Commutators, Hilbert and Riesz transforms, Shifts, Harmonic extensions and Martingales

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 and acts as a frequency filter. In this role, it is used in the treatment of AM FM technology. We use elementary elements called wavelets. There are significant applications to wavelets, such as in image processing, material science. In this project we bring together several important areas in mathematics, such as numeric analysis and probability, 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.

1) Failure of the matrix weighted bilinear Carleson embedding theorem
Authors: Komla Domelevo, Stefanie Petermichl, Kristina Ana Škreb
2) A matrix weighted bilinear Carleson Lemma and Maximal Function
Authors: Stefanie Petermichl, Sandra Pott, Maria Carmen Reguera
3)Continuous sparse domination and dimensionless weighted estimates for the Bakry Riesz vector
Authors: Komla Domelevo, Stefanie Petermichl, Kristina 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 Hytönen, Stefanie Petermichl, Alexander Volberg
6) Weighted little bmo and two-weight inequalities for Journé 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)Continuous-time sparse domination
Authors: Komla Domelevo, Stefanie Petermichl
9) The dyadic and the continuous Hilbert transforms with values in Banach spaces.
Authors: Komla Domelevo, Stefanie Petermichl
10) The dyadic and the continuous Hilbert transforms with values in Banach spaces. Part2
Authors: Komla Domelevo, Stefanie Petermichl
11) The matrix-weighted dyadic convex body maximal operator is not bounded
Authors: F. Nazarov, S. Petermichl, K. A. Škreb, S. Treil
12) Boundedness of Journé operators with matrix weights
Authors: Komla Domelevo, Spyridon Kakaroumpas, Stefanie Petermichl, Odí Soler i Gibert
13) Dyadic lower little BMO estimates
Authors: Komla Domelevo, Spyridon Kakaroumpas, Stefanie Petermichl, Odí Soler i Gibert
14) H∞ calculus for submarkovian semigroups on weighted L2 spaces
Authors: Komla Domelevo, Christoph Kriegler, Stefanie Petermichl
15) Dyadic product BMO in the Bloom setting
Authors: Spyridon Kakaroumpas, Odí Soler i Gibert
16) Approximation in the Zygmund and Hölder classes on R^n
Authors: Eero Saksman and Odí Soler i Gibert

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 a first positive, optimal result and 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. Recently the well known dyadic shift operator was modified and it lead to new proofs. We expect further important results or reductions to open problems in high dimensions via continuous sparse domination. We shed new light on a conjecture in Banach space geometry via a new and precise model that significantly reduces a well known conjecture in the area. We developed further connections between functional analysis, H infinity functional calculus and probability.

Periodic Reporting for period 5 - CHRiSHarMa (Commutators, Hilbert and Riesz transforms,Shifts, Harmonic extensions and Martingales)

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