Oscillatory integrals related to Carleson's theorem: fractional monomials

TL;DR

This paper extends the polynomial Carleson operator bounds to fractional monomials in one dimension and explores their connections with Carleson's theorem and Hilbert transforms along vector fields.

Contribution

It generalizes Stein and Wainger's $L^p$ bounds from integer polynomials to fractional monomials in one dimension.

Findings

Established $L^p$ bounds for fractional monomials

Connected fractional monomial operators with Carleson's theorem

Discussed relations with Hilbert transforms along vector fields

Abstract

Stein and Wainger proved the $L^p$ bounds of the polynomial Carleson operator for all integer-power polynomials without linear term. In the present paper, we partially generalise this result to all fractional monomials in dimension one. Moreover, the connections with Carleson's theorem and the Hilbert transform along vector fields or (variable) curves are also discussed in details.