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Singular integrals on Carleson measure spaces $ {CMO}^p$ on product spaces of homogeneous type


Authors: Ji Li and Lesley A. Ward
Journal: Proc. Amer. Math. Soc. 141 (2013), 2767-2782
MSC (2010): Primary 42B35; Secondary 42B20
DOI: https://doi.org/10.1090/S0002-9939-2013-11604-7
Published electronically: April 25, 2013
MathSciNet review: 3056567
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Abstract: In the setting of product spaces $ \widetilde {M}$ of homogeneous type, we prove that every product non-isotropic smooth (NIS) operator $ T$ is bounded on the generalized Carleson measure space  $ {\rm CMO}^p(\widetilde {M})$ of Han, Li and Lu for $ p_0 < p < 1$. Here $ p_0$ depends on the homogeneous dimensions of the measures on factors of the product space  $ \widetilde {M}$ and on the regularity of the quasi-metrics on factors of $ \widetilde {M}$. The $ L^p$ boundedness for $ 1<p<\infty $ of the class of NIS operators was developed in both the one-parameter case and the multiparameter case by Nagel and Stein, and the $ H^p$ boundedness was established in the multiparameter case by Han, Li and Lu.


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Additional Information

Ji Li
Affiliation: Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, People’s Republic of China
Email: liji6@mail.sysu.edu.cn

Lesley A. Ward
Affiliation: School of Mathematics and Statistics, University of South Australia, Mawson Lakes SA 5095, Australia
Email: Lesley.Ward@unisa.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2013-11604-7
Keywords: Carleson measure space, Calder\'on's reproducing formula, singular integral operators, non-isotropic smooth operators, product spaces of homogeneous type
Received by editor(s): November 1, 2011
Published electronically: April 25, 2013
Additional Notes: The first author was supported by the National Natural Science Foundation (grant No. 11001275), by the China Postdoctoral Science Foundation (grant No. 20100480819), and by a postdoctoral fellowship from the University of South Australia.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2013 American Mathematical Society