  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.
