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Quantitative Estimates for Square Functions with New Class of Weights

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Abstract

Let L be a non-negative self-adjoint operator on L2(X) where X is a metric space with a doubling measure. Assume that the kernels of the semigroup generated by − L satisfy a suitable upper bound related to a critical function ρ but these kernels are not assumed to satisfy any regularity conditions on spacial variables. In this paper, we prove the quantitative weighted estimates for some square functions associated to L which include the vertical square function, the conical square function and the g-functions. The novelty of our results is that the square functions associated to L might have rough kernels, hence do not belong to the Calderón-Zygmund class, and the class of weights is larger than the class of Muckenhoupt weights. Our results have applications in various settings of Schrödinger operators such as magnetic Schrödinger operators on the Euclidean space \(\mathbb {R}^n\) and Schrödinger operators on doubling manifolds.

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Acknowledgements

Xuan Thinh Duong was supported by Australian Research Council through the ARC grant DP190100970. T. Q. Bui was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.020–2019.327.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Macquarie University, NSW, 2109, Australia

    The Anh Bui & Xuan Thinh Duong

  2. Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    The Quan Bui

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  1. The Anh Bui
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Bui, T.A., Bui, T.Q. & Duong, X.T. Quantitative Estimates for Square Functions with New Class of Weights. Potential Anal 57, 545–569 (2022). https://doi.org/10.1007/s11118-021-09927-y

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