论文标题
多线性利特伍德 - 帕利 - 斯泰因运算师在非殖民空间上
Multilinear Littlewood-Paley-Stein Operators on Non-homogeneous Spaces
论文作者
论文摘要
让$κ\ ge 2，λ> 1 $，然后通过$$ g_ {λ，μ}^*（\ vec {f}）定义多线性littlewood-paley-stein操作员（\ x）= \ bigG（\ bigg（\ bigG（\ iint _ {\ iint _ {\ mathbb { \ bigg | \ int _ {\ Mathbb {r}^{nk}} s_t（y，\ vec {z}）\ prod_ {i = 1}^κf_i（z_i）\dμ（z_i）\dμ（z_i） dt} {t^{m + 1}}} \ bigG）^{\ frac12}，$$其中$ \ vartheta_t（x，x，y）= \ big（\ frac {t} {t + | x -y | x -y | |} \ big）^{mλ} $。在本文中，我们的主要目的是研究非均匀空间上$ g_ {λ，μ}^*$的界限。通过概率和二元技术，以及非均匀分析，我们表明$ g_ {λ，μ}^*$从$ l^{p_1}（μ）\ times \ cdots \ cdots \ times \ times \ times \ times l^times l^{p_κ}（p_κ}（μ）$ l^p（μ）$ l^p（μ）$ pats Pustions $ pusity Pustions $下来。多线性非卷积类型内核$ s_t $仅需要比多线性calderón-Zygmund型内核的标准条件满足某些弱条件，而这些措施$μ$仅被认为是上层两倍的措施（不加倍）。即使在勒布斯格措施下，上述结果也是新的。这是通过首先考虑到基于终点假设的$ g_ {λ，μ}^*$的强类型界限的足够条件来完成的，然后直接从弱类型假设上推导出强限制的大块。
Let $κ\ge 2, λ> 1$ and define the multilinear Littlewood-Paley-Stein operators by $$g_{λ,μ}^*(\vec{f})(x) = \bigg(\iint_{\mathbb{R}^{n+1}_{+}} \vartheta_t(x, y) \bigg|\int_{\mathbb{R}^{n κ}} s_t(y,\vec{z}) \prod_{i=1}^κ f_i(z_i) \ dμ(z_i)\bigg|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{\frac12}, $$ where $\vartheta_t(x, y)=\big(\frac{t}{t + |x - y|}\big)^{m λ}$. In this paper, our main aim is to investigate the boundedness of $g_{λ,μ}^*$ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that $g_{λ,μ}^*$ is bounded from $L^{p_1}(μ) \times \cdots \times L^{p_κ}(μ)$ to $L^p(μ)$ 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 $μ$ 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 $g_{λ,μ}^*$ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.
