Abstract
We introduce the Besov space \(\dot{B}^{0,L}_{1,1}\) associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space \(H_{L}^{1}\) associated with the Schrödinger operator via the Littlewood–Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that \(\dot{B}^{0,L}_{1,1}\) is a subspace of \(H_{L}^{1}\).
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Acknowledgements
The authors would like to thank Professor L.X. Yan for showing us the original idea of Theorem 1.6, and thank Dr. Lesley Ward for improving the writing of the manuscript. The authors would like to express their deep gratitude to the referee for a careful reading of the manuscript, and for offering numerous valuable suggestions to improve its mathematical accuracy.
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Communicated by Richard Rochberg.
Project is supported by NNSF of China (Grant No. 11001275) and (Grant No. 11001276), Xinmiao Project of Guangzhou University (Grant No. GRM1-101101) and Science Research Start Foundation of Guangzhou University (Grant No. GRM1-101001).
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Gong, R., Li, J. & Song, L. Besov and Hardy Spaces Associated with the Schrödinger Operator on the Heisenberg Group. J Geom Anal 24, 144–168 (2014). https://doi.org/10.1007/s12220-012-9331-3
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DOI: https://doi.org/10.1007/s12220-012-9331-3
Keywords
- Schrödinger operator
- Heisenberg group
- Besov space
- Hardy space
- Littlewood–Paley square function
- molecule
- atom