We define a new function space , which contains in particular BMO, BV, and , . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving norms of integer-valued functions in . We introduce a significant closed subspace, , of , containing in particular VMO and , . The above mentioned estimates imply in particular that integer-valued functions belonging to 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. 2083–2101DOI 10.4171/JEMS/551