New $L^p$ bounds for Bochner-Riesz multipliers associated with convex planar domains with rough boundary

Abstract: We consider generalized Bochner-Riesz multipliers of the form $(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the "additive energy" of the boundary of a convex domain. Second, we associate a set of directions to a convex domain and define a sequence of Nikodym-type maximal operators corresponding to this set of directions. We show that domains that have low higher order additive energy, as well as those which have asymptotically good $L^q$ bounds for the corresponding sequence of Nikodym-type maximal operators where $q=(p^{\prime}/2)^{\prime}$, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.