Multiplier transformations associated to convex domains in ${\mathbb{R}}^2$

Abstract

We consider Fourier multipliers in ${\mathbb{R}}^2$ of the form $m\circ \rho$ where $\rho$ is the Minkowski functional associated to a convex set in ${\mathbb{R}}^2$, 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 $\frac{e^{i\rho (\xi )}}{(1+|\xi |{)}^{-a}}$ associated to wave equations, which we use to derive results for more general multiplier transformations.

arXiv comment43 pages, 3 figures