I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?

Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider the space $$W=L^2(I;H^2(\Omega))\cap H^1(I;L^2(\Omega))=\{u\in L^2(I;H^2(\Omega))\mid u_t\in L^2(I;L^2(\Omega))\}. $$ Then is it possible to find some $r>2$, such that $W\hookrightarrow L^r(I;W^{1,r}(\Omega))$ continuously?


This should work according to the embedding $$W \hookrightarrow L^{\frac2{1-2s}}\bigl((L^2(\Omega),H^2(\Omega))_{\theta,1}\bigr)$$ where $0 < s < \frac12$ and $0 \leq \theta < 1-s$ as in Amann: Linear parabolic problems involving measures, Theorem 3.

To elaborate a bit more: By a relation between real and complex interpolation spaces, interpolation identitites for Bessel potential spaces and the supposed smoothness of $\Omega$, $$\bigl(L^2(\Omega),H^2(\Omega)\bigr)_{\theta,1} \hookrightarrow \bigl[L^2(\Omega),H^2(\Omega)\bigr]_\theta = H^{2\theta}(\Omega),$$ and $H^{2\theta}(\Omega) \hookrightarrow W^{1,r}(\Omega)$ where $2\theta - \frac{n}2 \geq 1-\frac{n}{r}$. (The standard textbooks by Triebel, there Thms. and and 4.6.2, and Bergh/Löfström contain this.) Now it remains to connect $r$ and the integrability $2/(1-2s)$ from the above embedding.

(My calculations gave that your desired embedding is correct for all $r \in [2,2+\frac4n)$ which corresponds to $s < \frac1{2+n}$ and $\theta = \frac12(ns+1)$, but of course you should double-check that..)

  • $\begingroup$ Thank you for pointing out the reference! I am not quite familiar with the interpolation space. May you kindly suggest a reference about the connection between $(L^2(\Omega),H^2(\Omega))_{\theta,1}$ and the classical Sobolev space? Wiki only states that for $s\in(k,k+1)$, $W^{s,p}(\Omega)=(W^{k,p}(\Omega),W^{k+1,p}(\Omega))_{\theta,p}$ for certain $\theta$. $\endgroup$ – John Jun 27 '19 at 0:53
  • $\begingroup$ @John I added some explanations, let me know if you need more. $\endgroup$ – Hannes Jun 27 '19 at 8:17
  • $\begingroup$ I found in Adams' book that $B^{2\theta}_{2,q}(\Omega)=(L^2(\Omega),W^{2}_{2}(\Omega))_{\theta,q}$ for $q\ge 1$. So essentially, the relation between real and complex interpolation spaces suggests that $B^{2\theta}_{2,1}(\Omega)\hookrightarrow H^{2\theta}(\Omega)$. Is it correct? $\endgroup$ – John Jun 27 '19 at 12:10
  • $\begingroup$ The above embedding is mentioned in Adams book for $\theta$ being an integer, i.e., $B^m_{p,1}(\Omega)\hookrightarrow W^m_{p}(\Omega)$, and the relation between real and complex interpolation spaces suggests it is true for arbitrary real $\theta\in (0,\infty)$. Is my understanding correct? $\endgroup$ – John Jun 27 '19 at 12:18
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    $\begingroup$ Yep, that's correct. (Watch out that $\theta \in (0,1)$ only though in the interpolation context! You'd need to replace $H^2$ by $H^k$ for $k$ large enough to get the embedding for all, also noninteger, smoothness parameters.) $\endgroup$ – Hannes Jun 27 '19 at 13:01

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