Let $\kappa \ge 2, \lambda > 1$ and define the multilinear Littlewood-Paley-Stein operators by $$g_{ \lambda,\mu}^*(\vec{ f})(x) = \bigg(\iint_{ \mathbb{ R}^{ n+1}_{ +}} \vartheta_t(x, y) \bigg|\int_{ \mathbb{ R}^{ n \kappa}} s_t(y,\vec{ z}) \prod_{ i=1}^{ \kappa} f_i(z_i) \ d\mu(z_i)\bigg|^2 \frac{ d\mu(y) dt}{ t^{ m+1}}\bigg)^{ \frac12}, $$ 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 $g_{ \lambda,\mu}^*$ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that $g_{ \lambda,\mu}^*$ 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 $g_{ \lambda,\mu}^*$ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.
Categories: math.CA