Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron Kerman ¹   ; Luboš Pick ²

¹ Department of Mathematics Brock University St. Catharines, Ontario Canada L2S 3A1
² Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovská 83 186%nbsp;75 Praha 8, Czech Republic

Studia Mathematica, Tome 206 (2011) no. 2, pp. 97-119

Résumé

We study imbeddings of the Sobolev space $$ W^{m,\varrho}({\mit\Omega}):= \{u:{\mit\Omega}\to\mathbb {R}\ \hbox{with}\ \varrho (\partial^{\alpha}u/ \partial x^{\alpha})\infty\ \text{when}\ |\alpha|\leq m\}, $$ in which ${\mit\Omega}$ is a%nbsp;bounded Lipschitz domain in $\mathbb R^{n}$, $\varrho$ is a%nbsp;rearrangement-invariant (r.i.) norm and $1\leq m\leq n-1$. For such a%nbsp;space we have shown there exist r.i.%nbsp;norms, $\tau_\varrho$ and $\sigma_\varrho$, that are optimal with respect to the inclusions $$ W^{m,\varrho}({\mit\Omega})\subset W^{m,\tau_\varrho}({\mit\Omega})\subset L_{\sigma_\varrho}({\mit\Omega}). $$ General formulas for $\tau_{\varrho}$ and $\sigma_{\varrho}$ are obtained using the $\mathcal K$-method of interpolation. These lead to explicit expressions when $\varrho$ is a%nbsp;Lorentz Gamma norm or an%nbsp;Orlicz norm.

Détail

DOI : 10.4064/sm206-2-1

Keywords: study imbeddings sobolev space varrho mit omega mit omega mathbb hbox varrho partial alpha partial alpha infty text alpha leq which mit omega nbsp bounded lipschitz domain mathbb varrho nbsp rearrangement invariant norm leq leq n nbsp space have shown there exist nbsp norms tau varrho sigma varrho optimal respect inclusions varrho mit omega subset tau varrho mit omega subset sigma varrho mit omega general formulas tau varrho sigma varrho obtained using mathcal k method interpolation these lead explicit expressions varrho nbsp lorentz gamma norm nbsp orlicz norm