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Geometric Harmonic Analysis III

Part of the Developments in Mathematics Book Series (volume 74)

DOI: 10.1007/978-3-031-22735-6

ISBN: 978-3-031-22734-9

Publisher: Springer International Publishing - Springer Journals

May 13, 2023

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Chapter 1Integral Representations and Integral Identities

Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius

Pages: 1 –265

[Working in the complex plane, in §1.1 we prove versions of the classical Cauchy-Pompeiu integral representation formula, Morera’s Theorem, the Residue Theorem, and the Schwarz-Pompeiu formula under minimal smoothness assumptions. This line of work continues in §1.2 where higher-dimensional...
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Chapter 2Calderón-Zygmund Theory on Uniformly Rectifiable Sets

Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius

Pages: 267 –632

[This chapter contains a powerful and nuanced Calderón-Zygmund theory for singular integral operators on uniformly rectifiable sets. The results here, which build on the work of many predecessors, including Calderón, Zygmund, Mikhlin, Coifman, McIntosh, Meyer, David, Jerison, Kenig, Semmes,...
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Chapter 3Quantitative Fatou-Type Theorems in Arbitrary UR Domains

Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius

Pages: 633 –767

[The trademark blueprint of a Fatou-type theorem is that size/integrability properties of the nontangential maximal operator for a null-solution of an elliptic equation in a certain domain implies the a.e. existence of the pointwise nontangential boundary trace of said function. It is natural to...
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Chapter 4Green Functions and Uniqueness for Boundary Problems for Second-Order Systems

Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius

Pages: 769 –827

[Any object G referred to as Green function (in relation to a given differential operator L and an open set Ω⊆Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}...
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Chapter 5Green Functions and Poisson Kernels for the Laplacian

Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius

Pages: 829 –879

[Good references for background material pertaining to the topics in this chapter are [71], [49], [51], [58], [79]. The developments here are motivated by questions in potential theory for the Laplacian in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}...
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Chapter 6Scattering by Rough Obstacles

Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius

Pages: 881 –920

[In this chapter the goal is to provide a unified approach to scattering theory in rough settings which is also aimed at identifying the broadest possible spectrum of radiation conditions for null-solutions of the vector Helmholtz operator Δ+k2\documentclass[12pt]{minimal}...

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