Maximal polynomial modulations of singular integrals

Abstract

Let $K$ be a standard Hölder continuous Calderón--Zygmund kernel on ${\mathbb{R}}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p({\mathbb{R}}^{\mathbf{d}})$ for $1<p<\infty$. This extends Sjölin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.

v6: small corrections following referee reports

References in corpus (5)

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• A sparse estimate for multisublinear forms involving vector-valued maximal functions
• Bounds for anisotropic Carleson operators
• A Note on the Polynomial Carleson Operator in higher dimensions
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Cited by in corpus (6)

• Operator-free sparse domination
• Discrete Analogoues in Harmonic Analysis: Maximally Monomially Modulated Singular Integrals Related to Carleson's Theorem
• On almost everywhere convergence of Malmquist-Takenaka series
• Modulation invariant operators
• On the endpoint behaviour of oscillatory maximal function
• Quantitative ergodic theorems for actions of groups of polynomial growth