Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·)
✍ Scribed by Lars Diening
- Publisher
- John Wiley and Sons
- Year
- 2004
- Tongue
- English
- Weight
- 210 KB
- Volume
- 268
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
We study the Riesz potentials I~α~f on the generalized Lebesgue spaces L^p(·)^(ℝ^d^), where 0 < α < d and I~α~f(x) ≔ ∫ |f(y)| |x – y|^α – d^ dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |x – y|) and is constant outside some large ball, we prove that I~α~ : L^p(·)^(ℝ^d^) → L^p^^♯^(·)(ℝ^d^), where $ {\textstyle {1 \over {p ^{\sharp} (x)}} = {1 \over {p(x)}} - {\alpha \over d}} $. If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to $ \tilde p $ on ℝ^d^ such that there exists a bounded linear extension operator ℰ : W^1,p(·)^(Ω) ↪ $ W^{1, {\tilde p}} $(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,p(·)^(ℝ^d^) ↪L^p*(·)^(R^d^) with $ {\textstyle {1 \over {p ^{\ast} (x)}} = {1 \over {p(x)}} - {k \over d}} $ and W^1,p(·)^(Ω) ↪ L^p*(·)^(Ω) for k = 1. We show compactness of the embeddings W^1,p(·)^(Ω) ↪ L^q(·)^(Ω), whenever q(x) ≤ p*(x) – ε for some ε > 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)