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The study analyzes fractal-local smoothing estimates in the wave equation, suggesting implications for Assouad spectrum and Legendre transform.
For any given set , we discuss a fractal frequency‐localized version of the local smoothing estimates for the half‐wave propagator with times in . A conjecture is formulated in terms of a quantity involving the Assouad spectrum of and the Legendre transform. We validate the conjecture for radial functions. We also prove a similar result for fractal‐time and square function bounds, for arbitrary functions and general time sets. We formulate a conjecture for generalizations.
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Beltrán et al. (2025) studied this question.
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