Mean lattice point discrepancy bounds, II: Convex domains in the plane

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.Comment: Revised version, to appear in Journal d'Analyse Mathematiqu

Two Weight Inequalities for Discrete Positive Operators

We characterize two weight inequalities for general positive dyadic operators. We consider both weak and strong type inequalities, and general (p,q) mapping properties. Special cases include Sawyers Fractional Integral operator results from 1988, and the bilinear embedding inequality of Nazarov-Treil-Volberg from 1999. The method of proof is an extension of Sawyer's argument.Comment: 20 pages, 3 figures. v2 correction of minor typos v3: Correction of typos. v4: Two references adde

A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis

Subject to a range of side conditions, the two weight inequality for the Hilbert transform is characterized in terms of (1) a Poisson A_2 condition on the weights (2) A forward testing condition, in which the two weight inequality is tested on intervals (3) and a backwards testing condition, dual to (2). A critical new concept in the proof is an Energy Condition, which incorporates information about the distribution of the weights in question inside intervals. This condition is a consequence of the three conditions above. The Side Conditions are termed 'Energy Hypotheses'. At one endpoint they are necessary for the two weight inequality, and at the other, they are the Pivotal Conditions of Nazarov-Treil-Volberg. This new concept is combined with a known proof strategy devised by Nazarov-Treil-Volberg. A counterexample shows that the Pivotal Condition are not necessary for the two weight inequality.Comment: 60 pages, 1 figure. v3. An important revision: The Energy Condition is reformulated, a key concept of the proof, is reformulated. The main result is unchanged. v4. important display corrected. v6: The earlier versions incorrectly claimed a characterization, as was pointed out to us by S. Treil v7. Corrections in Section

A simplified proof of an improved NTV conjecture for the Hilbert transform

We give a simplified proof of the NTV conjecture for the Hilbert transform that was proved by T. Hyt\"onen, M. Lacey, E.T. Sawyer, C.-Y. Shen and I. Uriarte-Tuero, building on previous work of F. Nazarov, S. Treil and A. Volberg. Our proof shows a bit more, namely that the Hilbert transform is bounded from one weighted space to another if and only if the two local testing conditions hold and the classical offset Muckenhoupt condition holds. The proof avoids the use of functional energy, two weight inequalities for Poisson integrals, and recursion of admissible collections of pairs of intervals, but retains the bottom-up corona construction, and a variant of the straddling lemmas, from M. Lacey. Finally, the proof can be extended to certain generalizations of fractional singular integral operators on the real line, without assuming a classical energy condition.Comment: 23 pages, more typos fixed and displays lengthened to fit the page. Control of the stopping form corrected. Thanks to Brett Wick for discussions. Results unchange