Abstract
In the past thirty years there has accumulated a large amount of information about conditions which are necessary and sufficient for various properties of spaces of Sobolev type to hold true. It is question of boundedness and compactness criteria for imbedding operators characterizing the domain or the weight functions, of tests for the possibility of extending functions from the domain to ℝn, of conditions asserting the density of one space of differentiable functions in another etc. An adequate description of the properties of function spaces has made it necessary to introduce new classes of domains of definition for the functions, or classes of measures entering in the norms. In this connection the universal importance of the notion of capacity of a set became manifest.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, D. R. [ 1981 ] Lectures on LP-potential theory. Dep. Math. Univ Umeâ Publ., 1981, no. 2.
Aubin, T. [ 1976 ] Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598. Zbl. 371. 46011
Burago, Yu. D., Zalgaller, V. A. [ 1980 ] Geometric inequalities. Nauka, Leningrad. Zbl. 436.52009. English translation: Grundlehren 285. Springer, Berlin - Heidelberg - New York 1988
Dyn’kin, E. M., Osilenker, B. P. [1983] Weighted estimates for singular integrals and their applications, in: Itogi Nauki Tekhn, Mat. Analiz., 21, 42–129. Zbl. 568.42009. English translation: J. Soy. Math. 30, 2094–2154 (1985)
Gol’dshtein, V. M., Reshetnyak, Yu. G. [ 1983 ] Introduction to the theory of functions with generalized derivatives and their applications. Nauka, Moscow [Russian]. Zbl. 591. 46021
Hedberg, L. I. [ 1981 ] Spectral synthesis in Sobolev spaces and uniqueness of solutions of the Dirichlet problem. Acta Math. 147, 237–264. Zbl. 504. 35018
Maz’ya, V. G. [ 1985 ] Sobolev spaces. LGU, Leningrad. English translation: Springer, Berlin - Heidelberg - New York - Tokyo 1985
Maz’ya, V. G., Shaposhnikova, T. O. [1985] Multipliers in spaces of differentiable functions. LGU: Leningrad. English translation: Monographs and Studies in Mathematics 23. Pitman, Boston 1985. Zbl. 645. 46031
Osserman, R. [ 1978 ] The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238. Zbl. 411. 52006
Pólya, G. Szegö, G. [ 1951 ] Isoperimetric inequalities in mathematical physics. Princeton University Press, Princeton. Zbl. 44, 383
Sawyer, E. T. [ 1982 ] Two weight norm inequalities for certain maximal and integral operators, in: Lecture Notes Math. 908, pp. 102–127. Springer, Berlin - Heidelberg - New York. Zbl. 508. 42024
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Maz’ya, V.G. (1991). Classes of Domains, Measures and Capacities in the Theory of Differentiable Functions. In: Nikol’skiĭ, S.M. (eds) Analysis III. Encyclopaedia of Mathematical Sciences, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09961-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-662-09961-2_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08083-8
Online ISBN: 978-3-662-09961-2
eBook Packages: Springer Book Archive