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Lebesgue space inequalities are proved for a variant of the triangular Hilbert transform involving curvature. The analysis relies on a crucial trilinear smoothing inequality developed herein, and on bounds for an anisotropic variant of the twisted paraproduct.

The trilinear smoothing inequality also leads to Lebesgue space bounds for a corresponding maximal function and a quantitative nonlinear Roth-type theorem concerning patterns in the Euclidean plane.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Advances in Mathematics, volume 390, in 2021 following peer review. This article may not exactly replicate the final published version. The definitive publisher-authenticated version is available online at

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