Let σ and ω be locally finite positive Borel measures on R n (possibly having common point masses), and let T α be a standard α-fractional Calderón-Zygmund operator on R n with 0 ≤ α < n. Suppose that Ω : R n → R n is a globally biLipschitz map, and refer to the images ΩQ of cubes Q as quasicubes. Furthermore, assume as side conditions the A α 2 conditions, punctured A α 2 conditions, and certain α-energy conditions taken over quasicubes. Then we show that T α is bounded from L 2 (σ) to L 2 (ω) 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 (σ) to L 2 (ω), then the quasitesting conditions hold, and the quasiweak boundedness condition holds. If the vector of α-fractional Riesz transforms R α σ (or more generally a strongly elliptic vector of transforms) is bounded from L 2 (σ) to L 2 (ω), then both the A α 2 conditions and the punctured A α 2 conditions hold. We do not know if our quasienergy conditions are necessary when n ≥ 2, except for certain situations in which one of the measures is one-dimensional, or both measures are sufficiently dispersed.
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