On the restriction of Laplace-Beltrami eigenfunctions and Cantor-type sets. (English) Zbl 1484.58018
Ciatti, Paolo (ed.) et al., Geometric aspects of harmonic analysis. Proceedings of the INdAM meeting, Cortona, Italy, June 25–29, 2018. Cham: Springer. Springer INdAM Ser. 45, 351-360 (2021).
Summary: Let \((M, g)\) denote a compact Riemannian manifold without boundary. This article is an announcement of Lebesgue norm estimates of Laplace-Beltrami eigenfunctions of \(M\) when restricted to certain fractal subsets \(\Gamma\) of \(M\). The proofs in their entirety appear in [the authors, Springer INdAM Ser. 45, 351–360 (2021; Zbl 07444057)]. The sets \(\Gamma\) that we consider are random and of Cantor-type. For large Lebesgue exponents \(p\), our estimates give a natural generalization of \(L^p\) bounds previously obtained in [L. Hörmander, Acta Math. 121, 193–218 (1968; Zbl 0164.13201); Ark. Mat. 11, 1–11 (1973; Zbl 0254.42010); C. D. Sogge, J. Funct. Anal. 77, No. 1, 123–138 (1988; Zbl 0641.46011); N. Burq et al., Duke Math. J. 138, No. 3, 445–486 (2007; Zbl 1131.35053)]. 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 [J. P. Kahane and J. Peyriere, Adv. Math. 22, 131–145 (1976; Zbl 0349.60051); I. Łaba and M. Pramanik, Geom. Funct. Anal. 19, No. 2, 429–456 (2009; Zbl 1184.28010); Duke Math. J. 158, No. 3, 347–411 (2011; Zbl 1242.42011); P. Shmerkin and V. Suomala, in: Recent developments in fractals and related fields. Conference on fractals and related fields III, Ile de Porquerolles, France, September 2015. Cham: Birkhäuser/Springer. 233–260 (2017; Zbl 1390.28022); Spatially independent martingales, intersections, and applications. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1435.60005)]. 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.
For the entire collection see [Zbl 1470.42001].
For the entire collection see [Zbl 1470.42001].
MSC:
| 58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
| 28A78 | Hausdorff and packing measures |
| 42A45 | Multipliers in one variable harmonic analysis |
| 60D05 | Geometric probability and stochastic geometry |
| 60G57 | Random measures |
| 28A80 | Fractals |
Citations:
Zbl 0164.13201; Zbl 0254.42010; Zbl 0641.46011; Zbl 1131.35053; Zbl 0349.60051; Zbl 1184.28010; Zbl 1242.42011; Zbl 1390.28022; Zbl 1435.60005References:
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