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

TL;DR

This paper establishes $L^p$ bounds for Fourier multipliers in $R^2$ associated with convex domains, especially those with non-smooth boundaries, using Minkowski dimension to measure boundary flatness.

Contribution

It introduces new $L^p$ bounds for multipliers linked to convex domains with irregular boundaries, utilizing Minkowski dimension to handle non-smooth cases.

Findings

Derived $L^p$ bounds for multipliers with non-smooth convex boundaries.

Analyzed oscillatory multipliers related to wave equations.

Connected boundary flatness to multiplier boundedness.

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.