Endpoint resolvent estimates for compact Riemannian manifolds

Abstract

We prove L p → L p ′ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension n in the endpoint case p = 2(n+1)/(n+3). It has the same behavior with respect to the spectral parameter z as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere.

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

“…Shao and Yao [41] proved the off-diagonal L p (M )-L q (M ) estimate of (1.16) for p, q satisfying 1/p − 1/q = 2/d, p ≤ 2(d+1) d+3 and q ≥ 2(d+1) (d−1) , but it is not known whether this range of p, q is optimal even for p, q which satisfy 1/p − 1/q = 2/d. In [19] Frank and Schimmer observed that the argument in [15] can be applied to establish L p (M )-L p (M ) analogue of (1.16) when 2d d+2 < p < 2(d+1) d+3 and d ≥ 2. They also obtained the estimate…”

$$(L^{r}, L^{s})$$(Lr,Ls) resolvent estimate for the sphere off the line $$\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$$1r-1s=2n

“…Frank and Schimmer's idea in their proof [6] of the bound for the Hada-mard parametrix actually directs us to a proof of Bourgain's interpolation technique in this special setting of Lebesgue spaces. We record their idea here.…”

“…However, a bound that depends on ζ would still be of great interest, especially if it is a negative power of ζ , as this bound will tend to 0 when |ζ | goes to infinity. A recent paper by Frank and Schimmer [6] contributed just to that. They provided the following estimate on compact manifolds:…”

“…Frank and Schimmer's result [6] is for general compact manifolds and concerns the exponents on the line of duality. We in this article, treat the spheres only, but consider more general exponents, those not necessarily on the line …”

(Lr,Ls) Resolvent estimate for the sphere off the line 1 r −1 s = 2 n

“…However, a bound that depends on ζ would still be of great interest, especially if it is a negative power of ζ, as this bound will tend to 0 when |ζ| goes to infinity. A recent paper by Frank and Schimmer [6] contributed just to that. They provided the following estimate on compact manifolds: where as in Ferreira-Kenig-Salo [5], Im √ ζ ≥ δ for a fixed δ, and C of course, is independent of ζ.…”