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arXiv · 1701.05249

Sparse Bounds for Maximally Truncated Oscillatory Singular Integrals

Ben Krause·Michael T. Lacey
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Abstract

For polynomial $ P (x,y)$, and any Calder\'{o}n-Zygmund kernel, $K$, the operator below satisfies a $ (1,r)$ sparse bound, for $ 1< r \leq 2$. $$ \sup _{\epsilon >0} \Bigl\lvert \int_{|y| > \epsilon} f (x-y) e ^{2 \pi i P (x,y) } K(y) \; dy \Bigr\rvert $$ The implied bound depends upon $ P (x,y)$ only through the degree of $ P$. We derive from this a range of weighted inequalities, including weak type inequalities on $ L ^{1} (w)$, which are new, even in the unweighted case. The unweighted weak-type estimate, without maximal truncations, is due to Chanillo and Christ (1987).

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Ben Krause, Michael T. Lacey. 2017-01-18. Sparse Bounds for Maximally Truncated Oscillatory Singular Integrals. https://arxiv.org/abs/1701.05249

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