Geometric Harmonic Analysis III

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

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.

"The complete set of volumes promises to deliver most of what is known about solving elliptic equations and systems on various kinds of flat domains under minimal conditions on the flatness (local and global) of the domains." (Raymond Johnson, zbMATH 1523.35001, 2023)

“The complete set of volumes promises to deliver most of what is known about solving elliptic equations and systems on various kinds of flat domains under minimal conditions on the flatness (local and global) of the domains.” (Raymond Johnson, zbMATH 1523.35001, 2023)