Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda I)^{-1}\Vert \leq \frac{C}{|\text{Im}(\lambda)|}$$ which parallel the corresponding estimates from the Euclidean space $\mathbb{R}^n$? A reference would be highly appreciated. Thanks.
3
-
2$\begingroup$ The proof is analogous to the Euclidian case. $\endgroup$– Denis SerreCommented May 21, 2015 at 5:27
-
2$\begingroup$ Doesn't such an estimate hold for any non-positive self-adjoint operator, without using the specifics? $\endgroup$– paul garrettCommented May 21, 2015 at 13:10
-
1$\begingroup$ As pointed out by Paul, you trivially have such an estimate, with $C=1$, for any self-adjoint operator. $\endgroup$– Christian RemlingCommented May 21, 2015 at 17:42
Add a comment
|