A class of bilinear multipliers given by Littlewood–Paley decomposition

Abstract

In this paper we study a particular class of bilinear multipliers which are given by Littlewood-Paley decompositions. In the first part of the paper, we show that if φ(ξ − η) is a bilinear multiplier for (p, q, r), 1 ≤ p, q ≤ ∞ satisfying the Hölder's condition 1 p + 1 q = 1 r and have support inside [0, 1), then its periodization φ (ξ) = j∈Z φ(ξ − j) is also a bilinear multiplier for the same triplet (p, q, r). Further, we show that for a given triplet (p, q, r) of exponents outside the local L 2 -range, there exists sequence {φ j } j∈Z of uniformly bounded bilinear multipliers so that the function σ(ξ) = j∈Z φ j (ξ) is not a bilinear multiplier for the triplet (p, q, r). In the second part, we describe several results for bilinear multipliers of the type m (ξ, η) which are similar to the first part in nature. In particular, we point out that the results described by P. Honzik (2014) [13] can be generalized to a more general setting.