Multiplier transformations associated to convex domains in ℝ²

2015/08/18 by Cladek, Laura
#42B15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.04280

Abstract

We consider Fourier multipliers in ℝ² of the form m∘ρ where ρ is the Minkowski functional associated to a convex set in ℝ², and prove L^p bounds for the corresponding multiplier operators. It is of interest to consider domains whose boundary is not smooth. Our results depend on a notion of Minkowski dimension introduced by Seeger and Ziesler that measures "flatness" of the boundary of the domain. Our methods analyze the case of oscillatory multipliers \frace^iρ(ξ)(1+|ξ|)^-a associated to wave equations, which we use to derive results for more general multiplier transformations.

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