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I am looking for a reference or a direct argument that shows the real interpolation space between $L_p$ and $B^s_{q,q}$ is $B^\alpha_{r,r}$, with the usual conditions on the indices. This result is stated in many places, and they refer to, e.g., a paper of DeVore and Popov on interpolation of Besov spaces, but this paper is concerned with only interpolation between two Besov spaces. When I try to replace one of the Besov spaces by an $L_p$ space, I run into the problem of writing $L_p$ as an $\ell_q^s(L_p)$-type space. I might be missing some simple argument. I would really appreciate if you give me some pointer.

Update: Below we have a couple of good answers. Thank you! However, I was hoping to see a more direct proof that does not go through Triebel-Lizorkin spaces. An approximation theory approach would be ideal for me. My real purpose is to have a similar interpolation result for certain approximation spaces, and Triebel-Lizorkin spaces do not seem to have analogues in approximation setting. If I have to, I will try to translate the proof through Triebel-Lizrokin into an approximation setting, but I wanted to check if a standard trick already exists in the literature. I apologize for not making it clear the first time.

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  • $\begingroup$ I suspect that this can be derived using the reiteration theorem for interpolation. $\endgroup$
    – Dunham
    Commented Aug 19, 2014 at 16:37
  • $\begingroup$ @Dunham: Reiteration approach seems to require writing $L_p$ as an interpolation space between Besov spaces, which I think is not possible (By interpolation you cannot get out of the Besov scale). $\endgroup$
    – timur
    Commented Aug 20, 2014 at 0:25
  • $\begingroup$ I haven't checked this carefully, but perhaps one could simply use the inclusions $B^0_{p,1} \subset L_p \subset B^0_{p,\infty}$ together with Besov space interpolation? $\endgroup$
    – Terry Tao
    Commented Aug 20, 2014 at 0:43
  • $\begingroup$ @TerryTao: I think in order for $[B^0_{p,t},B^s_{q,q}]_{\theta,r}=B^\alpha_{r,r}$ to happen we must have $t=p$. $\endgroup$
    – timur
    Commented Aug 20, 2014 at 1:10
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    $\begingroup$ Generally the situation with $p \leq 1$ is quite pathological, particularly the non-convex, non-locally integrable case $p<1$. Things are better if one uses Hardy spaces instead of Lebesgue spaces though. $\endgroup$
    – Terry Tao
    Commented Aug 23, 2014 at 2:48

2 Answers 2

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See Equation (8.33) of Triebel, Spaces of distributions of Besov type on Euclidean n-space. Duality, interpolation, 1973. (MR0348483)

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Theorem 8 on page 53 in these lecture notes gives a number of interpolation results for Besov and Triebel spaces. The results are partially proven, and a citation is given for the rest (Triebel's book Interpolation theory, function spaces, di fferential operators). Since $L^p=F^0_{p2}$ and $B^s_{qq}=F^s_{qq}$, the first equality of part (b) is what you need.

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  • $\begingroup$ These are all good answers, but I was hoping to see a more direct proof that does not go through the Triebel-Lizorkin spaces, for instance, by approximation theory. $\endgroup$
    – timur
    Commented Aug 20, 2014 at 0:17
  • $\begingroup$ Has there been any update on this question? I am interested in the real interpolation between $L^p(\mathbb{R}^N)$ and $W^{1,q}(\mathbb{R}^N)$, again without using the Triebel-Lizorkin spaces. Thanks! $\endgroup$
    – Gio67
    Commented Jan 2, 2021 at 16:42
  • $\begingroup$ @Gio67 I recommend asking a separate new question and linking to this one. $\endgroup$
    – Joonas Ilmavirta
    Commented Jan 2, 2021 at 17:00
  • $\begingroup$ I think that the book by Besov, Il'in, and Nikol'ski, "Integral representations of functions and imbedding theorems." Vol. II. has what I want. They deal with anisotropic Besov and Sobolev spaces and allow the functions and their derivatives to belong to different $L^p$ spaces. It's just excruciating to read. $\endgroup$
    – Gio67
    Commented Jan 6, 2021 at 2:36

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