Embeddings for spaces of maximal regularity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:40:27Z http://mathoverflow.net/feeds/question/70894 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70894/embeddings-for-spaces-of-maximal-regularity Embeddings for spaces of maximal regularity Marc 2011-07-21T11:22:40Z 2011-08-26T20:41:51Z <p>Let $T\in(0,\infty)$ and $\Omega\subset\mathbb R^n$ be a smooth domain. In terms of maximal regularity it can be very beneficial to know for which $s_i,p,n$ the following holds true</p> <p>$W^{s_1,p}(0,T;L^p(\Omega))\cap L^p(0,T;W^{s_2,p}(\Omega))\cdot W^{s_3,p}(0,T;L^p(\Omega))\cap L^p(W^{s_4,p}(\Omega))\subset$</p> <p>$\qquad\qquad\qquad\qquad\qquad W^{t_1,p}(0,T;L^p(\Omega))\cap L^p(W^{t_2,p}(\Omega)),\quad t_1,t_2\in(0,1)$</p> <p>In this respect I'm interested in sharp choices of $s,p,n\in\mathbb R_+^4\times(1,\infty)\times \mathbb N$. However due to very complicated Sobolev-Slobodeckii norms the calculations are quite involved, so I was wondering if anybody is aware of a reference in the literature for this problem.</p> http://mathoverflow.net/questions/70894/embeddings-for-spaces-of-maximal-regularity/71069#71069 Answer by lvb for Embeddings for spaces of maximal regularity lvb 2011-07-23T14:12:33Z 2011-08-26T20:41:51Z <p>One possible approach to prove embeddings of a similar kind is provided by Sobolevskii's <em>Mixed Derivative Theorem</em>, see for instance Denk, Saal, Seiler, <em>Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity.</em> (MR2410829), Lemma 4.1, or Denk, Hieber, Prüss, <em>Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data</em>, Proposition 3.2 for two different versions of the theorem.</p> <p>Applying the Mixed Derivative Theorem to suitable powers of the operators $1+\partial_t$ and $1-\Delta$ on $L^p(\mathbb{R};L^p(\mathbb{R}^d))$ with $p\in(1,\infty)$ and using interpolation in their Lemma 4.3 Denk, Saal and Seiler prove among others the embedding</p> <p>$$W^{s_1,p}(\mathbb{R};L^p(\mathbb{R}^d))\cap L^p(\mathbb{R};W^{s_2,p}(\mathbb{R}^d))\hookrightarrow W^{s_1 \kappa,p}(\mathbb{R};W^{s_2 (1-\kappa),p}(\mathbb{R}^d))$$</p> <p>for $0\leq\kappa\leq 1$ and $s_1,s_2\geq 0$. Now applying embedding theorems for pointwise products of functions in Sobolev spaces (e.g. Thm. 4.6.1.1 of Runst, Sickel, <em>Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations</em> (MR1419319)) one can prove a corresponding result for your problem, at least on the full space. Using extension/restriction one should be able to deal with the case where $\Omega$ is a domain.</p>