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

This paper is a sequel to our paper Sawyer et al. (Revista Mat Iberoam 32(1):79-174, 2016). Let sigma and omega be locally finite positive Borel measures on R-n (possibly having common point masses), and let T-alpha be a standard alpha-fractional Calderon-Zygmund operator on R-n with 0 <= alpha < n. Suppose that Omega : R-n -> 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 A(2)(alpha) conditions, punctured A(2)(alpha) conditions, and certain alpha-energy conditions taken over quasicubes. Then we show that T-alpha is bounded from L-2 (sigma) to L-2 (omega) if the quasicube testing conditions hold for T-alpha and its dual, and if the quasiweak boundedness property holds for T-alpha. Conversely, if T-alpha is bounded from L-2 (sigma) to L-2(omega), then the quasitesting conditions hold, and the quasiweak boundedness condition holds. If the vector of alpha-fractional Riesz transforms R-sigma(alpha) (or more generally a strongly elliptic vector of transforms) is bounded from L-2 (sigma) to L-2 (omega), then both the A(2)(alpha) conditions and the punctured A(2)(alpha) 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 R to 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 R to C+, arXiv:1310.4820v1, versions 2 and 3).