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

Abstract

Let $$m\ge 2, \lambda > 1$$ and define the multilinear Littlewood–Paley–Stein operators by $$ g_{\lambda ,\mu }^*(\vec {f})(x) = (\iint _{{\mathbb {R}}^{n+1}_{+}} (\frac{t}{t + |x - y|})^{m \lambda } |\int _{({{\mathbb {R}}^n})^{\kappa }} s_t(y,\vec {z}) \prod _{i=1}^{\kappa } f_i(z_i) \mathrm{d}\mu (z_1) \cdots \mathrm{d}\mu (z_{\kappa })|^2 \frac{\mathrm{d}\mu (y) \mathrm{d}t}{t^{m+1}})^{1/2}.$$ 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 Calderon–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.