Abstract
In this paper we study the \(L^p\)-\(L^q\) boundedness of the Fourier multipliers in the setting where the underlying Fourier analysis is introduced with respect to the eigenfunctions of an anharmonic oscillator A. Using the notion of a global symbol that arises from this analysis, we extend a version of the Hausdorff–Young–Paley inequality that guarantees the \(L^p\)-\(L^q\) boundedness of these operators for the range \(1<p \le 2 \le q <\infty \). The boundedness results for spectral multipliers acquired, yield as particular cases Sobolev embedding theorems and time asymptotics for the \(L^p\)-\(L^q\) norms of the heat kernel associated with the anharmonic oscillator. Additionally, we consider functions f(A) of the anharmonic oscillator on modulation spaces and prove that Linskĭi’s trace formula holds true even when f(A) is simply a nuclear operator.
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Notes
The singular values of a compact operator T between Hilbert spaces are the square roots of non-negative eigenvalues of the self-adjoint operator \(T^{*}T\), where \(T^{*}\) denotes the adjoint of T.
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Acknowledgements
The authors would like to thank the reviewers for their valuable comments and suggestions which helped to improve the manuscript. The authors are also grateful to Professor Michael Ruzhansky for his valuable suggestions. The authors are supported by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations, the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant Number 01M01021). Marianna Chatzakou is a Postdoctoral Fellow of the Research Foundation—Flanders (FWO) under the Postdoctoral Grant No. 12B1223N. Vishvesh Kumar is also supported FWO Senior Research Grant G011522N.
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Communicated by Pencho Petrushev.
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Chatzakou, M., Kumar, V. \(L^p\)-\(L^q\) Boundedness of Fourier Multipliers Associated with the Anharmonic Oscillator. J Fourier Anal Appl 29, 73 (2023). https://doi.org/10.1007/s00041-023-10047-x
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DOI: https://doi.org/10.1007/s00041-023-10047-x
Keywords
- Anharmonic oscillator
- Fourier multipliers
- Spectral multipliers
- Paley inequality
- Hausdorff–Young–Paley inequality
- \(L^p\)-\(L^q\) boundedness
- Heat kernels