Sparse Bounds for Maximally Truncated Oscillatory Singular Integrals

Abstract

For polynomial $P(x,y)$, and any Calderón-Zygmund kernel, $K$, the operator below satisfies a $(1,r)$ sparse bound, for $1<r\le 2$. $$\underset{\epsilon >0}{\sup }|{\int }_{|y|>\epsilon }f(x-y)e^{2\pi iP(x,y)}K(y)dy|$$ 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).

20 pages, one figure. Accepted to Annali SNS

Cited by in corpus (2)

• Weighted Estimates for Rough Bilinear Singular Integrals via Sparse Domination
• On the composition for rough singular integral operators