Journal of the European Mathematical Society

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Volume 17, Issue 9, 2015, pp. 2083–2101
DOI: 10.4171/JEMS/551

Published online: 2015-10-29

A new function space and applications

Jean Bourgain[1], Haïm Brezis[2] and Petru Mironescu[3]

(1) Institute for Advanced Study, Princeton, United States
(2) Rutgers University, Piscataway, United States
(3) Université Lyon 1, Villeurbanne, France

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.

Keywords: BMO, VMO, BV, Sobolev spaces, integer-valued functions, constant function, isoperimetric inequality

Bourgain Jean, Brezis Haïm, Mironescu Petru: A new function space and applications. J. Eur. Math. Soc. 17 (2015), 2083-2101. doi: 10.4171/JEMS/551