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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 Calderon-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 A2 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 .

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