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The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals

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We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform \(L^p\) bounds for maximal modulations of the associated operators. Our methods consist of identifying where to use effectively the polynomial Carleson theorem, and where we can take advantage of the presence of oscillation to obtain decay through the \(TT^*\) method.

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Acknowledgements

The author is indebted to his doctoral advisor Prof. Dr. Christoph Thiele for having suggested investigating Theorem 1. He also thanks Joris Roos, for helpful discussions, in both early and late stages of this project, and Pavel Zorin–Kranich for discussions on the main results of this manuscript and insights on his article [29]. Finally, the author ackowledges finantial support by the Deutscher Akademischer Austauschdienst (DAAD).

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  1. Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil

    João P. G. Ramos

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Correspondence to João P. G. Ramos.

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Communicated by Loukas Grafakos.

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Ramos, J.P.G. The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals. Math. Ann. 379, 159–185 (2021). https://doi.org/10.1007/s00208-020-02075-5

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