## The Structure of Functions (Google eBook)This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces, domains, manifolds and fractals. This approach combines the advantages of atomic and wavelet representations. It paves the way to sharp inequalities and embeddings in function spaces, spectral theory of fractal elliptic operators, and a regularity theory of some semi-linear equations. The book is self-contained, although some parts may be considered as a continuation of the author's book "Fractals and Spectra" (MMA 91). It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated. |

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3-quarks according to Definition apply arguments assertion assume atomic decomposition B-spaces ball condition Banach spaces Besov spaces bounded domain classical Sobolev spaces coincides compact d-set compact support context continuous embedding convergence Corollary counterpart cubes Definition 2.6 described Dirichlet Laplacian discussion eigenvalues elliptic operators embedding entropy numbers equivalence constants equivalent quasi-norm estimate F-spaces fceN finite Radon measure follows Fpg(R Fpq(R fractal function spaces Furthermore given Hardy inequalities Hence introduced in Definition isomorphic map j e N0 left-hand side Lipschitz Lipschitz continuous Lp(R Math non-negative notation obtain optimal coefficients particular positive numbers problem proof of Theorem Proof Step Proposition prove quarkonial decompositions radius Radon measure Recall refer Remark resolution of unity respect restriction Riemannian manifold right-hand side satisfies the ball Section Sobolev spaces spaces Bpq(R spectral theory sub-critical super-critical Theorem 2.9 Tri7 unconditional convergence usual modification Weyl measure

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Page ii - Managing Editors: H. Amann Universität Zurich, Switzerland J.-P. Bourguignon IHES, Bures-sur-Yvette, France K. Grove University of Maryland, College Park, USA P.-L. Lions Université de Paris-Dauphine, France Associate Editors: H. Araki, Kyoto University F. Brezzi, Università di Pavia KC Chang, Peking University N. Hitchin, University of Warwick H. Hofer, Courant Institute, New York H. Knörrer, ETH Zurich K. Masuda, University of Tokyo D. Zagier, Max-Planck-Institut Bonn Hans Triebel

Page 10 - be a cube in R¿ with sides parallel to the axes of coordinates, centred at 2¿'m, and with side length 2”, where mC

Page xii - made valuable suggestions which have been incorporated in the text. I am especially indebted to

Page 10 - be the Schwartz space of all complex-valued, rapidly decreasing, infinitely differentiable functions on R

Page xii - It is a pleasure to acknowledge the great help I have received from my

Page xi - which cover many wellknown classical spaces, such as (fractional) Sobolev spaces, HOlder-Zygmund spaces, Besov spaces and (inhomogeneous) Hardy spaces. The theory of these spaces has been developed systematically