Focusing on the theory of distributions, this book provides a quick yet comprehensive overview of its applications in partial differential equations and harmonic analysis. Each chapter is designed to enhance understanding through exercises, with solutions provided for many, facilitating self-study and deeper engagement with the material.
This monograph focuses on developing tools for establishing well-posedness results for elliptic differential operators in diverse geometric contexts. It introduces a new generation of Calderón-Zygmund theory applicable to variable coefficient singular integral operators, proving especially effective for boundary value problems involving the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds through boundary layer methods. The text addresses both absolute and relative boundary conditions for differential forms and examines the Hodge-Laplacian with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Positioned at the crossroads of partial differential equations, harmonic analysis, and differential geometry, this work is ideal for PhD students, researchers, and professionals. Key contents include an introduction and main results, geometric concepts and tools, harmonic layer potentials associated with the Hodge-de Rham formalism and the Levi-Civita connection on UR domains, boundary value problems for the Hodge-Laplacian, Fatou theorems, integral representations, and solvability of boundary problems. Additional insights and applications from various mathematical disciplines are also provided, along with a bibliography and index.