Regularity properties of commutators and BMO-Triebel-Lizorkin spaces

摘要

In this paper we consider the regularity problem for the commutators ([b,R_k])_(1≤k≤n) where b is a locally integrable function and (R_j)_(1≤j≤n) are the Riesz transforms in the n-dimensional euclidean space R~n. More precisely, we prove that these commutators ([b, R_k])_(1≤k≤n) are bounded from L~p into the Besov space B_p~(s,p) for 1 < p < +∞ and 0 < s < 1 if and only if b is in the BMO-Triebel-Lizorkin space F_∞~(s,p) The reduction of our result to the case p = 2 gives in particular that the commutators ([b, R_k])_(1≤k≤n) are bounded form L~2 into the Sobolev space H~s if and only if b is in the BMO-Sobolev space F_∞~(s,2).%Nous nous intéressons à la régularité des commutateurs ([b, R_k])_(1≤k ≤n) ou b est une fonction localement intégrable et (R_j)_(i≤j≤n) dé signent les transformées de Riesz. Nous montrons que si 1 < p < +∞ et 0 < s < 1, alors les commutateurs ([b, R_k])_(1≤k≤n_ sont continus de L~p(R~n) dans l'espace de Besov B_p~(s,p) si et seulement si b appartient à l'espace BMO-Triebel-Lizorkin F_∞ (s,p). En particulier, si p = 2, les commutateurs ([b,R_k])1≤k