L^{p}(\mu)-L^{q}(\nu) characterization for well localized operators

TL;DR

This paper characterizes two-weight inequalities for well localized operators between L^p and L^q spaces using a new square function testing condition, extending previous results beyond the case p=q=2.

Contribution

It introduces a novel square function testing condition that successfully characterizes two-weight inequalities for well localized operators when p and q are not necessarily equal to 2.

Findings

The new testing condition characterizes the two-weight inequality.

Counterexamples show failure of direct analogues for p=q≠2.

Application to positive dyadic operators demonstrates the condition's effectiveness.

Abstract

We consider a two weight $L^{p}(\mu) \to L^{q}(\nu)$-inequality for well localized operators as defined and studied by F. Nazarov, S. Treil and A. Volberg when $p=q=2$. A counterexample of F. Nazarov shows that the direct analogue of these results fails for for $p=q\not=2$. Here a new square function testing condition is introduced and applied to characterize the two weight norm inequality. The use of the square function testing condition is also demonstrated in connection with certain positive dyadic operators.