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Improvement of eigenfunction estimates on manifolds of nonpositive curvature | | Andrew Hassell
; Melissa Tacy
; | | Date: |
11 Dec 2012 | | Abstract: | Let $(M,g)$ be a compact, boundaryless manifold of dimension $n$ with the
property that either (i) $n=2$ and $(M,g)$ has no conjugate points, or (ii) the
sectional curvatures of $(M,g)$ are nonpositive. Let $Delta$ be the positive
Laplacian on $M$ determined by $g$. We study the $L^{2} o{}L^{p}$ mapping
properties of a spectral cluster of $sqrt{Delta}$ of width $1/loglambda$.
Under the geometric assumptions above, cite{berard77} B’{e}rard obtained a
logarithmic improvement for the remainder term of the eigenvalue counting
function which directly leads to a $(loglambda)^{1/2}$ improvement for
H"ormander’s estimate on the $L^{infty}$ norms of eigenfunctions. In this
paper we extend this improvement to the $L^p$ estimates for all
$p>frac{2(n+1)}{n-1}$. | | Source: | arXiv, 1212.2540 | | Services: | Forum | Review | PDF | Favorites | |
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