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Journal Articles J. Eur. Math. Soc. (JEMS) 6 Year : 2015

A new function space and applications


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.
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hal-00992422 , version 1 (17-05-2014)


  • HAL Id : hal-00992422 , version 1


Jean Bourgain, Haïm Brezis, Petru Mironescu. A new function space and applications. J. Eur. Math. Soc. (JEMS) 6, 2015, 17 (9), pp.2083--2101. ⟨hal-00992422⟩
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