Bessel Potentials with Logarithmic Components and Sobolev-Type Embeddings

Abstract

In a recent paper Edmunds, Gurka, and Opic [5] showed that Sobolev spaces of order k, based on the Zygmund spaces L n/k (log L)α (R n ), are continuously embedded into L ∞ (R n ) if α > 1/p′, p n/k. In this paper we replace L n/k (log L)α (R n ) by the Lebesgue space L n/k (R n ) and increase the smoothness of the functions involved by a "logarithmic" order α > 1/p′ to obtain the continuous embedding into L ∞ (R n ). Both approaches turn out to be equivalent. We also derive results of Trudinger-type [16] on embeddings into Orlicz spaces in the limit case k = n/p as well as results of Brezis-Wainger-type [2] on almost Lipschitz continuity in the superlimiting case k = n/p + 1.