Boundedness of the twisted paraproduct

Abstract

We prove L p estimates for a two-dimensional bilinear operator of paraproduct type. This result answers a question posed by Demeter and Thiele in [3].

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.…”

T(1) theorem for dyadic singular integral forms associated with hypergraphs

“…The most interesting case in [13] is obtained by taking F j 1 ,j 2 ,...,jr = ½ R r whenever j 1 + j 2 + · · · + j r ≥ 2, which leaves us with only r + 1 nontrivial functions. For the remaining functions we need to take p e = ∞, which makes the range of exponents empty unless (r + 1)…”

“…As a step in this program we take results from the papers [12] and [16], where the forms are dyadic and indexed by bipartite graphs, and generalize them to r-partite r-uniform hypergraphs. Some higher-dimensional instances of dyadic entangled forms were already discussed by Kovač [13] and Durcik [5], but our hypergraph generalization prefers a combinatorial description of the structure over a geometric one. Consequently, we can study less symmetric entangled forms and show their estimates in an open range of L p spaces.…”

Sobolev norm estimates for a class of bilinear multipliers

“…Suppose that σ ∈ C ∞ R 4 satisfies Estimates (9) and define bilinear multiplier operator T σ by (10). If there exists a constant c > 0 such that for every…”

“…We wanted to point out analogies between bilinear operators defined by (6) and (10) on the one side, and the Coifman-Meyer multipliers [7], [8] and bilinear pseudodifferential operators [1], [2], [4] on the other. Even though most of the reductions we perform can already be found in the literature, the auxiliary results we rely on were established by somewhat different techniques in [5] and [10], so the remarked connections might not be instantly visible.…”