A $p$-specific spectral multiplier theorem with sharp regularity bound for Grushin operators

TL;DR

This paper improves the spectral multiplier theorem for Grushin operators by establishing sharp regularity bounds for a wider range of p, using geometric analysis instead of weighted restriction estimates.

Contribution

It proves $L^p$-boundedness of spectral multipliers for Grushin operators under sharp regularity conditions, extending previous results and simplifying the approach.

Findings

Achieves sharp regularity condition $s>(d_1+d_2)(1/p-1/2)$ for $L^p$-boundedness.

Extends the spectral multiplier theorem to a broader p-range.

Introduces a geometric analysis approach avoiding weighted restriction estimates.

Abstract

In a recent work, P. Chen and E. M. Ouhabaz proved a $p$-specific $L^p$-spectral multiplier theorem for the Grushin operator acting on $\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$ which is given by \[ L =-\sum_{j=1}^{d_1} \partial_{x_j}^2 - \bigg( \sum_{j=1}^{d_1} |x_j|^2\bigg) \sum_{k=1}^{d_2}\partial_{y_k}^2. \] Their approach yields an $L^p$-spectral multiplier theorem within the range $1< p\le \min\{ \frac{2d_1}{d_1+2},\frac{2(d_2+1)}{d_2+3} \}$ under a regularity condition on the multiplier which is sharp only when $d_1\ge d_2$. In this paper, we improve on this result by proving $L^p$-boundedness under the expected sharp regularity condition $s>(d_1+d_2)(1/p-1/2)$. Our approach avoids the usage of weighted restriction type estimates which played a key role in the work of P. Chen and E. M. Ouhabaz, and is rather based on a careful analysis of the underlying sub-Riemannian geometry and restriction type estimates where the multiplier is truncated along the spectrum.