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A Local Tb Theorem for Square Functions and Parabolic Layer Potentials

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Abstract

In this paper, we give a locally parabolic version of Tb theorem for a class of vector-valued operators with off-diagonal decay in L2 and certain quasi-orthogonality on a subspace of L2, in which the testing functions themselves are also vector-valued. As an application, we establish the boundedness of layer potentials related to parabolic operators in divergence form, defined in the upper half-space ℝ n+2+ ≔ {(x, t, λ) ∈ ℝn+1 × (0, ∞)}, with uniformly complex elliptic, L∞, t, λ-independent coefficients, and satisfying the De Giorgi/Nash estimates.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, P. R. China

    Zhi Dan Wang

  2. Frontiers Science Center for Nonlinear Expectations, Ministry of Eduction, Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, 266237, P. R. China

    Guo Ming Zhang

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  1. Zhi Dan Wang
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  2. Guo Ming Zhang
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Correspondence to Guo Ming Zhang.

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Conflict of Interest The authors declare no conflict of interest.

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Supported by Natural Science Foundation of Jiangsu Province of China (Grant No. BK20220324), Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 22KJB110016)

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Wang, Z.D., Zhang, G.M. A Local Tb Theorem for Square Functions and Parabolic Layer Potentials. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-2576-5

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  • DOI: https://doi.org/10.1007/s10114-024-2576-5

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