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arXiv · 2501.12805

A fractal local smoothing problem for the wave equation

David Beltran·Joris Roos·Alex Rutar·Andreas Seeger
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Abstract

For any given set $E\subset [1,2]$, we discuss a fractal frequency-localized version of the $L^p$ local smoothing estimates for the half-wave propagator with times in $E$. A conjecture is formulated in terms of a quantity involving the Assouad spectrum of $E$ and the Legendre transform. We validate the conjecture for radial functions. We also prove a similar result for fractal-time $L^2\to L^q$ and square function bounds, for arbitrary $L^2$ functions and general time sets. We formulate a conjecture for $L^p\to L^q$ generalizations.

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David Beltran, Joris Roos, Alex Rutar, Andreas Seeger. 2025-01-22. A fractal local smoothing problem for the wave equation. https://arxiv.org/abs/2501.12805

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