Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities
Tom 206 / 2011
Streszczenie
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.
