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1993 | 106 | 1 | 77-92
Tytuł artykułu

Pointwise inequalities for Sobolev functions and some applications

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EN
Abstrakty
EN
We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer's theorem which states that Sobolev functions can be approximated by $C^m$ functions both in norm and capacity.
Twórcy
  • Institute of Mathematics, Polish Academy Of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland.
Bibliografia
  • [B1] B. Bojarski, Remarks on local function spaces, in: Lecture Notes in Math. 1302, Springer, 1988, 137-152.
  • [B2] B. Bojarski, Remarks on some geometric properties of Sobolev mappings, in: Functional Analysis & Related Topics, S. Koshi (ed.), World Scientific, 1991.
  • [B3] B. Bojarski, Remarks on Sobolev imbedding inequalities, in: Lecture Notes in Math. 1351, Springer, 1988, 52-68.
  • [Bu] V. I. Burenkov, Sobolev's integral representation and Taylor's formula, Trudy Mat. Inst. Steklov. 131 (1974), 33-38 (in Russian).
  • [CZ] A. P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171-225.
  • [F] H. Federer, Geometric Measure Theory, Springer, 1969.
  • [Fr] O. Frostman, Potentiel d'équilibre et capacité, Meddel. Lunds. Univ. Mat. Sem. 3 (1935).
  • [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.
  • [GR] V. M. Goldshteĭn and Yu. G. Reshetnyak, Quasiconformal Mappings and Sobolev Spaces, Nauka, Moscow 1983 (in Russian); English transl.: Kluwer Acad. Publ., 1990.
  • [H1] P. Hajłasz, Geometric theory of Sobolev mappings, Master's thesis, Warsaw Univ., 1990 (in Polish).
  • [H2] P. Hajłasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), 93-101.
  • [H3] P. Hajłasz, Note on Meyers-Serrin's theorem, Exposition. Math., to appear.
  • [He] L. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510.
  • [KA] L. Kantorovich and G. Akilov, Functional Analysis, 3rd ed., Nauka, Moscow 1984 (in Russian).
  • [La] N. S. Landkof, Foundations of Modern Potential Theory, Springer, 1972.
  • [L] F.-C. Liu, A Lusin type property of Sobolev functions, Indiana Univ. Math. J. 26 (1977), 645-651.
  • [M] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, London 1966.
  • [Ma] V. G. Maz'ya, Sobolev Spaces, Springer, 1985.
  • [Me] N. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255-292.
  • [MS] N. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 1055-1056.
  • [MZ] J. Michael and W. Ziemer, A Lusin type approximation of Sobolev functions by smooth functions, in: Contemp. Math. 42, Amer. Math. Soc., 1985, 135-167.
  • [R] Yu. G. Reshetnyak, Remarks on integral representations, Sibirsk. Mat. Zh. 25 (5) (1984), 198-200 (in Russian).
  • [S] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
  • [W] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math Soc. 36 (1934), 63-89.
  • [Z1] W. Ziemer, Uniform differentiability of Sobolev functions, Indiana Univ. Math. J. 37 (1988), 789-799.
  • [Z2] W. Ziemer, Weakly Differentiable Functions, Springer, 1989.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv106i1p77bwm
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