Trilinear smoothing inequalities and a variant of the triangular Hilbert transform

Lebesgue space inequalities are proved for a variant of the triangular Hilbert transform involving curvature. The analysis relies on crucial trilinear smoothing inequality developed herein, and bounds an anisotropic twisted paraproduct. also leads to corresponding maximal function quantitative nonlinear Roth-type theorem concerning patterns in Euclidean plane.