Space between $L^1$ and $BV$?

Question

I am looking for a function space $X_s$ such that this space has following properties:

1. $X_s$ is a Banach space, and has lower semi-continuous properties with respect to $L^p$ strong convergence.

2. I want $BV\subset X_s\subset L^1$, where $BV$ is the bounded variation space defined in Evans and Gariepy, i.e., $$ BV(\Omega):=\{u\in L^1(\Omega),\,TV(u,\Omega)<\infty\} $$ where $TV$ denote the total variation.

3. I want $0<s<1$ service as some sort of indictor so that as $s\to 0$, $X_s\to L^1$ in "some" sense; $s\to 1$ and $X_s\to BV$ in some sense.

I am thinking Besov space. Would this be a good solution? Any other function space you would recommend? What about BMO space?

Thank you!

Answer 1

It seems that what you're looking for is an interpolation space : https://en.wikipedia.org/wiki/Interpolation_space

I don't remember if these interpolation spaces between $BV$ and $L^1$ are "usual" spaces or not