Multiplier transformations associated to convex domains in R²

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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 Lᵖ 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 eⁱρ⁽ξ⁾(1+|ξ|)⁻ᵃ associated to wave equations, which we use to derive results for more general multiplier transformations.