Restriction theorem for the Fourier-Hermite transform associated with the normalized Hermite polynomials and the Ornstein-Uhlenbeck-Schr\"odinger equation

TL;DR

This paper establishes restriction theorems for Fourier-Hermite transforms linked to Hermite polynomials and derives Strichartz estimates for the Ornstein-Uhlenbeck operator, revealing optimal constants as the number of functions grows large.

Contribution

It introduces analogues of Stein-Tomas and Strichartz theorems for Fourier-Hermite transforms and provides optimal bounds for the associated Strichartz estimates.

Findings

Proved restriction theorems for Fourier-Hermite transforms.

Derived Strichartz estimates for the Ornstein-Uhlenbeck operator.

Identified optimal behavior of constants in the estimates.

Abstract

In this article, we prove the analogue theorems of Stein-Tomas and Srtichartz on the discrete surface restrictions of Fourier-Hermite transforms associated with the normalized Hermite polynomials and obtain the Strichartz estimate for the system of orthonormal functions for the Ornstein-Uhlenbeck operator $L=-\frac{1}{2}\Delta+\langle x, \nabla\rangle$ on $\mathbb{R}^n$. Further, we show an optimal behavior of the constant in the Strichartz estimate as limit of a large number of functions.