Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces

Youssfi, Abdellah. "Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces." Annales de l'institut Fourier 45.3 (1995): 795-807. <http://eudml.org/doc/75138>.

@article{Youssfi1995,
abstract = {In this paper we consider the regularity problem for the commutators $([b,R_k])_\{1\le k\le n\}$ where $b$ is a locally integrable function and $(R_j)_\{1\le j\le n\}$ are the Riesz transforms in the $n$-dimensional euclidean space $\{\Bbb R\}^n$. More precisely, we prove that these commutators $([b,R_k])_\{1\le k\le n\}$ are bounded from $L^p$ into the Besov space $\dot\{B\}_p^\{s,p\}$ for $1&lt; p&lt; +\infty $ and $0&lt; s&lt; 1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space $\dot\{F\}_\infty ^\{s,p\}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $([b,R_k])_\{1\le k\le n\}$ are bounded form $L^2$ into the Sobolev space $\dot\{H\}^s$ if and only if $b$ is in the $BMO$-Sobolev space $\dot\{F\}_\infty ^\{s,2\}$.},
author = {Youssfi, Abdellah},
journal = {Annales de l'institut Fourier},
keywords = {regularity; commutators; Riesz transforms; Besov space; BMO-Triebel- Lizorkin space; Sobolov space; BMO-Sobolev space},
language = {eng},
number = {3},
pages = {795-807},
publisher = {Association des Annales de l'Institut Fourier},
title = {Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces},
url = {http://eudml.org/doc/75138},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Youssfi, Abdellah
TI - Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 3
SP - 795
EP - 807
AB - In this paper we consider the regularity problem for the commutators $([b,R_k])_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j)_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${\Bbb R}^n$. More precisely, we prove that these commutators $([b,R_k])_{1\le k\le n}$ are bounded from $L^p$ into the Besov space $\dot{B}_p^{s,p}$ for $1&lt; p&lt; +\infty $ and $0&lt; s&lt; 1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space $\dot{F}_\infty ^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $([b,R_k])_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space $\dot{H}^s$ if and only if $b$ is in the $BMO$-Sobolev space $\dot{F}_\infty ^{s,2}$.
LA - eng
KW - regularity; commutators; Riesz transforms; Besov space; BMO-Triebel- Lizorkin space; Sobolov space; BMO-Sobolev space
UR - http://eudml.org/doc/75138
ER -