# A new function space and applications

Jean Bourgain; Haïm Brezis; Petru Mironescu

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 9, page 2083-2101
- ISSN: 1435-9855

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topBourgain, 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 -

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