.$ Applying the fact that $\left\|a\right\|_{\infty}\leq M_{N}^{1/p}$ and Lemma 2 we find that

[TABLE]

Let us set $\varrho:=\min_{k\in\mathbb{N}}\left\langle\alpha_{k}\right\rangle.$ Then, in view of (8) and (5) we can conclude that

[TABLE]

and

[TABLE]

By the definition of $\varrho$ there exists at least one index $k_{{0}}\in\mathbb{N}_{+}$ such that $\varrho=\left\langle\alpha_{k_{{0}}}\right\rangle.$ By using contition (5) we can conclude that

[TABLE]

Let us set $m_{*}:=\sup_{k}m_{k}$.

Let $0<p<1.$ According to (1) and using (9), (10) and (5) we obtain that

[TABLE]

Let $p=1.$ We combine (9)-(5) and invoke identity (1) to obtain that

[TABLE]

The proof of part a) is complete.

Now, we prove the second part of our theorem. Since,

[TABLE]

under condition (6), there exists an increasing subsequence $\left\{n_{k}:\ k\in\mathbb{N}\right\}\subset\left\{\alpha_{k}:\ k\in\mathbb{N}\right\}$ such that $n_{0}\geq 3$ and

[TABLE]

and

[TABLE]

Let $f=\left(f_{n}:n\in\mathbb{N}\right)$ be a martingale defined by

[TABLE]

where

[TABLE]

and

[TABLE]

It is easily seen that $a$ is a $p$-atom. Under condition (12) we can conclude that $f\in H_{p}$. (Theorem W immediately yields that $\|f\|_{H_{p}}\leq c_{p}<\infty$.)

According to (13) we readily see that

[TABLE]

Since, $M_{\left|n_{k}\right|}<n_{k}$, we get

[TABLE]

According to part a) of Theorem 1 for $I$ we have that

[TABLE]

Moreover, under condition (6) we can conclude that

[TABLE]

Let $x\in I_{\left\langle n_{k}\right\rangle+1}\left(e_{\left\langle n_{k}\right\rangle}\right).$ Applying Lemma 3 we obtain that

[TABLE]

Thus, we immediately have

[TABLE]

It follows that

[TABLE]

Hence, for large enough $k$,

[TABLE]

The proof is complete. ∎