Hölder type quasicontinuity. (English) Zbl 0803.46037

Summary: It is proved that a function \(u\in L^{m,p}(\mathbb{R}^ n)\) (which coincides with the Sobolev space \(W^{1,p}(\mathbb{R}^ n)\) if \(m=1\)) coincides with a Hölder continuous function \(w\) outside a set of small \(m\), \(q\)-capacity, where \(q< p\). Moreover, if \(m=1\), then the function \(w\) can be chosen to be close to \(u\) in the \(W^{1,p}\)-norm.


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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[1] Calderón, A. P.: Lebesgue spaces of differentiable functions and distributions,Proc. Symp. Pure Math. 4 (1961), 33-49. · Zbl 0195.41103
[2] Calderón, A.P. and Zygmund, A.: Local properties of solutions of elliptic partial differential operators,Studia Math. 20 (1961), 171-225. · Zbl 0099.30103
[3] Deny, J. and Lions, J. L.: Les espaces du type de Beppo Levi,Ann. Inst. Fourier (Grenoble) 5 (1953/54), 305-370.
[4] Liu, Fon-Che: A Lusin type property of Sobolev functions,Indiana Univ. Math. J. 26 (1977), 645-651. · Zbl 0368.46036 · doi:10.1512/iumj.1977.26.26051
[5] Maz’ya, V. G. and Khavin, V. P.: Nonlinear potential theory,Uspekhi Mat. Nauk 27(6) (1972), 67-138. English translation:Russian Math. Surveys 27 (1972), 71-148.
[6] Meyers, N. G.: A theory of capacities for potentials of functions in Lebesgue classes,Math. Scand. 26 (1970), 255-292. · Zbl 0242.31006
[7] Meyers, N. G.: Continuity properties of potentials,Duke Math. J. 42 (1975), 157-166. · Zbl 0334.31004 · doi:10.1215/S0012-7094-75-04214-3
[8] Michael, J. L. and Ziemer, W. P.: A Lusin type approximation of Sobolev functions by smooth functions,Contemporary Math., Amer. Math. Soc. 42 (1985), 135-167. · Zbl 0592.41031
[9] Reshetnyak, Yu. G.: On the concept of capacity in the theory of functions with generalized derivatives,Sibirsk. Mat. Zh. 10 (1969), 1109-1138.
[10] Triebel, H.:Theory of Function Spaces, Akademische Verlagsgesellschaft & Portig K.-G., Leipzig, 1983. · Zbl 0546.46028
[11] Ziemer, W. P.:Weakly Differentiable Functions, Sobolev Spaces and Function of Bounded Variation, Graduate Text in Mathematics 120, Springer-Verlag, 1989. · Zbl 0692.46022
[12] Ziemer, W. P.: Uniform differentiability of Sobolev functions,Indiana Univ. Math. J. 37(4)(1988), 789-799. · Zbl 0677.41033 · doi:10.1512/iumj.1988.37.37038
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