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Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces
The Journal of Geometric Analysis ( IF 1.2 ) Pub Date : 2020-08-10 , DOI: 10.1007/s12220-020-00491-2
Mingming Cao , Qingying Xue

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.



中文翻译:

非齐次空间上的多线性Littlewood-Paley-Stein算子

\(m \ ge 2,\ lambda> 1 \)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。

更新日期:2020-08-10
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