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Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

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Abstract

Let L be a non-negative self-adjoint operator acting on L 2(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e −tL whose kernel p t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L p-norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L p-norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L 2 estimates of the kernels of the spectral multipliers.

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Acknowledgements

The authors would like to thank the referee for carefully reading the manuscript and for making several useful suggestions. The authors thank X.T. Duong and A. Sikora for helpful discussions. R.M. Gong is supported by Xinmiao Project of Guangzhou University (Grant No. GRM1-101101) and Science Research Start Foundation of Guangzhou University (Grant No. GRM1-101001). L.X. Yan is supported by NNSF of China (Grant No. 10925106), Guangdong Province Key Laboratory of Computational Science and the Fundamental Research Funds for the Central Universities (Grant No. 09lgzs610) and Grant for Senior Scholars from the Association of Colleges and Universities of Guangdong.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P.R. China

    Ruming Gong

  2. Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, P.R. China

    Ruming Gong

  3. Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou, 510275, P.R. China

    Lixin Yan

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  1. Ruming Gong
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  2. Lixin Yan
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Correspondence to Lixin Yan.

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Communicated by Loukas Grafakos.

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Gong, R., Yan, L. Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces. J Geom Anal 24, 873–900 (2014). https://doi.org/10.1007/s12220-012-9359-4

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  • DOI: https://doi.org/10.1007/s12220-012-9359-4

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