Topics in Pure and Applied Harmonic Analysis

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This project is devoted to blending and intertwining methods typically attributable to either applied or theoretical aspect of modern Harmonic Analysis, in an effort to solve several long-standing problems, develop important cross-field techniques, and produce meaningful applications. One part of the project deals with the behavior of certain approximation/interpolation processes as their order tends to infinity, a recently developed variant of the Poisson summation formula involving irregular samples, and the regularity properties of solutions of certain boundary value problems. The other part applies the Bellman function method to computing the operator norms of the maximal function on Lebesgue spaces in non-martingale settings, the Riesz transforms on the space BMO, and the generalized Beurling-Ahlfors transform on differential forms; in addition, an extension of earlier sharp results for the John-Nirenberg inequality is studied.Among the practical areas that may be directly affected by the project are the fields of image and signal processing, including applications to medical tomography. The successful completion of this research should lead to improved methods for accumulation, transmission, and representation of various types of data. Studying the regularity properties of solutions of partial differential equations (PDE) is a fundamental step on the way to using those PDE to describe physical phenomena; such use is widespread and necessary in physics, chemistry, biology, material science, etc. In turn, the computation of norms of important operators, such as the Riesz transforms, allows one to estimate the size of solutions of various PDE. The Bellman function method itself links such computation to the existence of positive solutions to certain differential equations. On the other hand, the theoretical synergy, which is at the heart of the project, will result in the development of novel powerful techniques, affecting applied as well as pure aspects of Fourier analysis. Another important impact of this research is on education: the results, diverse and far-reaching, will be presented in courses and seminars on different levels, from advanced undergraduate to the doctoral.