A fractal local smoothing problem for the wave equation

TL;DR

This paper investigates fractal frequency localization in local smoothing estimates for the wave equation, proposing conjectures involving fractal spectra and validating them for radial functions, with extensions to fractal-time bounds.

Contribution

It introduces a fractal local smoothing problem for the wave equation, formulates related conjectures involving Assouad spectrum, and proves results for radial functions and fractal-time bounds.

Findings

Validated conjecture for radial functions.

Proved fractal-time $L^2\to L^q$ bounds for arbitrary $L^2$ functions.

Formulated conjectures for $L^p\to L^q$ generalizations.

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.