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.
Author(s): Dorina Mitrea, Irina Mitrea, Marius Mitrea
Series: Developments in Mathematics, 75
Publisher: Springer
Year: 2023
Language: English
Pages: 1003
City: Cham
Prefacing the Full Series
Description of Volume IV
Contents
Layer Potential Operators on Lebesgue and Sobolev Spaces
Comments on History and Physical Interpretations of Harmonic Layer Potentials
``Tangential'' Singular Integral Operators
A First Look at Layer Potential Operators
Examples and Alternative Points of View
Calderón-Zygmund Function Theory for Boundary Layer Potentials
Cauchy and Cauchy-Clifford Operators on Lebesgue and Sobolev Spaces
Kernels and Images of Boundary Layer Potentials
Modified Boundary Layer Potential Operators
Layer Potential Operators on Hardy, BMO, VMO, and Hölder Spaces
Double Layer Potential Operators on Hardy, BMO, VMO, and Hölder Spaces
Single Layer Operators Acting from Hardy Spaces
Integral Operators of Layer Potential Type on Hardy-Based Sobolev Spaces and BMO-1
Layer Potential Operators on Calderón, Morrey-Campanato, and Morrey Spaces
Boundary Layer Potentials on Calderón Spaces
Boundary Layer Potentials on Morrey-Campanato Spaces and Their Pre-Duals
Boundary Layer Potential Operators on Morrey Spaces and Their Pre-Duals
Layer Potential Operators Acting from Boundary Besov and Triebel-Lizorkin Spaces
Boundary-to-Boundary Layer Potentials from Besov and Triebel-Lizorkin Spaces into Themselves
Boundary-to-Domain Layer Potentials from Besov Spaces into Weighted Sobolev Spaces
Boundary-to-Domain Layer Potentials from Besov Spaces into Besov and Triebel-Lizorkin Spaces
Integral Representation Formulas of Layer Potential Type, and Consequences
Generalized Double Layers in Uniformly Rectifiable Domains
Theory of Generalized Double Layers
Generalized Double Layers with Matrix-Valued Kernels, and Chord-Dot-Normal SIO's
Another Look at Standard and Modified Riesz Transforms
Green Formulas and Layer Potential Operators for the Stokes System
Green-Type Formulas for the Stokes System
Boundary Layer Potential Operators for the Stokes System: Lebesgue, Sobolev, and Hardy Spaces
Other Integral Representations and Fatou-Type Results for the Stokes System
Layer Potentials for the Stokes System on Besov, Triebel-Lizorkin, and Weighted Sobolev Spaces
Applications to Analysis in Several Complex Variables
CR-Functions and Differential Forms on Boundaries of Locally Finite Perimeter Sets
Integration by Parts Formulas Involving the Operator on Sets of Locally Finite Perimeter
The Bochner-Martinelli Integral Operator
A Sharp Version of the Bochner-Martinelli-Koppelman Formula and Related Topics
The Extension Problem for Hölder CR-Functions on Boundaries of Ahlfors Regular Domains
The Extension Problem for Lp/BMO/VMO/Morrey Functions on Boundaries of Uniformly Rectifiable Domains
The Operator and the Dolbeault Complex on Uniformly Rectifiable Sets
Hardy Spaces for Second-Order Weakly Elliptic Operators in the Complex Plane
Null-Solutions and Boundary Traces for Bitsadze's Operator z2 in the Unit Disk
Null-Solutions and Boundary Traces for the Operator z2-2z2
Terms and notation used in Volume IV
References
