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
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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