On L ^p-Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds

Abstract

We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo (Forum Math, 2014) about regions ${\mathcal{R}_g \subset \mathbb{C}}$ for which there can be uniform ${L^{\frac{2n}{n+2}}\to L^{\frac{2n}{n-2}}}$ resolvent estimates for ${\Delta_g + \zeta}$ , ${\zeta \in \mathcal{R}_g}$ , where ${\Delta_g}$ is the Laplace-Beltrami operator with metric g on a given compact boundaryless Riemannian manifold of dimension ${n \geq 3}$ . This is related to earlier work of Kenig, Ruiz and the third author (Duke Math J 55:329–347, 1987) for the Euclidean Laplacian, in which case the region is the entire complex plane minus any disc centered at the origin. Presently, we show that for the round metric on the sphere, S ⁿ, the resolvent estimates in (Dos Santos Ferreira et al. in Forum Math, 2014), involving a much smaller region, are essentially optimal. We do this by establishing sharp bounds based on the distance from ${\zeta}$ to the spectrum of ${\Delta_{S^n}}$ . In the other direction, we also show that the bounds in (Dos Santos Ferreira et al. in Forum Math, 2014) can be sharpened logarithmically for manifolds with nonpositive curvature, and by powers in the case of the torus, ${\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n}$ , with the flat metric. The latter improves earlier bounds of Shen (Int Math Res Not 1:1–31, 2001). The work of (Dos Santos Ferreira et al. in Forum Math, 2014) and (Shen in Int Math Res Not 1:1–31, 2001) was based on Hadamard parametrices for ${(\Delta_g + \zeta)^{-1}}$ . Ours is based on the related Hadamard parametrices for ${\cos t \sqrt{-\Delta_g}}$ , and it follows ideas in (Sogge in Ann Math 126:439–447, 1987) of proving L ^p-multiplier estimates using small-time wave equation parametrices and the spectral projection estimates from (Sogge in J Funct Anal 77:123–138, 1988). This approach allows us to adapt arguments in Bérard (Math Z 155:249–276, 1977) and Hlawka (Monatsh Math 54:1–36, 1950) to obtain the aforementioned improvements over (Dos Santos Ferreira et al. in Forum Math, 2014) and (Shen in Int Math Res Not 1:1–31, 2001). Further improvements for the torus are obtained using recent techniques of the first author (Bourgain in Israel J Math 193(1):441–458, 2013) and his work with Guth (Bourgain and Guth in Geom Funct Anal 21:1239–1295, 2011) based on the multilinear estimates of Bennett, Carbery and Tao (Math Z 2:261–302, 2006). Our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of ${\sqrt{-\Delta_g}}$ , and, moreover, a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals, as opposed to those in (Sogge in J Funct Anal 77:123–138, 1988) for unit-length intervals. We show that the resolvent estimates are sensitive to clustering within the spectrum, which is not surprising given Sommerfeld’s original conjecture (Sommerfeld in Physikal Zeitschr 11:1057–1066, 1910) about these operators.