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Zbl 0692.46023
Maz'ya, Vladimir G.
(Mazja, V.G.; Maz'ja, V.G.)
Sobolev spaces. Transl. from the Russian by T. O. Shaposhnikova. (English)
[B] Berlin etc.: Springer-Verlag. xix, 486 p. (1985). ISBN 3-540-13589-8

From the author's preface: ``In the present monograph we consider various aspects of Sobolev space theory. Attention is paid mainly to the so called imbedding theorems.... We list some questions considered in the book.\par 1. What are the requirements on the measure $\mu$ for the inequality $(\int \vert u\vert\sp q d\mu)\sp{1/q}\le C\Vert u\Vert\sb{S\sp{\ell}\sb p},$ where $S\sp{\ell}\sb p$ is the Sobolev space or its generalization, to hold? \par 2. What are the minimal assumptions on the domain for the Sobolev imbedding theorem to remain valid? How do these theorems vary under the degeneration of the boundaries? How does the class of admissible domains depend on additional requirements placed upon the behavior of the function near the boundary? \par 3. How ``massive'' must a subset e of the domain $\Omega$ be in order that ``the Friedrichs inequality'' $\Vert u\Vert\sb{L\sb q(\Omega)}\le C\Vert \nabla\sb{\ell}u\Vert\sb{L\sb p(\Omega)}$ hold for all smooth functions that vanish in a neighborhood of e?''\par The book represents the state of art in the modern theory of Sobolev spaces. Based on sophisticated tools conditions for inequalities of the above type have often necessary and sufficient character. The underlying (non-smooth) domains are treated with great care. \par The book has 12 chapters. 1: Basic properties of Sobolev spaces (highlights are generalized Sobolev theorems and Hardy inequalities). 2: Inequalities for gradients of functions that vanish on the boundary. 3: On summability of functions in the space $L\sp 1\sb 1(\Omega)$. 4: On the summability of functions in the spaces $L\sp 1\sb p(\Omega)$. 5. On continuity and boundedness of functions in Sobolev spaces. 6: On functions in the space BV($\Omega)$ (generalized derivatives are measures). 7: Certain function spaces, capacities and potentials (Riesz and Bessel potentials, Besov spaces). 8: On summability with respect to an arbitrary measure of functions with fractional derivatives. 9: A variant of capacity. 10: An integral inequality for functions on a cube. 11: Imbedding of the space $\overset\circ\to L\sp{\infty}\sb p(\Omega)$ into other function spaces. 12: The imbedding $\overset\circ\to L\sp{\ell}\sb p(\Omega,\nu)\subset W\sp m\sb r(\Omega)$.
[H.Triebel]

MSC 2000:: *46E35 Sobolev spaces and generalizations
46-02 Research monographs (functional analysis)

Keywords: Sobolev space; imbedding theorems; Friedrichs inequality; capacities; Riesz and Bessel potentials; Besov spaces; integral inequality

Cited in: Zbl 1218.53037 Zbl 1217.46002 Zbl 1190.26010 Zbl 1175.31009 Zbl 1173.46015 Zbl 1127.26016 Zbl 1104.26018 Zbl 1036.42006 Zbl 0957.49001 Zbl 0926.31006 Zbl 0918.46033 Zbl 0712.31002 Zbl 0702.46016 Zbl 0727.46017

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