Geometric Harmonic Analysis IV Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis

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.

Traditionally, the label “Calderón-Zygmund theory” has been applied to a distinguished body of works primarily pertaining to the mapping properties of singular integral operators on Lebesgue spaces, in various geometric settings. Volume IV amounts to a versatile Calderón-Zygmund theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries, considered on a diverse range of function spaces. Novel applications to complex analysis in several variables are also explored here.

Introduction and Statement of Main Results Concerning the Divergence Theorem.- Examples, Counterexamples, and Additional Perspectives.- Tools from Geometric Measure Theory, Harmonic Analysis, and functional Analysis.- Open Sets with Locally Finite Surface Measures and Boundary Behavior.- Proofs of the Main Results Pertaining to the Divergence Theorem.- Applications to Singular Integrals, Function Spaces, Boundary Problems, and Further Results.