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 R² 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, L² bounds for partial operators along curves imply the full strength of the L² bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator.