On bilinear Littlewood-Paley square functions

Abstract

In this paper, we study the bilinear Littlewood-Paley square function introduced by M. Lacey. We give an easy proof of its boundedness from L p (R d) × L q (R d) into L r (R d), d ≥ 1, for all possible values of exponents p, q, r, i.e. for 2 ≤ p, q ≤ ∞, 1 ≤ r ≤ ∞ satisfying 1 p + 1 q = 1 r. We also prove analogous results for bilinear square functions on the torus group T d .

Norm variation of ergodic averages with respect to two commuting transformations

“…Bernicot [4] has verified this conjecture for a particular case of equidistant intervals of the same length, such as Ω = {[j, j + 1) : j ∈ Z}. The problem becomes simpler if we replace ½ ω with a smooth bump function adapted to ω, as was already observed by Lacey [28] in the case of the intervals [j, j + 1), see also [5], [32], [33]. The above bilinear square function S is associated with smooth truncations of the lacunary intervals Ω = {[2 j , 2 j+1 ) : j ∈ Z}.…”

Certain Multi(sub)linear Square Functions

“…The case m = 2 of this theorem was obtained by Ratnakumar and Shrivastava [11]. The new ingredient of this note is the extension of this result to the case where m ≥ 3, where a multilinear version of Young's inequality is needed.…”

“…The proofs of these results were based on rather complicated timefrequency analysis. However, Ratnakumar and Shrivastava [11] provided a proof for the boundedness of a smooth bilinear square functions, which is based on more elementary techniques. Motivated by the work of [11] and the increasing interest in multilinear operators, the aim of this paper is to study the L p boundedness properties of smooth m-(sub)linear Littlewood-Paley square functions.…”