Weighted norm inequalities for multisublinear maximal operator in martingale spaces

TL;DR

This paper characterizes weighted inequalities for a multisublinear maximal operator in martingale spaces, extending one-weight theory to a bilinear setting under certain conditions on the weights.

Contribution

It provides a characterization of weights ensuring boundedness of the multisublinear maximal operator in martingale spaces, including a partial bilinear one-weight theory.

Findings

Identifies conditions on weights for boundedness of the operator

Extends one-weight theory to bilinear case in martingale spaces

Provides criteria for weak and strong type inequalities

Abstract

Let $v,~\omega_1, ~\omega_2$ be weights and $1<p_1, ~p_2<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $(\omega_1, \omega_2)\in RH(p_1, p_2).$ For the multisublinear maximal operator $\mathfrak{M}$ in martingale spaces, we characterize the weights for which $\mathfrak{M}$ is bounded from $L^{p_1}(\omega_1)\times L^{p_2}(\omega_2)$ to $L^{p,\infty}(v)\hbox{or}L^p(v).$ If $v=\omega_2^{\frac{p}{p_2}}\omega_2^{\frac{p}{p_2}},$ we partially give the bilinear version of one-weight theory.