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

In a recent work, Chen and Ouhabaz proved a p-specific L-p-spectral multiplier theorem for the Grushin operator acting on R-d1 x R-d2 which is given by L = -Sigma(d1 )(j=1)partial derivative(2)(xj) - (Sigma(d1 )(j)(=1)vertical bar x(j)vertical bar(2))( )Sigma(d2 )(k)(=1)partial derivative(2)(yk). Their approach yields an L-p-spectral multiplier theorem within the range 1 < p <= min (2d(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) >= 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 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 R-d(2).