Maximal polynomial modulations of singular integrals

@article{ZorinKranich2017MaximalPM,
  title={Maximal polynomial modulations of singular integrals},
  author={Pavel Zorin-Kranich},
  journal={arXiv: Classical Analysis and ODEs},
  year={2017}
}
Let $K$ be a standard H\"older continuous Calder\'on--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 Sj\"olin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem. 
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