Mixed martingale Hardy spaces

TL;DR

This paper introduces and analyzes five types of mixed martingale Hardy spaces using mixed $L_{p}$-norms, establishing fundamental properties and inequalities including atomic decompositions and a generalized Burkholder-Davis-Gundy inequality.

Contribution

It defines new mixed martingale Hardy spaces and proves key properties and inequalities, extending classical results to the mixed-norm setting.

Findings

Atomic decompositions for the spaces

Doob's inequality in mixed-norm context

Generalized Burkholder-Davis-Gundy inequality

Abstract

In this paper we consider the martingale Hardy spaces defined with the help of the mixed $L_{\pv}$-norm. Five mixed martingale Hardy spaces will be investigated: $H_{\pv}^{s}$, $H_{\pv}^S$, $H_{\pv}^M$, $\cP_{\pv}$ and $\cQ_{\pv}$. Several results are proved for these spaces, like atomic decompositions, Doob's inequality, boundedness, martingale inequalities and the generalization of the well-known Burkholder-Davis-Gundy inequality.