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Multilinear commutators for fractional integrals in non-homogeneous spaces. (English) Zbl 1076.42009
Let $\mu$ be a positive Radon measure on $\Bbb R^d$ which satisfies the growth condition $\mu(B(x,r))\le Cr^n$ for all balls $B(x,r)$, where $n$ is a fixed number $0<n\le d$. For $0<\alpha<n$, let $I_\alpha$ be the fractional integral operator defined by $I_\alpha f(x)=\int_{\Bbb R^n}\vert x-y\vert ^{\alpha-n}f(y)\,d\mu(y)$. And for $m\in \Bbb N$ and suitable $b_j(x)$, $j=1,2,\ldots, m$, define the multilinear commutator $$I_{\alpha;b_j}f(x)=\int_{\Bbb R^n}\prod_{j=1}^{m}[b_j(x)-b_j(y)] \vert x-y\vert ^{\alpha-n}f(y)\,d\mu(y).$$ The authors give the following: Let $1<p<n/\alpha$ and $1/q=1/p-\alpha/n$. Then if $b_j\in \roman{RBMO}(\mu)$ ($\roman{BMO}$ space introduced by Tolsa for a non-homogeneous space), $j=1,\dots, m$, there exists $C>0$ such that $\Vert I_{\alpha;b_j}\Vert _{L^q(\mu)}\le C \prod_{j=1}^{m}\Vert b_j\Vert _{\roman{RBMO}(\mu)} \Vert f\Vert _{L^p(\mu)}$. When $m=1$, this was given by {\it W. Chen} and {\it E. Sawyer} [Ill. J. Math. 46, No. 4, 1287--1298 (2002; Zbl 1033.42008)]. The authors also give a weak type endpoint estimate for this multilinear commutator when $p=1$, for $b_j\in \roman{Osc}_{{\exp}L^r(\mu)}$ (Orlicz type function space).

42B20Singular and oscillatory integrals, several variables
47B47Commutators, derivations, elementary operators, etc.
42B25Maximal functions, Littlewood-Paley theory
47A30Operator norms and inequalities
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