Mathematical Research Letters

Volume 21 (2014)

Number 3

Two weight norm inequalities for the $g$ function

Pages: 521 – 536

DOI: https://dx.doi.org/10.4310/MRL.2014.v21.n3.a9

Authors

Michael T. Lacey (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

Kangwei Li (School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China)

Abstract

Given two weights $\sigma , w$ on $\mathbb{R}^n$, the classical $g$-function satisfies the norm inequality ${\lVert g(f \sigma) \rVert}_{L^2(w)} \lesssim {\lVert f \rVert}_{L^2(\sigma)}$ if and only if the two weight Muckenhoupt $A_2$ condition holds, and a family of testing conditions holds, namely\begin{equation*}\iint_{Q(I)} (\nabla P_t (\sigma \mathbf{1}_I) (x, t))^2 \; dw \: tdt \lesssim \sigma (I)\end{equation*}uniformly over all cubes $I \subset \mathbb{R} ^n$, and $Q (I)$ is the Carleson box over $I$. A corresponding characterization for the intrinsic square function of Wilson also holds.

Keywords

two weight inequalities, square functions

2010 Mathematics Subject Classification

42B20, 42B25

Accepted 24 February 2014

Published 13 October 2014