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

Abstract

We consider boundedness of a certain positive dyadic operator $$T^{\sigma }:L^p(\sigma ;{\ell }^2)\to L^p(\omega ),$$ 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.

18 pages. v2: Incorporates referees comments. Added more discussion concerning the conjecture about the two-weight inequality of the Hilbert transform between $L^p$ spaces

References in corpus (2)

• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II
• Two-weight $L^p$-$L^q$ bounds for positive dyadic operators: unified approach to $p\le q$ and $p>q$