Martingale Inequalities in Variable Exponent Hardy spaces with $0<p^-\leq p^+<\infty$

TL;DR

This paper explores martingale inequalities within variable exponent Hardy spaces, establishing atomic decompositions, embedding relations, and extending results to a broad range of exponents, advancing the understanding of variable Lebesgue spaces.

Contribution

It introduces atomic decompositions for variable exponent martingale Hardy spaces and extends embedding relations to the full range of exponents from zero to infinity.

Findings

Atomic decompositions for variable exponent martingale Hardy spaces.

Embedding relations between these spaces for small exponents.

Extension of results to all exponents with $0<p^- ext{ and } p^+<\infty$.

Abstract

We investigate the properties of the variable Lebesgue spaces with quasi-norm on a probability space, and give the atomic decompositions suited to the variable exponent martingale Hardy spaces. Using the decompositions and the harmonic mean of a variable exponent, we obtain several continuous embedding relations between martingale Hardy spaces with small exponent. Finally, we extend these results to the cases $0<p^-\leq p^+<\infty.$