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

Let $\kappa \ge 2,\lambda >1$ and define the multilinear Littlewood-Paley-Stein operators by $$g_{\lambda ,\mu }^{\ast }(\vec{f})(x)=({\iint }_{{\mathbb{R}}_{+}^{n+1}}{\vartheta }_t(x,y)\mid {\int }_{{\mathbb{R}}^{n\kappa }}{s}_t(y,\vec{z})\prod _{i=1}^{\kappa }{f}_i(z_i)d\mu (z_i){\mid }^2\frac{d\mu (y)dt}{t^{m+1}}{)}^{\frac{1}{2}},$$ where $\vartheta_t(x, y)=\big(\frac{t}{t + |x - y|}\big)^{m \lambda}$. In this paper, our main aim is to investigate the boundedness of $g_{\lambda ,\mu }^{\ast }$ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that $g_{\lambda ,\mu }^{\ast }$ is bounded from $L^{p_1}(\mu )\times \cdots \times L^{p_{\kappa }}(\mu )$ to $L^p(\mu )$ 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 Calder'{o}n-Zygmund type kernels and the measures $\mu$ 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 $g_{\lambda ,\mu }^{\ast }$ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.