Georgia Institute of Technology College of Sciences

We consider generalized Bochner-Riesz multipliers $(1-\rho(\xi))_+^{\lambda}$ 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. For integers $m\ge 2$, we find domains such that $(1-\rho(\xi))_+^{\lambda}\in M^p(\mathbb{R}^2)$ for all $\lambda>0$ in the range $\frac{m}{m-1}\le p\le 2$, but for which $\inf\{\lambda:\,(1-\rho)_+^{\lambda}\in M_p\}>0$ when $p<\frac{m}{m-1}$. 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 energy, as well as those which have asymptotically good $L^p$ bounds for the corresponding sequence of Nikodym-type maximal operators, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.