Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin