Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski. "Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces." Bulletin of the Polish Academy of Sciences. Mathematics 59.1 (2011): 65-75. <http://eudml.org/doc/286291>.

@article{B2011,
abstract = {For a function $f ∈ L_\{loc\}^\{p\}(ℝⁿ)$ the notion of p-mean variation of order 1, $₁^\{p\}(f,ℝⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^\{1,p\}(ℝⁿ)$ in terms of $₁^\{p\}(f,ℝⁿ)$ is directly related to the characterisation of $W^\{1,p\}(ℝⁿ)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.},
author = {B. Bojarski},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Sobolev spaces; mean variation},
language = {eng},
number = {1},
pages = {65-75},
title = {Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces},
url = {http://eudml.org/doc/286291},
volume = {59},
year = {2011},
}

TY - JOUR
AU - B. Bojarski
TI - Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2011
VL - 59
IS - 1
SP - 65
EP - 75
AB - For a function $f ∈ L_{loc}^{p}(ℝⁿ)$ the notion of p-mean variation of order 1, $₁^{p}(f,ℝⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1,p}(ℝⁿ)$ in terms of $₁^{p}(f,ℝⁿ)$ is directly related to the characterisation of $W^{1,p}(ℝⁿ)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.
LA - eng
KW - Sobolev spaces; mean variation
UR - http://eudml.org/doc/286291
ER -