# Boundedness of certain commutators over non-homogeneous metric measure spaces

Boundedness of certain commutators over non-homogeneous metric measure spaces Let $$(\mathcal {X},d,\mu )$$ ( X , d , μ ) 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 $$b\in \widetilde{\mathrm{RBMO}}(\mu )$$ b ∈ RBMO ~ ( μ ) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $$T_b:=bT-Tb$$ T b : = b T - T b generated by T and b from the atomic Hardy space $$\widetilde{H}^1(\mu )$$ H ~ 1 ( μ ) with the discrete coefficient into the weak Lebesgue space $$L^{1,\,\infty }(\mu )$$ L 1 , ∞ ( μ ) . From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator $$T_b$$ T b is bounded on $$L^p(\mu )$$ L p ( μ ) for all $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) . Moreover, the boundedness of the commutator generated by the generalized fractional integral $$T_\alpha \,(\alpha \in (0,1))$$ T α ( α ∈ ( 0 , 1 ) ) and the $$\widetilde{\mathrm{RBMO}}(\mu )$$ RBMO ~ ( μ ) function from $$\widetilde{H}^1(\mu )$$ H ~ 1 ( μ ) into $$L^{1/{(1-\alpha )},\,\infty }(\mu )$$ L 1 / ( 1 - α ) , ∞ ( μ ) is also presented. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Boundedness of certain commutators over non-homogeneous metric measure spaces

, Volume 7 (2) – Jun 1, 2016
32 pages

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-016-0136-6
Publisher site
See Article on Publisher Site

### Abstract

Let $$(\mathcal {X},d,\mu )$$ ( X , d , μ ) 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 $$b\in \widetilde{\mathrm{RBMO}}(\mu )$$ b ∈ RBMO ~ ( μ ) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $$T_b:=bT-Tb$$ T b : = b T - T b generated by T and b from the atomic Hardy space $$\widetilde{H}^1(\mu )$$ H ~ 1 ( μ ) with the discrete coefficient into the weak Lebesgue space $$L^{1,\,\infty }(\mu )$$ L 1 , ∞ ( μ ) . From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator $$T_b$$ T b is bounded on $$L^p(\mu )$$ L p ( μ ) for all $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) . Moreover, the boundedness of the commutator generated by the generalized fractional integral $$T_\alpha \,(\alpha \in (0,1))$$ T α ( α ∈ ( 0 , 1 ) ) and the $$\widetilde{\mathrm{RBMO}}(\mu )$$ RBMO ~ ( μ ) function from $$\widetilde{H}^1(\mu )$$ H ~ 1 ( μ ) into $$L^{1/{(1-\alpha )},\,\infty }(\mu )$$ L 1 / ( 1 - α ) , ∞ ( μ ) is also presented.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jun 1, 2016