Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces

Let m≥ 2 , λ> 1 and define the multilinear Littlewood–Paley–Stein operators by gλ,μ∗(f→)(x)=(∬R+n+1(tt+|x-y|)mλ|∫(Rn)κst(y,z→)∏i=1κfi(zi)dμ(z1)⋯dμ(zκ)|2dμ(y)dttm+1)1/2. In this paper, our main aim is to investigate the boundedness of gλ,μ∗ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that gλ,μ∗ is bounded from Lp1(μ)×⋯×Lpκ(μ) to L^p(μ) under certain weak type assumptions. The multilinear non-convolution type kernels s_t only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures μ are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of gλ,μ∗ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.