Embeddings for spaces of maximal regularity

Question

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

$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 $

$\qquad\qquad\qquad\qquad\qquad W^{t_1,p}(0,T;L^p(\Omega))\cap L^p(W^{t_2,p}(\Omega)),\quad t_1,t_2\in(0,1)$

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.

Answer 1

One possible approach to prove embeddings of a similar kind is provided by Sobolevskii's Mixed Derivative Theorem, see for instance Denk, Saal, Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity. (MR2410829), Lemma 4.1, or Denk, Hieber, Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Proposition 3.2 for two different versions of the theorem.

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

$$W^{s_1,p}(\mathbb{R};L^p(\mathbb{R}^d))\cap L^p(\mathbb{R};W^{s_2,p}(\mathbb{R}^d))\hookrightarrow W^{s_1 \kappa,p}(\mathbb{R};W^{s_2 (1-\kappa),p}(\mathbb{R}^d))$$

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, Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations (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.