An $\mathrm{L}^4$ 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 Kovač, 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 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 L p bounds remain an open problem.

Power-type cancellation for the simplex Hilbert transform

“…intertwined applications of the Cauchy-Schwarz inequality (2.8) and an integration by parts identity (2.15). The base case is closely related to the quadrilinear forms studied in [1] and [3].…”

Norm variation of ergodic averages with respect to two commuting transformations

“…Our inequality did not preserve the tensor structure in the variables x ′ and y ′ which will be needed later in (3.13). For that purpose we further estimate (3.5) by a superposition of Gaussians as it was done in [14]. Denote so σ(s) is comparable to |s| −λ for large |s|.…”

“…The range of exponents p 1 , p 2 , p, ̺ in the above discussion is likely not exhausted as the analogous work [24] in the simplified setting suggests. This paper, while self-contained, builds on a technique for bounding multi-linear and multiscale singular integral operators gradually developed by the authors in [14], [15], [22], [23], [24], [25], [26]. We consider the present application to quantitative norm convergence for double ergodic averages a milestone in these efforts.…”

𝐿^{𝑝} estimates for a singular entangled quadrilinear form

“…This article is a continuation of [3]. We are concerned with a quadrilinear singular integral form involving the entangled product of four functions on R 2 F (F 1 , F 2 , F 3 , F 4 )(x, y, x ′ , y ′ ) := F 1 (x, y)F 2 (x ′ , y)F 3 (x ′ , y ′ )F 4 (x, y ′ ).…”

“…This theorem is a consequence of the restricted type estimates given by Theorem 3 below. By the decomposition performed in [3], it suffices to prove Theorem 1 for m reduced to a single cone in the frequency plane (ξ, η). More precisely, it is enough to consider the form…”