Sparse bounds for maximal monomial oscillatory Hilbert transforms
Volume 242 / 2018
Studia Mathematica 242 (2018), 217-229
MSC: Primary 42B20; Secondary 42B25, 42B35.
DOI: 10.4064/sm8699-7-2017
Published online: 5 February 2018
Abstract
For each $ d \geq 2$, the maximal truncation of the Hilbert transform with a polynomial oscillation, $$ H _{ \ast } f (x) = \sup_{\epsilon }\biggl|\int _{|y| \gt \epsilon } f (x-y) \frac { e ^{2 \pi i y ^d }} y\,dy\biggr|, $$ satisfies a $ (1, r )$ sparse bound for all $ r \gt 1$. 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 without maximal truncations but with arbitrary polynomials is due to Chanillo and Christ (1987).
