Tb Theorems, Singular Integrals, Poisson Kernels, and Regularity of Boundaries

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The PI plans to work on problems in harmonic analysis linked by the interplay among local Tb Theorems, singular integral estimates, Poisson kernel estimates, square function estimates, and the regularity of boundaries. The goals of the proposed research are:1) to develop and apply ``local" Tb theorems to study the regularity of free boundaries and the solvability of elliptic boundary value problems; 2) to develop techniques to study the solvability of boundary value problems for complex elliptic equations, or more generally, for strongly elliptic systems, with bounded measurable coefficients;3) to investigate the relationships among boundedness of layer potentials, properties of harmonic measure, and uniform rectifiability;4) to continue to develop the theory of Hardy spaces adapted to a second order divergence form elliptic operator; in particular, this work may be viewed as an attempt to find a sharp solution to the Kato square root problem ``below the critical exponent."The project lies within the field of harmonic analysis and its application to, and interaction with, geometric measure theory and the theory of elliptic partial differential equations and systems. Roughly speaking, in harmonic analysis one investigates properties of functions and ``operators" (i.e., mappings which transform one function into another) by decomposing them into smaller, constituent pieces, which are easier to understand, and then reassembling the pieces. The name itself arose by analogy to the decomposition of a musical sound into its various frequency components (``harmonics"). Geometric measure theory involves the study of the relationship between geometric properties of sets, and their ``measures" (the latter are generalizations of the notions of length, area, and volume). Partial differential equations and systems of elliptic type describe a wide variety of phenomena in the real world, including electrostatics, and steady-state temperature distributions and elastic deformations. In the last decade the interplay between these different subfields of mathematics has turned out to be a fertile ground for investigation, with much exciting work remaining to be done. Progress on the problems to be considered would in all likelihood open up further avenues of investigation in these areas. All such progress will be disseminated by the PI via lectures at conferences, seminars and graduate courses, and via electronic preprints posted on his website and on the ArXiv. The PI plans to involve two postdocs and two graduate students in work related to this project.