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Details of Grant 

EPSRC Reference: EP/P009239/1
Title: Harmonic Analysis in rough environments
Principal Investigator: Reguera Rodriguez, Dr M
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: School of Mathematics
Organisation: University of Birmingham
Scheme: First Grant - Revised 2009
Starts: 01 March 2017 Ends: 31 January 2020 Value (£): 101,170
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
07 Sep 2016 EPSRC Mathematical Sciences Prioritisation Panel September 2016 Announced
Summary on Grant Application Form
One of the fundamental questions in Harmonic Analysis concerns the study of boundedness of singular integrals in various contexts. Classically, the question of boundedness has been considered and extensively studied in particularly smooth environments. The current project is concerned with the development of a theory for singular integrals in rougher environments, namely in Lebesgue spaces where the underlying measure can be quite irregular. This theory is fundamentally motivated by questions such as the David--Semmes Conjecture in Geometric Measure Theory, the characterization of removable singularities for bounded Lipschitz harmonic functions in Complex Analysis, the study of properties of model spaces in Operator Theory and the geometric properties of harmonic measure in any domain in PDE.

Despite being the source of much international activity and great accomplishments in the last two decades, the theory of singular integrals in rough environments is far from fully developed. The reason is the lack of robust methods that fully exploit the interplay between the cancellation of the operator and the roughness of the underlying measure. The aim of this project is to find such robust methods. The proposed project aims to do so by studying two different but very compelling settings.

In the first setting, the project aims to characterize the two-weight boundedness for a particular singular integral, the Bergman projection. The Bergman projection is a fundamental operator acting on the space of square integrable analytic functions on the disc. The questions in this setting lie at the interface of Harmonic Analysis and Complex Analysis. It is anticipated that a good understanding of this case would lead to progress on the understanding of more general singular integrals. So far the only fully understood case is that of the Hilbert transform, and the method of proof is particularly designed to suit that and only that operator.

The second setting lies at the intersection of Harmonic Analysis and Geometric Measure Theory. The main question is concerned with retrieving geometrical properties of the rough underlying measure from the boundedness of non-integer dimensional Riesz transforms. Riesz transforms are fundamental operators in Harmonic Analysis and PDE. The existing results in the area have a dimensional restriction due to the use of a proof strategy that has not found any extensions in other dimensions. The main goal in this setting is to find a robust technique that would be applicable in all non-integer dimensions.

It is expected that the better understanding of the interaction between the cancellation of a singular integral and the underlying rough measure will open up new avenues of research in Harmonic Analysis. The programs proposed would also have impact on the work of researchers in cognate areas such as Function Theory and PDE, among others.
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Organisation Website: http://www.bham.ac.uk