Restriction and spectral multiplier theorems on asymptotically conic manifolds

By C. Guillarmou, Andrew Hassell, Adam S. Sikora. 2010

74 citations recorded by the source corpus.

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(\lambda)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq 2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm scaling as $\lambda^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete $L^2$ restriction theorem, which is an $O(\lambda^{n(1/p - 1/p') - 1})$ estimate on the $L^p \to L^{p'}$ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner-Riesz summability results, which are sharp for $p$ in the range above.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/10e1c114590885c6042f96d22df5efbbfd0e6261

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In Applied Mathematics

  1. Lixin Yan
  2. T. Coulhon
  3. Adam S. Sikora
  4. El-Maati Ouhabaz
  5. Peng Chen
  6. Andrew Hassell
  7. Salahaddine Boutayeb
  8. Xuan Thinh Duong

In Mathematical Physics

  1. Lixin Yan
  2. T. Coulhon
  3. Adam S. Sikora
  4. El-Maati Ouhabaz
  5. Peng Chen
  6. Andrew Hassell
  7. Salahaddine Boutayeb
  8. Melissa Tacy

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