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# Mathematical Research Letters

## Volume 22 (2015)

### Number 5

### An $\mathrm{L}^4$ estimate for a singular entangled quadrilinear form

Pages: 1317 – 1332

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n5.a3

#### Author

#### 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. $\mathrm{L}^p$ boundedness of the twisted paraproduct is due to Kovač, who in parallel established estimates for the dyadic model of a closely related quadrilinear form. We prove an $(\mathrm{L}^4 , \mathrm{L}^4 , \mathrm{L}^4 , \mathrm{L}^4)$ bound for the continuous model of the latter by adapting the technique of Kovač 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 $\mathrm{L}^p$ bounds remain an open problem.

#### 2010 Mathematics Subject Classification

Primary 42B15. Secondary 42B20.