Two-weight estimates for the square function and $t$-Haar multipliers

TL;DR

This paper characterizes when dyadic weighted square functions and $t$-Haar multipliers are bounded between weighted $L^2$ spaces, providing necessary and sufficient conditions based on weight triples.

Contribution

It introduces a complete characterization of weight conditions for the boundedness of dyadic square functions and $t$-Haar multipliers, advancing understanding of weighted inequalities.

Findings

Derived necessary and sufficient conditions for boundedness of the dyadic weighted square function.

Established criteria for the boundedness of $t$-Haar multipliers based on the square function.

Provided a unified framework linking square functions and multipliers in weighted $L^2$ spaces.

Abstract

We present necessary and sufficient conditions on triples of weights $(u,v,w)$ for the boundedness of the dyadic weighted square function $S_w$ from $L^2(u)$ into $L^2(v)$. We use this characterization to obtain necessary and sufficient conditions for the boundedness of the $t$-Haar multipliers from $L^2(u)$ into $L^2(v)$ in terms of boundedness of the dyadic weighted square function.