# A new function space and applications

### Jean Bourgain

Institute for Advanced Study, Princeton, United States### Haïm Brezis

Rutgers University, Piscataway, United States### Petru Mironescu

Université Lyon 1, Villeurbanne, France

## Abstract

We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1 < p < \infty$. 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 $W^{1/p,p}$, $1 \le p < \infty$. 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.

## Cite this article

Jean Bourgain, Haïm Brezis, Petru Mironescu, A new function space and applications. J. Eur. Math. Soc. 17 (2015), no. 9, pp. 2083–2101

DOI 10.4171/JEMS/551