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.

Let and w be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a Poisson A 2 condition, and satisfy the testing conditions below, for the Hilbert transform H , Z I H. 1 I / 2 dw .I /; Z I H.w1 I / 2 d w.I /;with constants independent of the choice of interval I . Then H. / maps L 2 . / to L 2 .w/, verifying a conjecture of Nazarov, Treil, and Volberg. The proof has two components, a global-to-local reduction, carried out in this article, and an analysis of the local problem, to be elaborated in a future Part II version of this article.

We consider the two weight problem for the Hilbert transform, namely the question of finding real-variable characterization of those pair of weights for which the Hilbert transform acts boundedly on L 2 of the weights. Such a characterization is known subject to certain side conditions. We give a new proof, simpler in many details, of the best such result. In addition, we analyze underlying assumptions in the proof, especially in terms of two alternate side conditions. A new characterization in the case of one doubling weight is given.2000 Mathematics Subject Classification. Primary: 42B20 Secondary: 42B25, 42B35.

For 1 < p < ∞, weight w ∈ A p and any L 2 -bounded Calderón-Zygmund

Let and w be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a Poisson A 2 condition, and satisfy the testing conditions below, for the Hilbert transform H , Z I H. 1 I / 2 dw .I /; Z I H.w1 I / 2 d w.I /; with constants independent of the choice of interval I . Then H. / maps L 2 . / to L 2 .w/, verifying a conjecture of Nazarov, Treil, and Volberg. The proof uses basic tools of nonhomogeneous analysis with two components particular to the Hilbert transform. The first component is a global-to-local reduction which is a consequence of prior work by Lacey, Sawyer, Shen, and Uriarte-Tuero. The second component, an analysis of the local part, is the particular contribution of this article.