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Limits of sequence of Tensor Product Operators associated with the Walsh-Paley system

Ushangi Goginava, Farrukh Mukhamedov

Abstract

It is well-known that to establish the almost everywhere convergence of a sequence of operators on $L_1$-space, it is sufficient to obtain a weak $(1,1)$-type inequality for the maximal operator corresponding to the sequence of operators. However, in practical applications, the establishment of the mentioned inequality for the maximal operators is very tricky and difficult job. In the present paper, the main aim is a novel outlook at above mentioned inequality for the tensor product of two weighted one-dimensional Walsh-Fourier series. Namely, our main idea is naturally to consider uniformly boundedness conditions for the sequences which imply the weak type estimation for the maximal operator of the tensor product. More precisely, we are going to establish weak type of inequality for the tensor product of two dimensional maximal operators while having the uniform boundedness of the corresponding one dimensional operators. Moreover, the convergence of the tensor product of operators is established at the two dimensional Walsh-Lebesgue points.

Limits of sequence of Tensor Product Operators associated with the Walsh-Paley system

Abstract

It is well-known that to establish the almost everywhere convergence of a sequence of operators on -space, it is sufficient to obtain a weak -type inequality for the maximal operator corresponding to the sequence of operators. However, in practical applications, the establishment of the mentioned inequality for the maximal operators is very tricky and difficult job. In the present paper, the main aim is a novel outlook at above mentioned inequality for the tensor product of two weighted one-dimensional Walsh-Fourier series. Namely, our main idea is naturally to consider uniformly boundedness conditions for the sequences which imply the weak type estimation for the maximal operator of the tensor product. More precisely, we are going to establish weak type of inequality for the tensor product of two dimensional maximal operators while having the uniform boundedness of the corresponding one dimensional operators. Moreover, the convergence of the tensor product of operators is established at the two dimensional Walsh-Lebesgue points.

Paper Structure

This paper contains 7 sections, 4 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Background history of one-dimensional Fourier series
  3. Background history of two-dimensional Fourier series
  4. Lebesgue points and their application in the pointwise convergence of sequences of operators
  5. Proof of Theorem \ref{['mt1']}
  6. Proof of Theorem \ref{['mt2']}
  7. Walsh System

Key Result

Theorem 1

Let $\mathbb{T}_{i}$, $(i=0,1)$ be matrices of transformation, and $\left\{ n_{a}^{i}:\ a\in \mathbb{N}\right\}$ be subsequences of natural numbers. Let $\mathcal{T}_{n_{a}^{0}}^{\mathbb{T}_{0}}$ and $\mathcal{T}_{n_{a}^{1}}^{\mathbb{T}_{1}}$ be the corresponding sequences of operators on $L_{1}\lef If then

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Remark 5
  • Example 1
  • Remark 6
  • Definition 1
  • Theorem 2
  • ...and 6 more