Bellman function technique for multilinear estimates and an application to generalized paraproducts
Vjekoslav Kovac
42B20paraproductmultilinear operatorBellman functioninteger partition
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.
Indiana University Mathematics Journal
2011
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10.1512/iumj.2011.60.4784
10.1512/iumj.2011.60.4784
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Indiana Univ. Math. J. 60 (2011) 813 - 846
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