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

Abstract

Let $κ\ge 2, λ> 1$ and define the multilinear Littlewood-Paley-Stein operators by $$g_{λ,μ}^*(\vec{f})(x) = \bigg(\iint_{\mathbb{R}^{n+1}_{+}} \vartheta_t(x, y) \bigg|\int_{\mathbb{R}^{n κ}} s_t(y,\vec{z}) \prod_{i=1}^κ f_i(z_i) \ dμ(z_i)\bigg|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{\frac12}, $$ where $\vartheta_t(x, y)=\big(\frac{t}{t + |x - y|}\big)^{m λ}$. In this paper, our main aim is to investigate the boundedness of $g_{λ,μ}^*$ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that $g_{λ,μ}^*$ is bounded from $L^{p_1}(μ) \times \cdots \times L^{p_κ}(μ)$ 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 $g_{λ,μ}^*$ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.