Abstract
In this paper we will define and investigate the imaginary powers \((-\triangle _{k,1})^{-i\sigma },\sigma \in {\mathbb {R}}\) of the (k, 1)-generalized harmonic oscillator \(-\triangle _{k,1}=-\left\| x\right\| \triangle _k+\left\| x\right\| \) and prove the \(L^p\)-boundedness \((1<p<\infty )\) and weak \(L^1\)-boundedness of such operators. It is a parallel result to the \(L^p\)-boundedness \((1<p<\infty )\) and weak \(L^1\)-boundedness of the imaginary powers of the Dunkl harmonic oscillator \(-\triangle _k+\left\| x\right\| ^2\). To prove this result, we develop the Calderón–Zygmund theory adapted to the (k, 1)-generalized setting by constructing the metric space of homogeneous type corresponding to the (k, 1)-generalized setting, and show that \(\left( -\triangle _{k,1}\right) ^{-i\sigma }\) are singular integral operators satisfying the corresponding Hörmander type condition.
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The author would like to thank his adviser Nobukazu Shimeno for valuable comments and advice. All data included in this study are available upon request by contact with the corresponding author.
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Communicated by Palle Gorgenson.
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This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Andreas Seeger, Franz Luef and Serap Oztop.
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Teng, W. Imaginary Powers of (k, 1)-Generalized Harmonic Oscillator. Complex Anal. Oper. Theory 16, 89 (2022). https://doi.org/10.1007/s11785-022-01249-0
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DOI: https://doi.org/10.1007/s11785-022-01249-0