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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Gradient estimates for Schrödinger operators with characterizations of $BMO_{\mathcal {L}}$ on Heisenberg groups
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by Qingze Lin
Proc. Amer. Math. Soc. 151 (2023), 2127-2142
DOI: https://doi.org/10.1090/proc/16308
Published electronically: February 10, 2023

Abstract:

Let $\mathcal {L}=-\Delta _{\mathbb {H}^n}+V$ be a Schrödinger operator with the nonnegative potential $V$ belonging to the reverse Hölder class $B_{Q}$, where $Q$ is the homogeneous dimension of the Heisenberg group $\mathbb {H}^n$. In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to $\mathcal {L}$. As an application, we characterize the space $BMO_{\mathcal {L}}(\mathbb {H}^n)$, associated to the Schrödinger operator $\mathcal {L}$, in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup $\{e^{-s\mathcal {L}}\}_{s>0}$ and the Poisson kernel of the semigroup $\{e^{-s\sqrt {\mathcal {L}}}\}_{s>0}$, respectively. At last, we pose a conjecture about the converse characterization of $BMO_{\mathcal {L}}(\mathbb {H}^n)$.
References
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Bibliographic Information
  • Qingze Lin
  • Affiliation: School of Mathematics, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China
  • ORCID: 0000-0001-9760-9223
  • Email: linqz@mail2.sysu.edu.cn
  • Received by editor(s): May 27, 2022
  • Received by editor(s) in revised form: September 4, 2022, September 18, 2022, and September 19, 2022
  • Published electronically: February 10, 2023
  • Additional Notes: The author was supported by the NNSF of China (No. 12071490).
  • Communicated by: Benoit Pausader
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 2127-2142
  • MSC (2020): Primary 35J10, 35K08, 43A80
  • DOI: https://doi.org/10.1090/proc/16308
  • MathSciNet review: 4556206