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

TL;DR

This paper investigates the limits of tensor product operators related to the Walsh-Paley system, proposing new conditions for weak type inequalities and convergence at Walsh-Lebesgue points in a two-dimensional setting.

Contribution

It introduces a novel approach to establish weak type inequalities for tensor product operators using uniform boundedness conditions, enhancing understanding of their convergence behavior.

Findings

Established weak type inequalities for tensor product maximal operators under uniform boundedness

Proved convergence of tensor product operators at Walsh-Lebesgue points in two dimensions

Provided a new perspective on the limits of Walsh-Paley tensor product operators

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.