Abstract
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 \( n \ge 3\). 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.
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Acknowledgements
Michael T. Lacey research supported in part by grant National Science Foundation grant DMS-1600693, and by Australian Research Council grant DP160100153. All Authors: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, CA, during the Spring 2017 Semester. We benefited from conversations with Andreas Seeger and Richard Oberlin, as well as careful readings by referees.
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Communicated by Hans G. Feichtinger.
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Lacey, M.T., Mena, D. & Reguera, M.C. Sparse Bounds for Bochner–Riesz Multipliers. J Fourier Anal Appl 25, 523–537 (2019). https://doi.org/10.1007/s00041-017-9590-2
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DOI: https://doi.org/10.1007/s00041-017-9590-2