Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(•) and Wk,p

被引:251
作者
Diening, L [1 ]
机构
[1] Univ Freiburg, Fak Math, D-79104 Freiburg, Germany
关键词
Riesz potential; Sobolev embedding; generalized Lebesgue space; generalized Sobolev space; variable exponent; extension operator;
D O I
10.1002/mana.200310157
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study the Riesz potentials I(alpha)f on the generalized Lebesgue spaces L-p(.)(R-d), where 0 < alpha < d and I(alpha)f(x) := integral(Rd) \f(y)\ \x - y\(alpha-d) dy. Under the assumptions that p locally satisfies \p(x) - p(x)\ less than or equal to C /(- ln \x - y\) and is constant outside some large ball, we prove that I-alpha : L-p(.)(R-d) --> Lp(#)((.))(R-d), where 1/p#(x) = 1/p(x) - alpha/d. If p is given only on a bounded domain Omega with Lipschitz boundary we show how to extend p to (p) over tilde on R-d such that there exists a bounded linear extension operator E : W-1,W-p(.) (Omega) hooked right arrow W-1,W-(p) over tilde(R-d), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings W-k,W-p(.) (R-d) hooked right arrow L-p*(.) (R-d) with 1/p*(x) = 1/p(x) - k/d and W-1,W-p(.) (Omega) hooked right arrow L-p*(.) (Omega) for k = 1. We show compactness of the embeddings W-1,W-p(.) (Omega) hooked right arrow L-q(.) (Omega), whenever q(x) less than or equal to p*(x) - epsilon for some epsilon > 0. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
引用
收藏
页码:31 / 43
页数:13
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