Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces <i>L</i>^<i>p</i>(·) and <i>W</i>^{<i>k,p</i>(·)}

<jats:title>Abstract</jats:title><jats:p>We study the Riesz potentials <jats:italic>I<jats:sub>α</jats:sub>f</jats:italic> on the generalized Lebesgue spaces <jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic>(·)</jats:sup>(ℝ<jats:sup><jats:italic>d</jats:italic></jats:sup>), where 0 < <jats:italic>α</jats:italic> < <jats:italic>d</jats:italic> and <jats:italic>I<jats:sub>α</jats:sub>f</jats:italic>(<jats:italic>x</jats:italic>) ≔ ∫<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="equation/tex2gif-inf-3.gif" xlink:title="urn:x-wiley:0025584X:media:MANA200310157:tex2gif-inf-3" /> |<jats:italic>f</jats:italic>(<jats:italic>y</jats:italic>)| |<jats:italic>x</jats:italic> – <jats:italic>y</jats:italic>|<jats:sup><jats:italic>α</jats:italic> – <jats:italic>d</jats:italic></jats:sup> <jats:italic>dy</jats:italic>. Under the assumptions that <jats:italic>p</jats:italic> locally satisfies |<jats:italic>p</jats:italic>(<jats:italic>x</jats:italic>) – <jats:italic>p</jats:italic>(<jats:italic>x</jats:italic>)| ≤ <jats:italic>C</jats:italic>/(– ln |<jats:italic>x</jats:italic> – <jats:italic>y</jats:italic>|) and is constant outside some large ball, we prove that <jats:italic>I<jats:sub>α</jats:sub></jats:italic> : <jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic>(·)</jats:sup>(ℝ<jats:sup><jats:italic>d</jats:italic></jats:sup>) → <jats:italic>L<jats:sup>p</jats:sup><jats:sup>♯</jats:sup>(·)</jats:italic>(ℝ<jats:sup><jats:italic>d</jats:italic></jats:sup>), where <jats:styled-content>$ {\textstyle {1 \over {p ^{\sharp} (x)}} = {1 \over {p(x)}} - {\alpha \over d}} $</jats:styled-content>. If <jats:italic>p</jats:italic> is given only on a bounded domain Ω with Lipschitz boundary we show how to extend <jats:italic>p</jats:italic> to <jats:styled-content>$ \tilde p $</jats:styled-content> on ℝ<jats:sup><jats:italic>d</jats:italic></jats:sup> such that there exists a bounded linear extension operator ℰ : <jats:italic>W</jats:italic><jats:sup>1,<jats:italic>p</jats:italic>(·)</jats:sup>(Ω) ↪ <jats:styled-content>$ W^{1, {\tilde p}} $</jats:styled-content>(ℝ<jats:sup><jats:italic>d</jats:italic></jats:sup>), while the bounds and the continuity condition of <jats:italic>p</jats:italic> are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings <jats:italic>W</jats:italic><jats:sup><jats:italic>k,p</jats:italic>(·)</jats:sup>(ℝ<jats:sup><jats:italic>d</jats:italic></jats:sup>) ↪<jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic>*(·)</jats:sup>(R<jats:sup><jats:italic>d</jats:italic></jats:sup>) with <jats:styled-content>$ {\textstyle {1 \over {p ^{\ast} (x)}} = {1 \over {p(x)}} - {k \over d}} $</jats:styled-content> and <jats:italic>W</jats:italic><jats:sup>1,<jats:italic>p</jats:italic>(·)</jats:sup>(Ω) ↪ <jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic>*(·)</jats:sup>(Ω) for <jats:italic>k</jats:italic> = 1. We show compactness of the embeddings <jats:italic>W</jats:italic><jats:sup>1,<jats:italic>p</jats:italic>(·)</jats:sup>(Ω) ↪ <jats:italic>L</jats:italic><jats:sup><jats:italic>q</jats:italic>(·)</jats:sup>(Ω), whenever <jats:italic>q</jats:italic>(<jats:italic>x</jats:italic>) ≤ <jats:italic>p</jats:italic>*(<jats:italic>x</jats:italic>) – <jats:italic>ε</jats:italic> for some <jats:italic>ε</jats:italic> > 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)</jats:p>