The restriction theorem for the Grushin operators

TL;DR

This paper proves a restriction theorem for Grushin operators, extending harmonic analysis techniques to a class of degenerate differential operators with applications similar to those on the Heisenberg group.

Contribution

The paper establishes a restriction theorem for Grushin operators, providing a new analytical tool analogous to the classical restriction theorem on the Heisenberg group.

Findings

Proved a restriction theorem for Grushin operators.

Extended harmonic analysis methods to degenerate operators.

Provided a framework for further analysis of similar operators.

Abstract

We study the Grushin operators acting on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_t$ and defined by the formula \begin{equation*} L=-\overset{d_1}{\underset{j=1}{\sum}}\partial_{x_j}^2-\left(\overset{d_1}{\underset{j=1}{\sum}}|x_j|^2\right)\overset{d_2}{\underset{k=1}{\sum}}\partial_{t_k}^2. \end{equation*} We establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. Muller.