A class of bilinear multipliers given by Littlewood–Paley decomposition

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)ϕ(ξ−η) is a bilinear multiplier for (p,q,r)(p,q,r), 1≤p,q≤∞1≤p,q≤∞ satisfying the Hölder's condition 1p+1q=1r and have support inside [0,1)[0,1), then its periodization ϕ♯(ξ)=∑j∈Zϕ(ξ−j)ϕ♯(ξ)=∑j∈Zϕ(ξ−j) is also a bilinear multiplier for the same triplet (p,q,r)(p,q,r). Further, we show that for a given triplet (p,q,r)(p,q,r) of exponents outside the local L2L2-range, there exists sequence {ϕj}j∈Z{ϕj}j∈Z of uniformly bounded bilinear multipliers so that the function σ(ξ)=∑j∈Zϕj(ξ)σ(ξ)=∑j∈Zϕj(ξ) is not a bilinear multiplier for the triplet (p,q,r)(p,q,r). In the second part, we describe several results for bilinear multipliers of the type m(ξ,η)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.