Further characterizations of Sobolev spaces

Nguyen, Hoai-Minh. "Further characterizations of Sobolev spaces." Journal of the European Mathematical Society 010.1 (2008): 191-229. <http://eudml.org/doc/277478>.

@article{Nguyen2008,
abstract = {Let $(F_n)_\{n\in \mathbb \{N\}\}$ be a sequence of non-decreasing functions from $[0,+\infty )$ into $[0,+\infty )$. Under some suitable hypotheses of $(F_n)_\{n\in \mathbb \{N\}\}$, we will prove that if $g\in L^p(\mathbb \{R\}^N)$, $1<p<+\infty $, satisfies $\liminf _\{n\rightarrow \infty \}\int _\{\mathbb \{R\}^N\}\int _\{\mathbb \{R\}^N\}F_n(|g(x)-g(y)|)/|x-y|^\{N+p\}dxdy<+\infty $, then $g\in W^\{1,p\}(\mathbb \{R\}^N)$ and moreover $\lim _\{n\rightarrow \infty \}\int _\{\mathbb \{R\}^N\}\int _\{\mathbb \{R\}^N\}F_n(|g(x)-g(y)|)/|x-y|^\{N+p\}dxdy=K_\{N,p\}\int _\{\mathbb \{R\}^N\}|\nabla g(x)|^pdx$, where $K_\{N,p\}$ is a positive constant depending only on $N$ and $p$. This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. I 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some partial results concerning the case $p=1$ and various open problems.},
author = {Nguyen, Hoai-Minh},
journal = {Journal of the European Mathematical Society},
keywords = {Sobolev spaces},
language = {eng},
number = {1},
pages = {191-229},
publisher = {European Mathematical Society Publishing House},
title = {Further characterizations of Sobolev spaces},
url = {http://eudml.org/doc/277478},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Nguyen, Hoai-Minh
TI - Further characterizations of Sobolev spaces
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 1
SP - 191
EP - 229
AB - Let $(F_n)_{n\in \mathbb {N}}$ be a sequence of non-decreasing functions from $[0,+\infty )$ into $[0,+\infty )$. Under some suitable hypotheses of $(F_n)_{n\in \mathbb {N}}$, we will prove that if $g\in L^p(\mathbb {R}^N)$, $1<p<+\infty $, satisfies $\liminf _{n\rightarrow \infty }\int _{\mathbb {R}^N}\int _{\mathbb {R}^N}F_n(|g(x)-g(y)|)/|x-y|^{N+p}dxdy<+\infty $, then $g\in W^{1,p}(\mathbb {R}^N)$ and moreover $\lim _{n\rightarrow \infty }\int _{\mathbb {R}^N}\int _{\mathbb {R}^N}F_n(|g(x)-g(y)|)/|x-y|^{N+p}dxdy=K_{N,p}\int _{\mathbb {R}^N}|\nabla g(x)|^pdx$, where $K_{N,p}$ is a positive constant depending only on $N$ and $p$. This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. I 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some partial results concerning the case $p=1$ and various open problems.
LA - eng
KW - Sobolev spaces
UR - http://eudml.org/doc/277478
ER -