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 Calderón–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.