Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, for $1 < p \le +\infty,$

$BV_p (\mathbb{R}) = \{ f:\mathbb{R}\to \mathbb{R}; \|f\|_{\infty} + \|f'\|_p < +\infty\},$

with the case $p=1$ being defined as

$BV(\mathbb{R}) = \{f: \mathbb{R} \to \mathbb{R}; \|f\|_{\infty} + \|df\|_{TV} < +\infty\},$

where $\|d\mu \|_{TV} = \int_{R} |d\mu|$ stands for the total variation of the measure $d\mu$. This concepts being defined, I was interested in a Marcinkiewicz-like interpolation theorem for those spaces:

"Let $T$ be an operator defined on some 'large' subset of measurable functions, $T$ sublinear. If $T$ preserves $BV_p$ and $BV_q$, then it should preserve all spaces $BV_s$, $p\le s \le q.$"

Well, I don't really know if such a statement could even hold, but any theorem that gives me some 'general' conditions on $T$ would be of use here; note also that I do not require $T$ to be linear, and so I would need any result of this kind to hold for sublinear operators, just like the Marcinkiewicz interpolation theorem.

I have searched a bit looking for something related to that, and the closer I could get was a 1979 paper by DeVore and Scherer where they find explicit bounds for the interpolating functional between Sobolev Spaces, and in the end they state a few interpolation results, but only for linear operators.

Unfortunately, I did not check the paper too thoroughly, and my knowledge of this is way too limited, so if anyone with a bit more experience could tell me if their results imply, for instance, BV-spaces results, or even if the techniques can be adapted straightforwardly, I would be really thankful!

I would also appreciate if anyone could tell me if any theorem of an 'interpolation kind' could hold in the context I mentioned, as it could be of importance for me.


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