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Full Description
This monograph presents state-of-the-art results at the intersection of Harmonic Analysis, Functional Analysis, Geometric Measure Theory, and Partial Differential Equations, providing tools for treating elliptic boundary value problems for systems of PDE's in rough domains. Largely self-contained, it develops a comprehensive Calderón-Zygmund theory for singular integral operators on many Herz-type spaces, and their associated Hardy and Sobolev spaces, in the optimal geometric-measure theoretic setting of uniformly rectifiable sets. The present work highlights the effectiveness of boundary layer potential methods as a means of establishing well-posedness results for a wide family of boundary value problems, including Dirichlet, Neumann, Regularity, and Transmission Problems. Graduate students, researchers, and professional mathematicians interested in harmonic analysis and boundary problems will find this monograph a valuable resource in the field.
Contents
Introduction.- Preliminary Matters.- Beurling Algebras.- Beurling-Hardy Spaces.- Functions of Bounded Central Mean Oscillations.- Calderon-Zygmund Theory on Beurling-Hardy Spaces and BCMO_p Spaces.- Weakly Elliptic Systems and Layer Potentials on UR Domains.- Layer Potentials on Beurling-Hardy Spaces and BCMO_p Spaces.- Boundary Value Problems on Beurling-Hardy Spaces and BCMO_p Spaces.- Herz Spaces of First Generation.- Herz-type Hardy Spaces of First Generation.- Functions of (p,q)-Bounded Central Mean Oscillations.- Calderon-Zygmund Theory on First-Generation Herz-type Spaces.- Herz-Based Sobolev Spaces of First Generation.- Layer Potentials on Herz-type Spaces of First Generation and Invertibility Results.- Boundary Value Problems on Herz-type Spaces of First Generation.- Measure Theoretic Herz Spaces.- Calderon-Zygmund Theory on Herz Spaces of Second Generation.- Boundary Value Problems on Herz Spaces of Second Generation.- Inhomogeneous Herz-type Hardy Spaces of Second Generation.- Singular Integral Operators on Herz-type Hardy Spaces of Second Generation.- Layer Potentials and Boundary Problems on Herz-type Hardy Spaces of Second Generation.- A New Class of Herz-type Spaces, Singular Integrals, and Boundary Problems.- Herz Spaces on Bounded Ahlfors Regular Sets.- Boundary Value Problems in Domains with Compact Boundary.
