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

Abstract

Abstract. We prove 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 [5] and studied in [1] and [6]. The method we use builds on the approach from [6] 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.

Norm variation of ergodic averages with respect to two commuting transformations

“…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.…”

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

“…For instance, for even kernels κ one has Λ(F 1 , F 2 , F 3 , F 4 ) = Λ(F 3 , F 4 , F 1 , F 2 ). Moreover, T and Λ can be seen as the smallest non-trivial examples of a family of entangled multilinear forms associated with bipartite graphs, whose dyadic models were studied in [4].…”

“…For the dyadic model Λ d , Kovač proved (L p 1 , L p 2 , L p 3 , L p 4 ) estimates whenever p j are Hölder-type exponents satisfying 2 < p j < ∞ for all j. See [5] and [4]. However, due to the more complex structure of the form, one cannot efficiently rewrite Λ and Λ d in a similar way as T and T d to exploit the mentioned square functions.…”

Quantitative norm convergence of double ergodic averages associated with two commuting group actions

“…We are able to make a slight progress on this question in the particular case r = 2 by adapting the technique that originated in the papers [17], [18], and [19] by Thiele and the author. However, due to several technical reasons explained later we need to work in a slightly different group setting, borrowed for instance from [4].…”