A Two Weight Fractional Singular Integral Theorem with Side Conditions, Energy and k-Energy Dispersed

Abstract

This paper is a sequel to our paper Sawyer et al. (Revista Mat Iberoam 32(1):79–174, 2016). Let σ and ω be locally finite positive Borel measures on $$\mathbb{R}^{n}$$ (possibly having common point masses), and let T α be a standard α-fractional Calderón-Zygmund operator on $$\mathbb{R}^{n}$$ with 0 ≤ α < n. Suppose that $$\Omega: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$$ is a globally biLipschitz map, and refer to the images $$\Omega Q$$ of cubes Q as quasicubes. Furthermore, assume as side conditions the $$\mathcal{A}_{2}^{\alpha }$$ conditions, punctured A 2 α conditions, and certain α -energy conditions taken over quasicubes. Then we show that T α is bounded from $$L^{2}\left (\sigma \right )$$ to $$L^{2}\left (\omega \right )$$ if the quasicube testing conditions hold for T α and its dual, and if the quasiweak boundedness property holds for T α . Conversely, if T α is bounded from $$L^{2}\left (\sigma \right )$$ to $$L^{2}\left (\omega \right )$$ , then the quasitesting conditions hold, and the quasiweak boundedness condition holds. If the vector of α-fractional Riesz transforms R σ α (or more generally a strongly elliptic vector of transforms) is bounded from $$L^{2}\left (\sigma \right )$$ to $$L^{2}\left (\omega \right )$$ , then both the $$\mathcal{A}_{2}^{\alpha }$$ conditions and the punctured A 2 α conditions hold. Our quasienergy conditions are not in general necessary for elliptic operators, but are known to hold for certain situations in which one of the measures is one-dimensional (Lacey et al., Two weight inequalities for the Cauchy transform from $$\mathbb{R}$$ to $$\mathbb{C}_{+}$$ , arXiv:1310.4820v4; Sawyer et al., The two weight T1 theorem for fractional Riesz transforms when one measure is supported on a curve, arXiv:1505.07822v4), and for certain side conditions placed on the measures such as doubling and k-energy dispersed, which when k = n − 1 is similar to the condition of uniformly full dimension in Lacey and Wick (Two weight inequalities for the Cauchy transform from $$\mathbb{R}$$ to $$\mathbb{C}_{+}$$ , arXiv:1310.4820v1, versions 2 and 3).

Locations

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