Sparse Bounds for Bochner-Riesz Multipliers

The Bochner-Riesz multipliers $B_{\delta }$ on $\mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< \delta < \frac {n-1}2$. The range of sparse bounds increases to the optimal range, as $\delta$ increases to the critical value, $\delta =\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.