Pointwise inequalities for Sobolev functions and some applications
Bogdan Bojarski; Piotr Hajłasz
Studia Mathematica (1993)
- Volume: 106, Issue: 1, page 77-92
- ISSN: 0039-3223
Access Full Article
topAbstract
topHow to cite
topBojarski, Bogdan, and Hajłasz, Piotr. "Pointwise inequalities for Sobolev functions and some applications." Studia Mathematica 106.1 (1993): 77-92. <http://eudml.org/doc/216004>.
@article{Bojarski1993,
abstract = {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.},
author = {Bojarski, Bogdan, Hajłasz, Piotr},
journal = {Studia Mathematica},
keywords = {Sobolev function; Taylor polynomial; approximation; integral representation; Bessel capacity; pointwise inequalities for Sobolev functions; Michael-Ziemer's theorem},
language = {eng},
number = {1},
pages = {77-92},
title = {Pointwise inequalities for Sobolev functions and some applications},
url = {http://eudml.org/doc/216004},
volume = {106},
year = {1993},
}
TY - JOUR
AU - Bojarski, Bogdan
AU - Hajłasz, Piotr
TI - Pointwise inequalities for Sobolev functions and some applications
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 1
SP - 77
EP - 92
AB - 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.
LA - eng
KW - Sobolev function; Taylor polynomial; approximation; integral representation; Bessel capacity; pointwise inequalities for Sobolev functions; Michael-Ziemer's theorem
UR - http://eudml.org/doc/216004
ER -
References
top- [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). Zbl0313.46032
- [CZ] A. P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171-225. Zbl0099.30103
- [F] H. Federer, Geometric Measure Theory, Springer, 1969. Zbl0176.00801
- [Fr] O. Frostman, Potentiel d'équilibre et capacité, Meddel. Lunds. Univ. Mat. Sem. 3 (1935). Zbl61.1262.02
- [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. Zbl0562.35001
- [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. Zbl0840.26009
- [H3] P. Hajłasz, Note on Meyers-Serrin's theorem, Exposition. Math., to appear. Zbl0799.46042
- [He] L. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. Zbl0283.26003
- [KA] L. Kantorovich and G. Akilov, Functional Analysis, 3rd ed., Nauka, Moscow 1984 (in Russian). Zbl0555.46001
- [La] N. S. Landkof, Foundations of Modern Potential Theory, Springer, 1972. Zbl0253.31001
- [L] F.-C. Liu, A Lusin type property of Sobolev functions, Indiana Univ. Math. J. 26 (1977), 645-651. Zbl0368.46036
- [M] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, London 1966. Zbl0177.17902
- [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. Zbl0242.31006
- [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). Zbl0597.26018
- [S] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl0207.13501
- [W] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math Soc. 36 (1934), 63-89. Zbl0008.24902
- [Z1] W. Ziemer, Uniform differentiability of Sobolev functions, Indiana Univ. Math. J. 37 (1988), 789-799. Zbl0677.41033
- [Z2] W. Ziemer, Weakly Differentiable Functions, Springer, 1989.
Citations in EuDML Documents
top- Agnieszka Kałamajska, On lower semicontinuity of multiple integrals
- Guozhen Lu, Richard Wheeden, High order representation formulas and embedding theorems on stratified groups and generalizations
- Takao Ohno, Tetsu Shimomura, Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces
- Takao Ohno, Tetsu Shimomura, Musielak-Orlicz-Sobolev spaces on metric measure spaces
- Irene Fonseca, Nicola Fusco, Paolo Marcellini, Topological degree, Jacobian determinants and relaxation
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.