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

Objective

This project aims to develop two arrays of questions at the heart of harmonic
analysis, probability and operator theory:

Multi-parameter harmonic analysis.

Through the use of wavelet methods in harmonic analysis, we plan to shed new
light on characterizations for boundedness of multi-parameter versions of
classical Hankel operators in a variety of settings. The classical Nehari's theorem on
the disk (1957) has found an important generalization to Hilbert space
valued functions, known as Page's theorem. A relevant extension of Nehari's
theorem to the bi-disk had been a long standing problem, finally solved in
2000, through novel harmonic analysis methods. It's operator analog remains
unknown and constitutes part of this proposal.

Sharp estimates for Calderon-Zygmund operators and martingale
inequalities.

We make use of the interplay between objects central to
Harmonic analysis, such as the Hilbert transform, and objects central to
probability theory, martingales. This connection has seen many faces, such as
in the UMD space classification by Bourgain and Burkholder or in the formula
of Gundy-Varapoulos, that uses orthogonal martingales to model the behavior of
the Hilbert transform. Martingale methods in combination with optimal control
have advanced an array of questions in harmonic analysis in recent years. In
this proposal we wish to continue this direction as well as exploit advances
in dyadic harmonic analysis for use in questions central to probability. There
is some focus on weighted estimates in a non-commutative and scalar setting, in the understanding of discretizations
of classical operators, such as the Hilbert transform and their role played
when acting on functions defined on discrete groups. From a martingale
standpoint, jump processes come into play. Another direction is the use of
numerical methods in combination with harmonic analysis achievements for martingale estimates.

Host institution

JULIUS-MAXIMILIANS-UNIVERSITAT WURZBURG
Net EU contribution
€ 1 024 526,21
Address
Sanderring 2
97070 Wuerzburg
Germany

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Region
Bayern Unterfranken Würzburg, Kreisfreie Stadt
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00

Beneficiaries (2)

JULIUS-MAXIMILIANS-UNIVERSITAT WURZBURG
Germany
Net EU contribution
€ 1 024 526,21
Address
Sanderring 2
97070 Wuerzburg

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Region
Bayern Unterfranken Würzburg, Kreisfreie Stadt
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00
UNIVERSITE PAUL SABATIER TOULOUSE III

Participation ended

France
Net EU contribution
€ 499 436,79
Address
Route De Narbonne 118
31062 Toulouse Cedex 9

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Region
Occitanie Midi-Pyrénées Haute-Garonne
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00