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

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.