Sparse Bounds for Maximal Monomial Oscillatory Hilbert Transforms

TL;DR

This paper establishes sparse bounds for maximal monomial oscillatory Hilbert transforms, leading to new weak-type inequalities for these operators with weights and unweighted cases, extending prior results.

Contribution

It proves sparse bounds for maximal monomial oscillatory Hilbert transforms, providing new weak-type inequalities for weighted and unweighted cases with arbitrary polynomials.

Findings

Sparse bounds hold for all r > 1.

Weak-type inequalities are valid for A_1 weights.

Results extend previous unweighted estimates to weighted settings.

Abstract

For each $ d \geq 2$, the Hilbert transform with a polynomial oscillation as below satisfies a $ (1, r )$ sparse bound, for all $ r>1$ $$ H _{ \ast } f (x) = \sup _{\epsilon } \Bigl\lvert \int_{|y| > \epsilon} f (x-y) \frac { e ^{2 \pi i y ^d }} y\; dy \Bigr\rvert. $$ This quickly implies weak-type inequalities for the maximal truncations, which hold for $A_1$ weights, but are new even in the case of Lebesgue measure. The unweighted weak-type estimate \emph{without maximal truncations} but with arbitrary polynomials, is due to Chanillo and Christ (1987).