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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - CHRiSHarMa (Commutators, Hilbert and Riesz transforms,Shifts, Harmonic extensions and Martingales)

Teaser

Summary

The Hilbert transform is central to the area of Harmonic analysis. It gives access to the harmonic conjugate function and as such it can predict the motion of an electron in a charged environment. It also transforms elementary waves. In this role, it is used in the treatment of AM FM technology. Last, there are significant applications to wavelets, image processing, material science. In this project we bring together several important areas in mathematics via the development of modern techniques in harmonic analysis. Then, problems in the vicinity of the important Hilbert transform are considered. Corner stones include the use of randomness for deterministic questions, either via the PI\'s Haar Shift, the method of Bellman functions, stochastic representations or the Sparse domination. In all these directions the project has seen significant progress. We achieve precise behaviour for certain phenomena in harmonic analysis.

Work performed

1) Failure of the matrix weighted bilinear Carleson embedding theorem
Authors: Komla Domelevo, Stefanie Petermichl, Kristina Ana Å kreb
2) A matrix weighted bilinear Carleson Lemma and Maximal Function
Authors: Stefanie Petermichl, Sandra Pott, Maria Carmen Reguera
3)Dimensionless Lp estimates for the Riesz vector on manifolds
Authors: Kamilia Dahmani, Komla Domelevo, Stefanie Petermichl
note: extension to the negative curvature case in the process of completion with Skreb.
4) On the failure of lower square function estimates in the non-homogeneous weighted setting
Authors: K. Domelevo, P. Ivanisvili, S. Petermichl, S. Treil, A. Volberg
5) The sharp square function estimate with matrix weight
Authors: Tuomas HytÃ¶nen, Stefanie Petermichl, Alexander Volberg
6) Weighted little bmo and two-weight inequalities for JournÃ© commutators
Authors: Irina Holmes, Stefanie Petermichl, Brett D. Wick
7) Various sharp estimates for semi-discrete Riesz transforms of the second order
Authors: Komla Domelevo, Adam Osekowski, Stefanie Petermichl
8) Convex body domination and weighted estimates with matrix weights
Authors: Fedor Nazarov, Stefanie Petermichl, Sergei Treil, Alexander Volberg
9) Continuous-time sparse domination
Authors: Komla Domelevo, Stefanie Petermichl
10) Differential subordination under change of law
Authors: Komla Domelevo, Stefanie Petermichl

in progress: counter example to the convex body valued matrix weighted Carleson Lemma with Nazarov, Treil, Skreb

in the final stages: weighted H infinity functional calculus with Domelevo and Kriegler

Final results

Significant progress was made in the area of sparse domination via the use of probabilistic methods with a continuous time parameter. Understanding of the puzzling so called matrix A2 conjecture was improved via some negative results that highlight the arising non-commutativity we see when multiplying matrices. Progress on commutator estimates allowed desired estimates but also, as a side effect, significantly simplified work done in the 80s by leading harmonic analysts. This progress inspired the work of several groups of junior researchers. We expect further important results or reductions to open problems in high dimensions via continuous sparse domination. There will be significant progress by the end of the project on the matrix A2 conjecture and its implications to uniform mixing. We expect to shed new light on a conjecture in Banach space geometry. We expect further connections between functional analysis, H infinity functional calculus and probability.