Boundedness of the bilinear Littlewood-Paley square function on variable Lorentz spaces

Abstract

Let m∈S (ℝ). For n∈ℤ define mn (ξ)=m (ξ−n) and let Sn be the bilinear multiplier operator associated with mn. In this paper, if p (∞) = r (∞), p (0) = r (0), 1p(t)+1s(t)≥1q(t)+1r(t), r (t) ≤ s (t), 1s(t)=1s1(t)+1s2(t) and 2r(t)+1=2r1(t)+2r2(t), then it is proved that the bilinear Littlewood-Paley square function S (f, g) is bounded on from Lr1(⋅),s1(⋅) (ℝ)×Lr2(⋅),s2(⋅) (ℝ) to Lp(⋅),q(⋅) (ℝ).