A new function space and applications

Bourgain, Jean, Brezis, Haïm, and Mironescu, Petru. "A new function space and applications." Journal of the European Mathematical Society 017.9 (2015): 2083-2101. <http://eudml.org/doc/277289>.

@article{Bourgain2015,
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.},
author = {Bourgain, Jean, Brezis, Haïm, Mironescu, Petru},
journal = {Journal of the European Mathematical Society},
keywords = {BMO; VMO; Sobolev spaces; integer-valued functions; constant function; isoperimetric inequality; $\mathrm \{BMO\}$; $\mathrm \{VMO\}$; $\mathrm \{BV\}$; Sobolev spaces; integer-valued functions; constant functions; isoperimetric inequality},
language = {eng},
number = {9},
pages = {2083-2101},
publisher = {European Mathematical Society Publishing House},
title = {A new function space and applications},
url = {http://eudml.org/doc/277289},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Bourgain, Jean
AU - Brezis, Haïm
AU - Mironescu, Petru
TI - A new function space and applications
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 9
SP - 2083
EP - 2101
AB - 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.
LA - eng
KW - BMO; VMO; Sobolev spaces; integer-valued functions; constant function; isoperimetric inequality; $\mathrm {BMO}$; $\mathrm {VMO}$; $\mathrm {BV}$; Sobolev spaces; integer-valued functions; constant functions; isoperimetric inequality
UR - http://eudml.org/doc/277289
ER -