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.

令\（m \ ge 2，\ lambda> 1 \）并通过\（g _ {\ lambda，\ mu} ^ *（\ vec {f}）（x）=（\\定义多线性Littlewood–Paley–Stein算子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}。\）本文的主要目的是研究非均匀空间上\（g _ {\ lambda，\ mu} ^ * \）的有界性。通过概率和二进位技术以及非均匀分析，我们证明\（g _ {\ lambda，\ mu} ^ * \）的界线是\（L ^ {p_1}（\ mu）\ times \ cdots \ times L ^ {p _ {\ kappa}}（\ mu \\）在某些弱类型假设下为\（L ^ p（\ mu）\）。多线性非卷积类型内核\（s_t \）仅需要比多线性Calderón–Zygmund类型内核的标准条件满足一些弱条件，并且仅将量度\（\ mu \）视为上倍量度（非翻倍）。即使在勒贝格措施下，上述结果也是新的。首先通过基于端点假设考虑\（g _ {\ lambda，\ mu} ^ * \）的强类型有界的充分条件，然后直接从弱类型推导大块上的强界假设。