Bellman Functions and Two Weight Inequalities for Haar Multipliers

We are going to give necessary and suucient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give suucient conditions for two weight norm inequalities for the Hilbert transform. 0. Introduction Weighted norm inequalities for singular integral operators appear naturally in many areas of analysis, probability, operator theory ect. The one-weight case is now pretty well understood, and the answers are given by the famous Helson{Szegg o theorem and the Hunt{Muckenhoupt{Wheden Theorem. The st one state that the Hilbert Transform H is bounded in the weighted space L 2 (w) if and only if w can be represented as w = expfu + Hvg, where u; v 2 L 1 , kuk 1 < =2. The Hunt{Muckenhoupt{Wheden Theorem states that the Hilbert transform H is bounded in L p (w) if and only if the weight w satisses the so-called Muckenhoupt A p condition sup I 1 jIj Z I w 1 jIj Z I w ?1=(p?1) p?1 < 1; (A p) where the supremum is taken over all intervals I. This condition is also necessary and suucient fo boundedness for a wide class of singular integral operators, as well as for the boundedness of the maximal operator M, Mf(x) = sup I3x 1 jIj Z I jfj; here supremum is taken over all intervals I containing x. It is worth mentioning, that there in no direct proof of equivalence the Helson{ Szegg o condition and the Muckenhoupt condition A 2. Two weight inequalities, i. e. the problem when an operator acts from L 2 (w) to L 2 (v) (one can also consider L p case, even with diierent exponents p, but the L 2 case is complicated enough, so we restrict our attention on it) appears naturally in many areas like the theory of Hankel and Toeplitz operators, perturbation theory, etc. Things look much more complicated in the two-weight case, and it is probably an agreement now that there is no simple (Muckenhoupt type) necessary and suucient condition of boundedness of the Hilbert Transform. It was a big surprise when Eric Sawyer S1] found necessary and suucient condition for a maximal operator M to be a bounded operator from L 2 (w) to L 2 (v): his theorem states that it is enough to test the boundedness on a very special class of test functions, namely only on functions I w …