Commutators with fractional integral operators

Irina Holmes¹ ; Robert Rahm² ; Scott Spencer¹

¹School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160, U.S.A.
²School of Mathematics Washington University in St. Louis One Brookings Drive St. Louis, MO 63130, U.S.A.

Studia Mathematica, Tome 233 (2016) no. 3, pp. 279-291

Résumé

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in A_{p,q}$ and $\alpha /n+1/q=1/p$, the norm $\|[b,I_\alpha ]:L^p(\mu ^p)\to L^q(\lambda ^q)\| $ is equivalent to the norm of $b$ in the weighted BMO space ${\rm BMO}(\nu )$, where $\nu =\mu \lambda ^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

Détail

DOI : 10.4064/sm8419-4-2016

Keywords: investigate weighted norm inequalities commutator fractional integral operator multiplication function particular lambda alpha norm alpha lambda equivalent norm nbsp weighted bmo space bmo where lambda work extends results topic existing literature continues line investigation which initiated bloom recently developed further first author lacey wick