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Let $X,Y,Z$ be Banach spaces such that $X,Y\Subset Z$. We know that if $X\Subset Y$ (The symbol "$\Subset$" means continuous embedding), then $(Z,X)_{\theta,p}\Subset (Z,Y)_{\theta,p}$ for all $\theta\in (0,1)$ and $p\in[1,\infty]$. How about the reverse results? Let us say, if $(Z,X)_{\theta,p}\Subset (Z,Y)_{\theta,p}$ for some $\theta\in (0,1)$ and $p\in (1,\infty)$, then what can we say about the inclusion relations about $X$ and $Y$? Can one show that $X\Subset Y$. If not, is it possible to make it valid under some additional requirements? References are appriciated.

Let $X,Y,Z$ be Banach spaces such that $X,Y\hookrightarrow Z$. We know that if $X\Subset Y$ (The symbol "$\hookrightarrow $" means continuous embedding), then $(Z,X)_{\theta,p}\hookrightarrow (Z,Y)_{\theta,p}$ for all $\theta\in (0,1)$ and $p\in[1,\infty]$. How about the reverse results? Let us say, if $(Z,X)_{\theta,p}\hookrightarrow (Z,Y)_{\theta,p}$ for some $\theta\in (0,1)$ and $p\in (1,\infty)$, then what can we say about the inclusion relations about $X$ and $Y$? Can one show that $X\hookrightarrow Y$. If not, is it possible to make it valid under some additional requirements? References are appriciated.

Let $X,Y,Z$ be Banach spaces such that $X,Y\Subset Z$. We know that if $X\Subset Y$ (The symbol "$\Subset$" means continuous embedding), then $(Z,X)_{\theta,p}\Subset (Z,Y)_{\theta,p}$ for all $\theta\in (0,1)$ and $p\in[1,\infty]$. How about the reverse results? Let us say, if $(Z,X)_{\theta,p}=(Z,Y)_{\theta,p}$ for some $\theta\in (0,1)$ and $p\in (1,\infty)$, then what can we say about the inclusion relations about $X$ and $Y$? Can one show that $X\Subset Y$. If not, is it possible to make it valid under some additional requirements? References are appriciated.

Let $X,Y,Z$ be Banach spaces such that $X,Y\Subset Z$. We know that if $X\Subset Y$ (The symbol "$\Subset$" means continuous embedding), then $(Z,X)_{\theta,p}\Subset (Z,Y)_{\theta,p}$ for all $\theta\in (0,1)$ and $p\in[1,\infty]$. How about the reverse results? Let us say, if $(Z,X)_{\theta,p}\Subset (Z,Y)_{\theta,p}$ for some $\theta\in (0,1)$ and $p\in (1,\infty)$, then what can we say about the inclusion relations about $X$ and $Y$? Can one show that $X\Subset Y$. If not, is it possible to make it valid under some additional requirements? References are appriciated.

Let $X,Y,Z$ be Banach spaces such that $X,Y\Subset Z$. We know that if $X\Subset Y$, then $(Z,X)_{\theta,p}\Subset (Z,Y)_{\theta,p}$ for all $\theta\in (0,1)$ and $p\in[1,\infty]$. How about the reverse results? Let us say, if $(Z,X)_{\theta,p}=(Z,Y)_{\theta,p}$ for some $\theta\in (0,1)$ and $p\in (1,\infty)$, then what can we say about the inclusion relations about $X$ and $Y$? Can one show that $X\Subset Y$. If not, is it possible to make it valid under some additional requirements? References are appriciated.

Let $X,Y,Z$ be Banach spaces such that $X,Y\Subset Z$. We know that if $X\Subset Y$ (The symbol "$\Subset$" means continuous embedding), then $(Z,X)_{\theta,p}\Subset (Z,Y)_{\theta,p}$ for all $\theta\in (0,1)$ and $p\in[1,\infty]$. How about the reverse results? Let us say, if $(Z,X)_{\theta,p}=(Z,Y)_{\theta,p}$ for some $\theta\in (0,1)$ and $p\in (1,\infty)$, then what can we say about the inclusion relations about $X$ and $Y$? Can one show that $X\Subset Y$. If not, is it possible to make it valid under some additional requirements? References are appriciated.