article

Title: Bellman function technique for multilinear estimates and an application to generalized paraproducts

Authors: Vjekoslav Kovac

Issue: Volume 60 (2011), Issue 3, 813-846

Abstract:

$We prove $\mathrm{L}^p$ estimates for a class of two-dimensional multilinear forms that naturally generalize (dyadic variants of) both classical paraproducts and the twisted paraproduct introduced in [C. Demeter and C. Thiele, \textit{On the two-dimensional bilinear Hilbert transform}, Amer. J. Math. \textbf{132} (2010), no. 1, 201--256] and studied in [F. Bernicot, \textit{Fiber-wise Calder\'on-Zygmund decomposition and application to a bi-dimensional paraproduct}, Illinois J. Math., to appear] and [V. Kova\{v}c, \textit{Boundedness of the twisted paraproduct}, Rev. Mat. Iberoam., to appear]. The method we use builds on the approach from [V. Kova\{v}c, \textit{Boundedness of the twisted paraproduct}, Rev. Mat. Iberoam., to appear], and we present it as a rather general technique for proving estimates on dyadic multilinear operators. In the particular application to "generalized paraproducts" this method is combined with combinatorics of integer partitions.$