I am having trouble finding any results concerning real interpolation of vector-valued Sobolev spaces. Namely, I would like to know if a continuous embedding of the type,

$$ L^p(0,T;X_1)\cap W^{1,p}(0,T;X_0) \hookrightarrow \mathcal{C}([0,T]; (X_0, X_1)_{\theta, p}) $$

can hold for some $\theta \in (0,1)$, where $X_1$ and $X_0$ are two Banach spaces that are compatible for (real) interpolation and $T > 0$.

Any reference or anything I could grab on to takle this will be greatly appreciated. Thank you very much.

Disclaimer : I am not very familiar with interpolation so I would prefer rather accessible sources if possible.

EDIT : I change the ambitions of my question which involved an inclusion inside a vector-valued fractional Sobolev space which seemed a little bit two complex to first tackle.

  • $\begingroup$ How is the vector valued Sobolev space $W^{\theta,p}(0,T;X)$ defined? (I ask because the sequence space version of the interpolation you want $(\ell^{0}_q(X_1),\ell^{1}_q(X_2))_{\theta,q} = \ell^{\theta}_q((X_1, X_2)_{\theta,q})$ is true, and so if you have a Littlewood Paley type representation of the vector-valued Sobolev spaces the same way you do for the scalar ones, then the result should hold.) // Incidentally, the paper you linked to is about the non-diagonal case of the interpolation; the result that you desire is firmly in the diagonal case (all 4 occurrences of $p$ agree). $\endgroup$ Apr 6 at 13:23
  • $\begingroup$ That is a good question. I didn't really think about it until now. I think I would have naturally opted for the definition with a Fourier transform approach with the multiplier $\xi \mapsto (1+|\xi|^2)^{\theta/2}$. Are you refering to the so called Lizorkin-Triebel spaces $F^s_{p,q}$ ? Thank you for your remark, indeed i am in the diagonal case. $\endgroup$
    – Theleb
    Apr 6 at 14:40

1 Answer 1


The desired embedding is indeed correct for $\theta = 1-1/p$. This is a classical result in interpolation theory and the theory of evolution equations. See for example the book of Amann [2], Theorem III.4.10.2.$^1$

In fact, the real interpolation space $$X:= \bigl(X_0,X_1\bigr)_{1-1/p,p}$$ can be defined (equivalently to several other methods) as the space of point evaluations of zero of functions in the intersection space. This is called the (Lions-Peetre) trace method which you will find in nearly any book on interpolation theory. (For instance in Bergh/Löfström [1], Corolllary 3.12.3) From there, one uses e.g. continuity of shift semigroups to obtain the desired embedding.

If, as quite often when dealing with the spaces in the question, $X_1 \hookrightarrow X_0$, then the embedding is true for all $0 \leq \theta \leq 1-1/p$ by factoring through $\theta=1-1/p$ and a natural embedding of interpolation spaces. ("Take more of the bigger space $X_0$".) However, if you are content with $\theta < 1-1/p$, then you can in fact do better and even have Hölder-continuity in time of order $\alpha = 1-1/p-\theta$.

Since I also checked the previous version of your question, let me mention that results as asked there are indeed also known. A common name is mixed derivative theorem. (Which is unfortunate, since web searches will not be very effective.) But naturally they become much more involved considering noninteger differentiability scales. Right now I know that such results (and more or less rigorous proofs) are considered e.g. in another work of Amann [3] and by Denk and Kaip; maybe rather see the accessible version for $X_0 = L^p$ in [4], Proposition 4.3. But there are surely more.

[1] Bergh, Jöran; Löfström, Jörgen, Interpolation spaces. An introduction, Grundlehren der mathematischen Wissenschaften. 223. Berlin-Heidelberg-New York: Springer-Verlag.

[2] Amann, Herbert, Linear and quasilinear parabolic problems. Vol. 1: Abstract linear theory, Monographs in Mathematics. 89. Basel: Birkhäuser.

[3] Amann, Herbert, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat., III. Ser. 35, No. 1, 161-177 (2000).

[4] Tolksdorf, Patrick, On the $\mathrm {L}^p$-theory of the Navier-Stokes equations on three-dimensional bounded Lipschitz domains, Math. Ann. 371, No. 1-2, 445-460 (2018).

$^1$ I am not quite sure whether this qualifies as an accessible source, since Amann's works are notoriously hard to read, but I hope this particular result should be accessible enough..

  • 1
    $\begingroup$ Thank you very much for your detailed answer. The mentionned trace method seems to have a fundamental role in interpolation theory. I will look further into that. $\endgroup$
    – Theleb
    Apr 22 at 20:29

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