Bounded operators on Martingale Hardy spaces

TL;DR

This thesis explores advanced harmonic analysis on martingale Hardy spaces, focusing on convergence of Fourier series, summability methods, and optimality of results with applications to classical and novel theorems.

Contribution

It develops new convergence criteria and optimality results for various summability methods of Vilenkin-Fourier series on martingale Hardy spaces.

Findings

Strong convergence results for partial sums of Vilenkin-Fourier series.

Necessary and sufficient conditions for norm convergence of subsequences of Fejér means.

Optimality of the derived convergence and summability results.

Abstract

The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fej\'er means is valid. Furthermore, we consider Riesz and N\"orlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are "inverse" summability methods of N\"orlund, but only in the case when their coefficients are monotone.