We prove the Kato conjecture for elliptic operators on R n. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = −div (A∇) with bounded measurable coefficients in R n is the Sobolev space H 1 (R n) in any dimension with the estimate √ Lf 2 ∼ ∇f 2 .

For the bilinear Hilbert transform given by:

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Abstract. We give a simplified proof that the Carleson operator is of weak type (2, 2). This estimate is the main ingredient in the proof of Carleson's theorem on almost everywhere convergence of Fourier series of functions in L 2 ([0, 1]).

Abstract. A martingale transform T , applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp A p bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L 1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A 2 bounds in that setting.

Abstract. Let R be the vector of Riesz transforms on R n , and let µ, λ ∈ A p be two weights on R n , 1 < p < ∞. The two-weight norm inequality for the commutatoris shown to be equivalent to the function b being in a BM O space adapted to µ and λ. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both λ and µ being Lebesgue measure, and Bloom in the case of dimension one.

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.

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