Sparse Bounds for Bochner-Riesz Multipliers

The Bochner-Riesz multipliers B on Rn are shown to satisfy a range of sparse bounds, for all 0<<. The range of sparse bounds increases to the optimal range, as increases to the critical value, =, even assuming only partial information on the Bochner-Riesz conjecture in dimensions n3. 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.