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We are going to give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform.1991 Mathematics Subject Classification. 42B20, 42A50, 47B35.

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We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderón-Zygmund operator on L 2 (µ). We do not assume any kind of doubling condition on the measure µ, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L ∞ . Thus we answer positively a question of Christ as to whether the L ∞ -assumption can be replaced by a BMO assumption.We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry. In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calderón-Zygmund operators with respect to very bad measures.

Abstract. In this paper we are proving that Sawyer type condition for boundedness work for individual Haar multipliers, as well as for the Haar shift and other "well localized" operators. IntroductionThe main question of this paper is two weight estimates for singular integral operators, i. e. the questions when an integral operator T is bounded operator from a weighted spaceOne of the most interesting cases is the case when T is the Hilbert trans-s−t dt. In this case necessary and sufficient conditions are not known, and the problem seems to be very hard. The reader can see some approaches to the problem and the partial results in [NTV3]. This paper also contains the conjecture, which, roughly speaking, says that a certain list of three simple conditions must be sufficient (and of course they are necessary) for two-weight boundedness of the Hilbert transform. In paper [NV] we showed that dropping any of the three conditions and having just two of them is not sufficient.In this paper we deal with dyadic analogues of the Hilbert transform, the so-called Haar multipliers and their generalizations. It turns out that for such operators it is possible to find necessary and sufficient condition for two weight estimates.Let us introduce the main object. The standard dyadic grid D = D 0 is the collection of all dyadic intervals [2 k · j, 2 k · (j + 1)), k, j ∈ Z. A general dyadic grid is an object obtained from D 0 by a dilation and a shift.For an interval I ⊂ R we define the (L 2 -normalized) Haar function h I by h I := |I| −1/2 (χ I + − χ I − ); here |I| stands for the length of the interval I; I + and I − denote its right and left halves respectively, and χ E denotes the characteristic function (indicator) of the set E.Given a sequence α = {α I } I∈D define the Haar multiplier T α by