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

We consider Fourier multipliers in $\mathbb{R}^2$ of the form $m\circρ$ where $ρ$ 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 $\frac{e^{iρ(ξ)}}{(1+|ξ|)^{-a}}$ associated to wave equations, which we use to derive results for more general multiplier transformations.

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