Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,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 ) ≔ ∫ | 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 1 p ^ (x) = 1 p(x) - α d . If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to p on ℝ d such that there exists a bounded linear extension operator ℰ : W 1, p (·) (Ω) ↪ W^1, 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 1 p ^ (x) = 1 p(x) - k 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)