Abstract
Consider the Hermite operator \(H=-\Delta +|x|^2\) on the Euclidean space \(\mathbb {R}^n\). The main aim of this article is to develop a theory of homogeneous and inhomogeneous Besov and Triebel–Lizorkin spaces associated to the Hermite operator. Our inhomogeneous Besov and Triebel–Lizorkin spaces are different from those introduced by Petrushev and Xu (J Fourier Anal Appl 14, 372–414 2008). As applications, we show the boundedness of negative powers and spectral multipliers of the Hermite operators on some appropriate Besov and Triebel–Lizorkin spaces.
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Notes
Actually, the theory of spaces of test functions \(\mathcal {D}_L\) and the spaces of distribution \(\mathcal {S}'_L\) in [10] still holds for nonnegative self-adjoint operators \(L\) which satisfy Gaussian upper bound for the small time only.
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Acknowledgments
The authors were supported by Australian Research Council. They would like to thank the referees for useful comments and suggestions to improve the paper.
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Communicated by Pencho Petrushev.
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Bui, T.A., Duong, X.T. Besov and Triebel-Lizorkin Spaces Associated to Hermite Operators. J Fourier Anal Appl 21, 405–448 (2015). https://doi.org/10.1007/s00041-014-9378-6
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DOI: https://doi.org/10.1007/s00041-014-9378-6