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arXiv 2007.13104math.CA
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非齐性空间上的多线性Littlewood-Paley-Stein算子

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

Mingming Cao, Qingying Xue

备注：
to appear in J. Geom. Anal. 33 pages
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摘要：

设$κ\ge 2, λ> 1$，用$$g_{λ,μ}^*(\vec{f})(x) = \bigg(\iint_{\mathbb{R}^{n+1}_{+}} \vartheta_t(x, y) \bigg|\int_{\mathbb{R}^{n κ}} s_t(y,\vec{z}) \prod_{i=1}^κ f_i(z_i) \ dμ(z_i)\bigg|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{\frac12}, $$定义多线性Littlewood-Paley-Stein算子，其中$\vartheta_t(x, y)=\big(\frac{t}{t + |x - y|}\big)^{m λ}$ .本文主要研究$g_{λ,μ}^*$在非齐性空间上的有界性。通过概率和并矢技术，以及非齐次分析，我们证明了$g_{λ,μ}^*$是有界的$L^{p_1}(μ) \times \cdots \times L^{p_κ}(μ)$到$L^p(μ)$在一定的弱型假设。多线性非卷积型核$s_t$只需要满足比多线性Calderón-Zygmund型核的标准条件更弱的条件，并且测度$μ$仅被假设为上加倍测度（非加倍测度）.上述结果是新的，即使在勒贝格措施。首先在端点假设的基础上给出了$g_{λ,μ}^*$强型有界的充分条件，然后由弱型假设直接推导出大片段上的强界.

英文摘要：

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.

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