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

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

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.

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.

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

$$ L^p(0,T;X_1)\cap W^{1,p}(0,T;X_0) \subset W^{\theta,p}(0,T; (X_0, X_1)_{\theta, p}) $$

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 $T > 0$. The closest I got in my research was this article https://www.ams.org/journals/proc/2005-133-06/S0002-9939-04-07714-7/S0002-9939-04-07714-7.pdf which solely tackles the problem in $L^p$ spaces and not for Sobolev spaces.

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 : The found article deals with the non-diagonal case, which really is not in the scope of my problem.

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

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

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.

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

$$ L^p(0,T;X_1)\cap W^{1,p}(0,T;X_0) \subset W^{\theta,p}(0,T; (X_0, X_1)_{\theta, p}) $$

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 $T > 0$. The closest I got in my research was this article https://www.ams.org/journals/proc/2005-133-06/S0002-9939-04-07714-7/S0002-9939-04-07714-7.pdf which solely tackles the problem in $L^p$ spaces and not for Sobolev spaces.

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.

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

$$ L^p(0,T;X_1)\cap W^{1,p}(0,T;X_0) \subset W^{\theta,p}(0,T; (X_0, X_1)_{\theta, p}) $$

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 $T > 0$. The closest I got in my research was this article https://www.ams.org/journals/proc/2005-133-06/S0002-9939-04-07714-7/S0002-9939-04-07714-7.pdf which solely tackles the problem in $L^p$ spaces and not for Sobolev spaces.

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 : The found article deals with the non-diagonal case, which really is not in the scope of my problem.