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mitrea d geometric harmonic analysis iv 2023

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Textbook in PDF format 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. Prefacing the Full Series Description of Volume IV 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

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mitrea d geometric harmonic analysis iv 2023

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