Commutators with fractional integral operators
Volume 233 / 2016
Studia Mathematica 233 (2016), 279-291
MSC: Primary 42A05, 42A50, 42B20; Secondary 42A61, 42B25.
DOI: 10.4064/sm8419-4-2016
Published online: 24 May 2016
Abstract
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.
