Bounded operators on Martingale Hardy spaces

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 Fejér means is valid. Furthermore, we consider Riesz and Nörlund 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 Nörlund, but only in the case when their coefficients are monotone.

Georgian PhD thesis. arXiv admin note: substantial text overlap with arXiv:1803.00627, arXiv:1503.05396 by other authors