We prove the uniform estimates for the resolvent
as a map
from
to
on real
hyperbolic space
,
where
and
.
In contrast with analogous results on Euclidean space
, the exponent
here can be
arbitrarily close to
.
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
than
that on
.
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