Ph.D. Thesis-Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series

TL;DR

This thesis advances harmonic analysis by establishing new bounds for Vilenkin-Fourier coefficients, exploring convergence conditions, and developing Hardy inequalities for various summability methods within Vilenkin systems.

Contribution

It introduces novel estimations, boundedness results, and convergence criteria for Vilenkin-Fourier series, including new methods for Hardy inequalities and analysis of Nörlund and Riesz means.

Findings

New bounds for Vilenkin-Fourier coefficients

Necessary and sufficient conditions for convergence

Optimality of the derived inequalities

Abstract

In this PhD thesis we discuss, develop and apply this fascinating theory connected to modern harmonic analysis. In particular we make new estimations of Vilenkin-Fourier coefficients and prove some new results concerning boundedness of maximal operators of partial sums. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of the partial sums is valid and develop new methods to prove Hardy type inequalities for the partial sums with respect to the Vilenkin systems. We also do the similar investigation for the Fej\'er means. Furthermore, we investigate some N\"orlund means but only in the case when their coefficients are monotone. Some well-know examples of N\"orlund means are Fej\'er means, Ces\`aro means and N\"orlund logarithmic means. In addition, we consider Riesz logarithmic means, which are not example of N\"orlund 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.