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taheri function spaces partial diff equations vol 1 2015

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Textbook in PDF format This book, which is the first volume of two, presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a quick review of basic properties of harmonic functions and Poisson integrals and then moves into a detailed study of Hardy spaces. The classical Dirichlet problem is considered and a variety of methods for its resolution ranging from potential theoretic (Perron’s method of sub-harmonic functions and Wiener’s criterion, Green’s functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) are presented. Parallel to this is the development of the necessary function spaces: Lorentz and Marcinkiewicz spaces, Sobolev spaces (integer as well as fractional order), Hardy spaces, the John-Nirenberg space BMO, Morrey and Campanato spaces, Besov spaces and Triebel-Lizorkin spaces. Harmonic analysis is deeply intertwined with the topics covered and the subjects of summability methods, Tauberian theorems, convolution algebras, Calderon-Zygmund theory of singular integrals and Littlewood-Paley theory that on the one hand connect to various PDE estimates (Calderon-Zygmund inequality, Strichartz estimates, Mihlin-Hormander multipliers, etc.) and on the other lead to a unified characterisation of various function spaces are discussed in great depth. The book ends by a discussion of regularity theory for second order elliptic equations in divergence form— first with continuous and next with measurable coefficients—and covers, in particular, De Giorgi’s theorem, Moser iteration, Harnack inequality and local boundedness of solutions. (The case of elliptic systems and related topics is discussed in the exercises.) Harmonic Functions and the Mean-Value Property Poisson Kernels and Green’s Representation Formula Abel-Poisson and Fejér Means of Fourier Series Convergence of Fourier Series: Dini vs. Dirichlet-Jordon Harmonic-Hardy Spaces hp(D) Interpolation Theorems of Marcinkiewicz and Riesz-Thorin The Hilbert Transform on Lp(T) and Riesz’s Theorem Harmonic-Hardy Spaces hp(Bn) Convolution Semigroups; The Poisson and Heat Kernels on Rn Perron’s Method of Sub-Harmonic Functions From Abel-Poisson to Bochner-Riesz Summability Fourier Transform on S′(Rn); The Hilbert-Sobolev Spaces Hs(Rn)

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Taheri A. Function Spaces and Partial Differential Equations Vol 1. 2015.pdf 18.3 MB

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taheri function spaces partial diff equations vol 1 2015

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