Two weight norm inequalities for the $g$ function

Abstract

Given two weights $\sigma ,w$ on ${\mathbb{R}}^n$, the classical $g$-function satisfies the norm inequality $\Vert g(f\sigma ){\Vert }_{L^2(w)}\lesssim \Vert f{\Vert }_{L^2(\sigma )}$ 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 (σ\mathbf 1_I)(x, t))^2 \; dw \, t dt \lesssim σ(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.

15 pages. Reflects the report from referee

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