For every p in (1, infinity), the maximal modulation of the line-restricted Hilbert transform along the parabola is bounded on Lp, uniformly over all lines.
Maximal polynomial modulations of singular integrals
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abstract
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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math.CA 1years
2019 1verdicts
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The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals
For every p in (1, infinity), the maximal modulation of the line-restricted Hilbert transform along the parabola is bounded on Lp, uniformly over all lines.