Uniform Sobolev Resolvent Estimates for the Laplace-Beltrami Operator on Compact Manifolds

In this paper we continue the study on the resolvent estimates of the Laplace-Beltrami operator $\Delta_g$ on a compact manifolds $M$ with dimension $n\geq3$. On the Sobolev line $1/p-1/q=2/n$ we can prove that the resolvent $(\Delta_g+\zeta)^{-1}$ is uniformly bounded from $L^p$ to $L^q$ when $(p,q)$ are within the admissible range $p\leq2(n+1)/(n+3)$ and $q\geq2(n+1)/(n-1)$ and $\zeta$ is outside a parabola opening to the right and a small disk centered at the origin. This naturally generalizes the previous results in \cite{Kenig} and \cite{bssy} which addressed only the special case when $p=2n/(n+2), q=2n/(n-2)$. Using the shrinking spectral estimates between $L^p$ and $L^q$ we also show that when $(p,q)$ are within the interior of the admissible range, one can obtain a logarithmic improvement over the parabolic region for resolvent estimates on manifolds equipped with Riemannian metric of non-positive sectional curvature, and a power improvement depending on the exponent $(p,q)$ for flat torus. The latter therefore partially improves Shen's work in \cite{Shen} on the $L^p\to L^2$ uniform resolvent estimates on the torus. Similar to the case as proved in \cite{bssy} when $(p,q)=(2n/(n+2),2n/(n-2))$, the parabolic region is also optimal over the round sphere $S^n$ when $(p,q)$ are now in the admissible range. However, we may ask if the admissible range is sharp in the sense that it is the only possible range on the Sobolev line for which a compact manifold can have uniform resolvent estimate for $\zeta$ being ouside a parabola.