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

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