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This paper is a sequel to our paper Rev. Mat. Iberoam. 32 (2016), no. 1,79-174. Let T be a standard fractional Calderon Zygmund operator. Assumeappropriate Muckenhoupt and quasienergy side conditions. Then we show that T isbounded from one weighted space to another if the quasicube testing conditionshold for T and its dual, and if the quasiweak boundedness property holds for T.Conversely, if T is bounded, then the quasitesting conditions hold, and thequasiweak boundedness condition holds. If the vector of fractional Riesztransforms (or more generally a strongly elliptic vector of transforms) isbounded, then the appropriate Muckenhoupt conditions hold. We do not know ifour quasienergy conditions are necessary in higher dimensions, except forcertain situations in which one of the measures is one-dimensional as inarXiv:1310.4820 and arXiv:1505.07822v4, and for certain side conditions placedon the measures such as doubling and k-energy dispersed, which when k=n-1 issimilar to the condition of uniformly full dimension in Lacey and WickarXiv:1312.6163v3.

A two weight fractional singular integral theorem with side conditions, energy and k-energy dispersed