Abstract
The aim of the article is to show the invalidity of the Strichartz estimate for the free Schrödinger equation associated with the Ornstein–Uhlenbeck operator \(L=-\frac{1}{2}\Delta +\langle x, \nabla \rangle \) in \(\mathbb {R}^n\). As a consequence we obtain a negative answer to the Fourier–Hermite restriction problem associated with the normalized Hermite polynomials for a discrete surface in \(\mathbb {Z}\times \mathbb {N}_0^n.\)
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Acknowledgements
The first author thanks Indian Institute of Technology Guwahati for the support provided during the period of this work.
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The first author thanks the Ministry of Human Resourse and Development (MHRD) of the Government of India for funding the research fellowship.
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Ghosh, S., Swain, J. Restriction Theorem Fails for the Fourier–Hermite Transform Associated with the Normalized Hermite Polynomials with Respect to a Discrete Surface. Results Math 79, 1 (2024). https://doi.org/10.1007/s00025-023-02030-1
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DOI: https://doi.org/10.1007/s00025-023-02030-1
Keywords
- Fourier–Hermite transform
- Restriction theorem
- Strichartz estimate
- Ornstein–Uhlenbeck operator
- Ornstein–Uhlenbeck–Schrödinger equation