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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 extension/restriction, the product rule embedding theorems for pointwise products of functions in Sobolev spaces (e.g. Thm. 4.6.1.1 of Runst, Hölder Sickel, Sobolev spaces of fractional order, Nemytskij operators and if necessary interpolation nonlinear partial differential equations (MR1419319)) one should be able to 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.

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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 extension/restriction, the product rule for Sobolev spaces, Hölder and if necessary interpolation one should be able to prove a corresponding result for your problem.

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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 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 $s_1,s_2\geq 0$. Using Now applying extension/restriction, the product rule for Sobolev spaces and interpolation one should be able to prove a corresponding result for your problem.

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