Geometric Harmonic Analysis III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering

Mathematics Subject Classification: 32A Holomorphic functions of several complex variables • 26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.) • 31B Higher-dimensional potential theory • 35J Elliptic equations and elliptic systems • 42B Harmonic analysis in several variables

Chapter Headings: • 1. Integral Representations and Integral Identities • 2. Calderón-Zygmund Theory on Uniformly Rectifiable Sets • 3. Quantitative Fatou-Type Theorems in Arbitrary UR Domains • 4. Green Functions and Uniqueness for Boundary Problems for Second-Order Systems • 5. Green Functions and Poisson Kernels for the Laplacian • 6. Scattering by Rough Obstacles

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations.

Volume III is concerned with integral representation formulas for nullsolutions of elliptic PDEs, Calderón-Zygmund theory for singular integral operators, Fatou type theorems for systems of elliptic PDEs, and applications to acoustic and electromagnetic scattering. Overall, this amounts to a powerful and nuanced theory developed on uniformly rectifiable sets, which builds on the work of many predecessors.