Boundedness of certain commutators over non-homogeneous metric measure spaces. (English) Zbl 1367.47043
Summary: Let \((\mathcal {X},d,\mu)\) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let \(T\) be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and \(b\in \widetilde{\mathrm{RBMO}}(\mu)\) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator \(T_b:=bT-Tb\) generated by \(T\) and \(b\) from the atomic Hardy space \(\widetilde{H}^1(\mu)\) with the discrete coefficient into the weak Lebesgue space \(L^{1,\,\infty}(\mu)\). From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator \(T_b\) is bounded on \(L^p(\mu)\) for all \(p\in (1,\infty)\). Moreover, the boundedness of the commutator generated by the generalized fractional integral \(T_\alpha \,(\alpha \in (0,1))\) and the \(\widetilde{\mathrm{RBMO}}(\mu)\) function from \(\widetilde{H}^1(\mu)\) into \(L^{1/{(1-\alpha)},\,\infty}(\mu)\) is also presented.
MSC:
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B35 | Function spaces arising in harmonic analysis |
| 30L99 | Analysis on metric spaces |
Keywords:
geometrically doubling measure; upper doubling measure; non-homogeneous metric measure space; commutator; Calderón-Zygmund operator; fractional integral; \(\widetilde{\mathrm{RBMO}}\) function; Hardy spaceReferences:
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