Distributions. Generalized functions with applications in Sobolev spaces (Q416001)

This heavy textbook (833 pages) concerns distributions, Sobolev spaces, and applications. It is divided into 9 chapters, namely: 1.) Schwartz distributions; 2.) Differentiation of distributions and application of distributional derivatives; 3.) Derivatives of piecewise smooth functions, Green\(^{\prime}\)s formula, elementary solutions, applications to Sobolev spaces; 4.) Additional properties of \(\mathcal D^{\prime}(\Omega)\); 5.) Local properties, restrictions, unification principle, space \(\mathcal E^{\prime}(\mathbb R^n)\) of distributions with compact support; 6.) Convolution of distributions; 7.) Fourier transforms of functions of \(L^1(\mathbb R^n)\) and \(\mathcal S(\mathbb R^n)\); 8.) Fourier transforms of distributions and Sobolev spaces of arbitrary order \(H^s(\mathbb R^n)\); 9.) Vector-valued distributions. In appendices, indispensable notions of analysis and geometry are recalled.NEWLINENEWLINE The author points out that all the basic results, concepts, theorems etc \(\dots\), on distributions, presented in this book, are due uniquely to NEWLINE[\textit{L. Schwartz}, Théorie des distributions. Nouvelle édition, entièrement corrigée, refondue et augmentée. Publ. Inst. Math. Univ. Strasbourg. IX-X. Paris: Hermann \& Cie (1966; Zbl 0149.09501); ibid., Méthodes mathématiques pour les sciences physiques. Enseign. des sciences. Paris: Hermann \& Cie (1961; Zbl 0101.41301); ibid., Œuvres scientifiques. I., II, III, Doc. Math., Paris 9. Paris: Société Mathématique de France (SMF) (2011; Zbl 1236.01038); Doc. Math., Paris 10. (2011; Zbl 1235.01039); Doc. Math., Paris 11 (2011; Zbl 1235.01040)].NEWLINENEWLINE The book is very well written. Notations are carefully recalled at the beginning of the book and an index is presented in the 12 last pages. It will be a helpful document for mathematicians who want to get used to distributions and for engineers and researchers in applied sciences.