Strong convergence of two--dimensional Vilenkin-Fourier series

TL;DR

This paper proves that specific means of quadratical partial sums of two-dimensional Vilenkin-Fourier series are uniformly bounded from Hardy space to Lebesgue space for 0<p≤1, establishing optimality of the sequence used.

Contribution

It demonstrates the boundedness of certain summation means of two-dimensional Vilenkin-Fourier series from Hardy space to Lebesgue space and confirms the optimality of the sequence involved.

Findings

Boundedness of means from Hardy space to Lp for 0<p≤1

Optimality of the denominator sequence in the means

Extension of results to two-dimensional Vilenkin-Fourier series

Abstract

We prove that certain means of the quadratical partial sums of the two-dimensional Vilenkin-Fourier series are uniformly bounded operators from the Hardy space $H_{p}$ to the space $L_{p}$ for $0<p\leq 1.$ We also prove that the sequence in the denominator cannot be improved.