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A mollifying operator with a variable radius, and an inverse theorem on traces. (English. Russian original) Zbl 0597.46032

Sib. Math. J. 26, 891-901 (1985); translation from Sib. Mat. Zh. 26, No. 6(154), 141-152 (1985).
Let \(\Omega\) be an open subset of \(R^ n\) with smooth boundary \(\Gamma\), \(\alpha\), \(p\in R\), \(1\leq p\leq \infty\) and \(r\in N\). The terminology and notations being as in S. M. Nikol’skij, Approximation of functions of several variables and imbedding theorems, (Russian) (1977; Zbl 0496.46020), it is well known that \(W^ r_{p,\alpha}\subset W^{r- \ell}_{p,\alpha +\ell}\) and the embedding operator is continuous provided that \(0<r+\alpha -p^{-1}<r\), \(s_ 0\in N\) is the integer part of \(r+\alpha -p^{-1}\) and \(\ell \in N\), \(1\leq \ell <r-s_ 0\). Obviously, the reverse inclusion is not true, but the author constructs a bounded linear operator T:\(W^ r_{p,\alpha}\to \cap^{\infty}_{\ell =0}W^{r+\ell}_{p,\alpha -\ell}\) such that \(\partial^ sTf/\partial \nu^ s|_{\Gamma}=\partial^ sf/\partial \nu^ s|_{\Gamma}\), \(s=0,...,s_ 0-1\), where \(\nu\) denotes the outer normal to \(\Gamma\). The operator T is an averaging operator with variable radius and an approximately chosen kernel.
Reviewer: S.Cobzaş

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Citations:

Zbl 0496.46020
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References:

[1] S. M. Nikol’skii, The Approximation of Functions of Several Variables, and Embedding Theorems [in Russian], Nauka, Moscow (1977).
[2] O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorem [in Russian], Nauka, Moscow (1975).
[3] S. V. Uspenskii, ?Embedding theorems for weighted classes,? Tr. Mat. Inst. Steklov,60, 282-303 (1961).
[4] V. I. Burenkov, ?The density of infinitely differentiable functions in Sobolev spaces for an arbitrary open set,? Tr. Mat. Inst. Steklov,131, 39-50 (1974). · Zbl 0313.46033
[5] V. I. Burenkov, ?Mollifying operators with variable step and their application to approximation by infinitely differentiable functions,? in: Nonlinear Analysis. Function Spaces and Applications, Vol. 2, Teubner, Leipzig (1982), pp. 5-37. · Zbl 0536.46021
[6] V. I. Burenko, ?Regularized distance,? Tr. Mosk. Inst. Radiotekh. Elektron i Avtomat, No. 67, Matematika, 113-117 (1973).
[7] L. D. Kudryavtsev, ?Direct and inverse embedding theorems. Application to the solution of elliptic equations by the variational method,? Trudy Mat. Inst. Steklov,55, 1-181 (1959).
[8] A. P. Calderon and A. Zygmund, ?Local properties of solutions of elliptic partial differential equations,? Stud. Math.,20, 217-225 (1961).
[9] E. Stein, Singular Integrals and Differential Properties of Functions [Russian translation], Mir, Moscow (1973).
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