THE SOBOLEV INEQUALITIES ON REAL HYPERBOLIC SPACES AND EIGENVALUE BOUNDS FOR SCHRÖDINGER OPERATORS WITH COMPLEX POTENTIALS

Abstract

We prove the uniform estimates for the resolvent ${(\mathrm{\Delta }-\alpha )}^{-1}$ as a map from $L^q$ to $L^{q^{\prime }}$ on real hyperbolic space ${ℍ}^n$, where $\alpha \in ℂ\setminus [{(n-1)}^2∕4,\infty )$ and . In contrast with analogous results on Euclidean space ${ℝ}^n$, the exponent $q$ here can be arbitrarily close to $2$. This striking improvement is due to two non-Euclidean features of hyperbolic space: the Kunze–Stein phenomenon and the exponential decay of the spectral measure. In addition, we apply this result to the study of eigenvalue bounds of the Schrödinger operator with a complex potential. The improved Sobolev inequality results in a better long-range eigenvalue bound on ${ℍ}^n$ than that on ${ℝ}^n$.