标题： 关于沿齐次曲线的多线性最大算子 显示英文标题

标题： On Multi-linear Maximal Operators Along Homogeneous Curves

摘要： 假设\[ \vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\]是一个齐次多项式曲线。 我们证明，当$p_1,\dots,p_n > 1$和$\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$时，存在一个绝对常数$0 < C = C_{p_1,\dots,p_n;\vec{\gamma}} < \infty$使得\[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-\gamma_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \]。我们的主要工具是一个平滑估计，改编自 Kosz-Mirek-Peluse-Wright 的工作。 显示英文摘要

摘要： Suppose that \[ \vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$, there exists an absolute constant $0 < C = C_{p_1,\dots,p_n;\vec{\gamma}} < \infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-\gamma_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright.