论文标题
奇异的球形最大运算符在一类退化两步的lie群上
Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups
论文作者
论文摘要
令$ g \ cong \ mathbb {r}^{d} \ ltimes \ mathbb {r} $是一个有限维的两步nilpotent组，群乘以$（x，x，u）\ cdot（u）\ cdot（y，v）满足$ 2 \ leq {\ rm stark} \，j <d $的堕落条件。考虑$$ {\ frak m} f（x，u）= \ sup_ {t> 0} \ big | \int_σf（x-ty，u- t x^{t} jy）dμ（y）dμ（y） $σ$的高斯曲率在supp $dμ$上不存在。在本文中，我们证明，当$ d \ geq 4 $时，最大运算符$ {\ frak m} $在$ l^{p}（g）$上的范围$（d-1）/（d-1）/（d-2）<p \ leq \ leq \ leq \ infty $。
Let $G\cong\mathbb{R}^{d} \ltimes \mathbb{R}$ be a finite-dimensional two-step nilpotent group with the group multiplication $(x,u)\cdot(y,v)\rightarrow(x+y,u+v+x^{T}Jy)$ where $J$ is a skew-symmetric matrix satisfying a degeneracy condition with $2\leq {\rm rank}\, J <d$. Consider the maximal function defined by $$ {\frak M}f(x, u)=\sup_{t>0}\big|\int_Σ f(x-ty, u- t x^{T}Jy) dμ(y)\big|, $$ where $Σ$ is a smooth convex hypersurface and $dμ$ is a compactly supported smooth density on $Σ$ such that the Gaussian curvature of $Σ$ is nonvanishing on supp $dμ$. In this paper we prove that when $d\geq 4$, the maximal operator ${\frak M}$ is bounded on $L^{p}(G)$ for the range $(d-1)/(d-2)<p\leq\infty$.
