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Restriction of Laplace-Beltrami eigenfunctions to Cantor-type sets on manifolds

Eswarathasan, Suresh and Pramanik, Malabika 2019. Restriction of Laplace-Beltrami eigenfunctions to Cantor-type sets on manifolds. ArXiv

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Given a compact Riemannian manifold (M,g) without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to certain fractal subsets Γ of M. The sets Γ that we consider are random and of Cantor-type. For large Lebesgue exponents p, our estimates give a natural generalization of Lp bounds previously obtained in \cite{Ho68, Ho71, Sog88, BGT07}. The estimates are shown to be sharp in this range. The novelty of our approach is the combination of techniques from geometric measure theory with well-known tools from harmonic and microlocal analysis. Random Cantor sets have appeared in a variety of contexts before, specifically in fractal geometry, multiscale analysis, additive combinatorics and fractal percolation \cite{{KP76}, {LP09}, {LP11}, {SS17}, {SS18}}. They play a significant role in the study of optimal decay rates of Fourier transforms of measures, and in the identification of sets with arithmetic and geometric structures. Our methods, though inspired by earlier work, are not Fourier-analytic in nature.

Item Type: Article
Date Type: Submission
Status: Submitted
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Last Modified: 10 Mar 2020 19:24

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