## 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. 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 vector-valued variant of the Carleson-Hunt theorem.

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- General Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0101 Pure Mathematics

### Citation

*Journal of Geometric Analysis*.

*Journal of Geometric Analysis*, 2017.

*Journal of Geometric Analysis*, Springer Verlag, 2017.

## Published In

## ISSN

## Publication Date

## Publisher

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0101 Pure Mathematics