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timur
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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.

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.

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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timur
timur
  • 3.3k
  • 1
  • 36
  • 42

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 thatof writing $L_p$ spaces are not of the correctas 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.

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 that $L_p$ spaces are not of the correct $\ell_q^s(L_p)$-type. I might be missing some simple argument. I would really appreciate if you give me some pointer.

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.

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timur
timur
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  • 36
  • 42

Interpolation between $L_p$ and $B^s_{q,q}$

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 that $L_p$ spaces are not of the correct $\ell_q^s(L_p)$-type. I might be missing some simple argument. I would really appreciate if you give me some pointer.