Abstract
We prove \(L^p\) bounds for partial polynomial Carleson operators along monomial curves \((t,t^m)\) in the plane \(\mathbb {R}^2\) with a phase polynomial consisting of a single monomial. These operators are “partial” in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, \(L^2\) bounds for partial operators along curves imply the full strength of the \(L^2\) bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator.
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Acknowledgements
We would like to thank C. Thiele and E. M. Stein for many helpful comments and discussions. Pierce is supported in part by NSF DMS-1402121. Roos is supported by the German National Academic Foundation. Yung is supported in part by the Hong Kong Research Grant Council Early Career Grant CUHK24300915. This collaboration was initiated at the Hausdorff Center for Mathematics and Oberwolfach and continued at the joint AMS-EMS-SPM 2015 international meeting at Porto. The authors thank all institutions involved for gracious and productive work environments.
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Guo, S., Pierce, L.B., Roos, J. et al. Polynomial Carleson Operators Along Monomial Curves in the Plane. J Geom Anal 27, 2977–3012 (2017). https://doi.org/10.1007/s12220-017-9790-7
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DOI: https://doi.org/10.1007/s12220-017-9790-7