Improvement of eigenfunction estimates on manifolds of nonpositive curvature

By Andrew Hassell, Melissa Tacy. 2012

70 citations recorded by the source corpus.

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 $L^{2}\to{}L^{p}$ mapping properties of a spectral cluster of $\sqrt{\Delta}$ of width $1/\log\lambda$. 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 $(\log\lambda)^{1/2}$ 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 $p>\frac{2(n+1)}{n-1}$.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/691a3053f43a6b0d55152a66980b67164a3ce6ce

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In Applied Mathematics

1. Matthew D. Blair
2. Christopher D. Sogge
3. Jason Metcalfe
4. Steve Zelditch
5. Boris Hanin
6. Y. Canzani

In Computational Theory and Mathematics

1. Matthew D. Blair
2. Christopher D. Sogge
3. Alexander Logunov
4. Steve Zelditch

Citations in the corpus

• Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions (2015, background): “Over the years, there have been many works on improving L-norms for relatively large exponents p > 2(n+1) n−1 , including [15], [23], [24], [28] and [29].”