Mathematical Research Letters

Volume 21 (2014)

Number 3

Two weight norm inequalities for the $g$ function

Pages: 521 – 536



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)


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.


two weight inequalities; square functions

2010 Mathematics Subject Classification

42B20, 42B25

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