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

Abstract

AbstractIn a recent work, Chen and Ouhabaz proved a p-specific $$L^p$$ L p -spectral multiplier theorem for the Grushin operator acting on $${\mathbb {R}}^{d_1}\times {\mathbb {R}}^{d_2}$$ R d 1 × R d 2 which is given by $$\begin{aligned} L =-\sum _{j=1}^{d_1} \partial _{x_j}^2 - \Bigg ( \sum _{j=1}^{d_1} |x_j|^2\Bigg ) \sum _{k=1}^{d_2}\partial _{y_k}^2. \end{aligned}$$ L = - ∑ j = 1 d 1 ∂ x j 2 - ( ∑ j = 1 d 1 | x j | 2 ) ∑ k = 1 d 2 ∂ y k 2 . Their approach yields an $$L^p$$ L p -spectral multiplier theorem within the range $$1< p\le \min \{ 2d_1/(d_1+2), 2(d_2+1)/(d_2+3) \}$$ 1 < p ≤ min { 2 d 1 / ( d 1 + 2 ) , 2 ( d 2 + 1 ) / ( d 2 + 3 ) } under a regularity condition on the multiplier which is sharp only when $$d_1\ge d_2$$ d 1 ≥ d 2 . In this paper, we improve on this result by proving $$L^p$$ L p -boundedness under the expected sharp regularity condition $$s>(d_1+d_2)(1/p-1/2)$$ 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 Chen and 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 of the Laplacian on $${\mathbb {R}}^{d_2}$$ R d 2 .