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A spectral multiplier theorem for Hardy spaces associated with Schrödinger operator on the Heisenberg group

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Abstract

Let \(\{T_t\}_{t>0}\) be the semigroup of linear operators generated by a Schrödinger operator \({\mathcal {L}}=-\Delta _{{\mathbb {H}}^n}+V\) on Heisenberg group \({\mathbb {H}}^n\), where \(\Delta _{{\mathbb {H}}^n}\) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class \(B_{Q/2}\) and Q is the homogeneous dimension of \({\mathbb {H}}^n\). Let \(\int _0^{\infty } \uplambda dE_{{\mathcal {L}}}(\uplambda )\) be the spectral resolution of \({\mathcal {L}}\), we prove that if a function F satisfies a Mihlin condition with exponent \(\alpha >Q/2\) then the operator \(F({\mathcal {L}})=\int _0^\infty F(\uplambda )dE_{{\mathcal {L}}}(\uplambda )\) is bounded on Hardy space \(H_{{\mathcal {L}}}^1({{\mathbb {H}}^n})\).

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Acknowledgements

The authors would like to thank Professor Heping Liu for some helpful advice and to express sincere thanks to the referees for the valuable comments and helpful suggestions.

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  1. Key Laboratory of Mathematics and Complex Systems, Ministry of Education Institution of Mathematics and Mathematical Education School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Nan Hu & Jiman Zhao

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  1. Nan Hu
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  2. Jiman Zhao
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Correspondence to Jiman Zhao.

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Jiman Zhao: The corresponding author, supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900).

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Hu, N., Zhao, J. A spectral multiplier theorem for Hardy spaces associated with Schrödinger operator on the Heisenberg group. J. Pseudo-Differ. Oper. Appl. 13, 5 (2022). https://doi.org/10.1007/s11868-021-00435-6

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