Sparse Bounds for Maximal Monomial Oscillatory Hilbert Transforms

Ben Krause; Michael T. Lacey

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arXiv · 1609.01564

Sparse Bounds for Maximal Monomial Oscillatory Hilbert Transforms

Ben Krause·Michael T. Lacey

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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).

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BibTeX RIS

Ben Krause, Michael T. Lacey. 2016-09-06. Sparse Bounds for Maximal Monomial Oscillatory Hilbert Transforms. https://arxiv.org/abs/1609.01564

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