An $L^4$ estimate for a singular entangled quadrilinear form

Abstract

The twisted paraproduct can be viewed as a two-dimensional trilinear form which appeared in the work by Demeter and Thiele on the two-dimensional bilinear Hilbert transform. $L^p$ boundedness of the twisted paraproduct is due to Kova\v{c}, who in parallel established estimates for the dyadic model of a closely related quadrilinear form. We prove an $(L^4,L^4,L^4,L^4)$ bound for the continuous model of the latter by adapting the technique of Kova\v{c} to the continuous setting. The mentioned forms belong to a larger class of operators with general modulation invariance. Another instance of such is the triangular Hilbert transform, which controls issues related to two commuting transformations in ergodic theory, and for which $L^p$ bounds remain an open problem.