Math @ Duke

Publications [#320661] of Lillian B. Pierce
Papers Published
 Guo, S; Pierce, LB; Roos, J; Yung, P, Polynomial Carleson operators along monomial curves in the plane,
arXiv:1605.05812 [math]
(May, 2016)
(last updated on 2017/12/11)
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 stoppingtime 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 onedimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vectorvalued variant of the CarlesonHunt theorem.


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