A capacitary weak type inequality for Sobolev functions and its applications

Tai, Wei-Shyan

doi:10.1090/S0002-9939-00-05976-1

A capacitary weak type inequality for Sobolev functions and its applications
HTML articles powered by AMS MathViewer

by Wei-Shyan Tai PDF

Proc. Amer. Math. Soc. 129 (2001), 699-711 Request permission

Abstract:

In this paper a capacitary weak type inequality for Sobolev functions is established and is applied to reprove some well-known results concerning Lebesgue points, Taylor expansions in the $L^p$-sense, and the Lusin type approximation of Sobolev functions.

References

David R. Adams, Maximal operators and capacity, Proc. Amer. Math. Soc. 34 (1972), 152–156. MR 350314, DOI 10.1090/S0002-9939-1972-0350314-1
David R. Adams, Quasi-additivity and sets of finite $L^{p}$-capacity, Pacific J. Math. 79 (1978), no. 2, 283–291. MR 531319
Thomas Bagby and William P. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129–148. MR 344390, DOI 10.1090/S0002-9947-1974-0344390-6
A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037
A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI 10.4064/sm-20-2-181-225
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Herbert Federer and William P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139–158. MR 435361, DOI 10.1512/iumj.1972.22.22013
Fon-Che Liu, A Lusin type property of Sobolev functions, Ind. Univ. Math. J., 26 (1977), 645–651.
Fon Che Liu and Wei Shyan Tai, Maximal mean steepness and Lusin type properties, Ricerche Mat. 43 (1994), no. 2, 365–384. MR 1324758
B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
Norman G. Meyers, Taylor expansion of Bessel potentials, Indiana Univ. Math. J. 23 (1973/74), 1043–1049. MR 348482, DOI 10.1512/iumj.1974.23.23085
J. H. Michael and William P. Ziemer, A Lusin type approximation of Sobolev functions by smooth functions, Classical real analysis (Madison, Wis., 1982) Contemp. Math., vol. 42, Amer. Math. Soc., Providence, RI, 1985, pp. 135–167. MR 807986, DOI 10.1090/conm/042/807986
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63–89.
William P. Ziemer, Uniform differentiability of Sobolev functions, Indiana Univ. Math. J. 37 (1988), no. 4, 789–799. MR 982830, DOI 10.1512/iumj.1988.37.37038
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3