Title: "Bochner-Riesz multipliers associated to convex planar domains with
rough boundary".

Abstract:
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.