The investigator plans to investigate several questions linked by the common themes of local Tb Theorems, and the interplay between singular integral estimates, Poisson kernel estimates, square function estimates, and the regularity of boundaries. Among the main directions of the proposed research are: 1. To investigate the relationships among boundedness of layer potentials, properties of harmonic measure, and uniform rectifiability. 2. To investigate the structure of uniformly rectifiable sets. 3. To develop and apply "local" Tb theorems to study the regularity of free boundaries and the solvability of elliptic boundary value problems. 4. 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.
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. 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 phenoma in the real world, including electrostatics, and steady-state temperature distributions and elastic deformations. A particular focus of the present proposal is to explore further the relationship between geometry and properties of "harmonic measure." The latter area of investigation has already found application in recent work in acoustical engineering, in particular in the design of a room with desirable acoustic properties. Progress on the problems to be considered would in all likelihood open up further avenues of investigation. All such progress will be disseminated by the investigator via lectures at conferences, seminars and graduate courses, and via electronic preprints posted on his website and on the ArXiv web site. The investigator will involve Ph.D. students and a postdoc on problems related to the proposed work. Two former postdocs are already involved in the project.
