2 results
L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 2 / May 2024
- Published online by Cambridge University Press:
- 15 March 2024, pp. 431-459
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We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include
$\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.
Bounds for spectral projectors on tori
- Part of
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- Journal:
- Forum of Mathematics, Sigma / Volume 10 / 2022
- Published online by Cambridge University Press:
- 12 May 2022, e24
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We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of
$L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.
