Polynomial Carleson Operators Along Monomial Curves in the Plane

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.