Convergence in Measure of Strong logarithmic means of double Fourier series

TL;DR

This paper investigates the convergence properties of strong logarithmic means of double Fourier series within certain function spaces, identifying the maximal Orlicz space where convergence in measure occurs.

Contribution

It determines the maximal Orlicz space for which Nörlund strong logarithmic means of double Fourier series converge in measure for functions in that space.

Findings

Identifies the maximal Orlicz space for convergence in measure.

Establishes convergence of strong logarithmic means in the specified space.

Provides conditions under which convergence in measure is guaranteed.

Abstract

N\"orlund strong logarithmic means of double Fourier series acting from space $% L\log L(\mathbb{T}^{2}) $ into space $L_{p}(\mathbb{T}% ^{2}), 0<p<1$ are studied. The maximal Orlicz space such that the N\"o% rlund strong logarithmic means of double Fourier series for the functions from this space converge in two-dimensional measure is found.