Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p (·) and W k,p (·)

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 ) ≔ ∫   ℝ   d| 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)