L^p-boundedness of the triangular Hilbert transform

Determine whether the triangular Hilbert transform, a trilinear singular integral operator in two dimensions obtained by integrating out one kernel dimension because it projects to zero in all function arguments, satisfies any L^{p_1} × L^{p_2} → L^{p_3} boundedness estimates under the Hölder scaling 1/p_1 + 1/p_2 + 1/p_3 = 1.

Background

Within the discussion of partially degenerate cases in dimension d = 2, the paper highlights the triangular Hilbert transform as a particularly challenging operator, obtained by integrating out one dimension of the kernel since it projects to zero in the arguments of all functions.

The authors state that no L^p bounds are currently known for this operator and note that existing techniques are insufficient to prove such bounds. They further remark that a version of their main theorem with uniformity in the quasiconformal distortion parameter K would imply bounds for the triangular Hilbert transform. Moreover, establishing such bounds would have implications for the Carleson operator in corresponding L^p spaces.