A new function space and applications

Abstract

We define a new function space $B$, which contains in particular BMO, BV and $W^{1/p,p}$, $p\in (1,\infty)$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $p\in [1,\infty)$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a ''common roof'' to various, seemingly unrelated, statements asserting that integer-valued functions satisfying some kind of regularity condition must be constant.