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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 \(\mathbb R^d\) which satisfies the growth condition \(\mu(B(x,r))\leq Cr^n\) for all balls \(B(x,r)\), where \(n\) is a fixed number \(0<n\leq d\). For \(0<\alpha<n\), let \(I_\alpha\) be the fractional integral operator defined by \(I_\alpha f(x)=\int_{\mathbb R^n}| x-y| ^{\alpha-n}f(y)\,d\mu(y)\). And for \(m\in \mathbb N\) and suitable \(b_j(x)\), \(j=1,2,\ldots, m\), define the multilinear commutator \[ I_{\alpha;b_j}f(x)=\int_{\mathbb R^n}\prod_{j=1}^{m}[b_j(x)-b_j(y)] | x-y| ^{\alpha-n}f(y)\,d\mu(y). \] 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 \(\| I_{\alpha;b_j}\| _{L^q(\mu)}\leq C \prod_{j=1}^{m}\| b_j\| _{\roman{RBMO}(\mu)} \| f\| _{L^p(\mu)}\). When \(m=1\), this was given by W. Chen and 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).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47B47 Commutators, derivations, elementary operators, etc.
42B25 Maximal functions, Littlewood-Paley theory
47A30 Norms (inequalities, more than one norm, etc.) of linear operators

Citations:

Zbl 1033.42008
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