A Two-weight inequality between $L^p(\ell^2)$ and $L^p$

TL;DR

This paper develops a two-weight inequality for a positive dyadic operator between vector-valued and scalar Lebesgue spaces, providing a characterization for boundedness in certain p ranges and introducing a new Carleson embedding theorem.

Contribution

It introduces a characterization of boundedness for a dyadic operator in two-weight settings and proves a new Carleson-type embedding theorem.

Findings

Boundedness characterized by testing conditions for p in [2, ∞)

Development of a new Carleson-type embedding theorem

Extension of two-weight theory to vector-valued operators

Abstract

We consider boundedness of a certain positive dyadic operator $$ T^\sigma \colon L^p(\sigma; \ \! \ell^2) \to L^p(\omega), $$ that arose during our attempts to develop a two-weight theory for the Hilbert transform in $L^p$. Boundedness of $T^\sigma$ is characterized when $p \in [2, \infty)$ in terms of certain testing conditions. This requires a new Carleson-type embedding theorem that is also proved.