论文标题
海森伯格集团的Kakeya最大不平等
Kakeya maximal inequality in the Heisenberg group
论文作者
论文摘要
我们通过在Heisenberg Group $ \ Mathbb {H}^1 $中配备了KorányiDance $ d Dance $ d $ D _ {我们表明，$$ \ |m_Δf\ | _ {l^3（s^1）} \ leq c（\ varepsilon）Δ^{ - 1/3- \ \ varepsilon} \ | f \ | | _ | _ | _ {l^3（l^3（l^3（\ nathbb {\ mathb {h}^1） l^3（\ mathbb {h}^1），$$对于所有$ \ varepsilon> 0 $。该证明是基于最近的一种变体，该变体是由于沃尔夫（Wolff）的圆形最大函数定理的Pramanik，Yang和Zahl，用于与Sogge的电影曲率条件相关的一类平面曲线。作为我们Kakeya最大不平等的应用，我们为Heisenberg Kakeya的Hausdorff尺寸恢复了$（\ Mathbb {h}^1，d _ {\ Mathbb {h}}}）$的尖锐下限。
We define the Heisenberg Kakeya maximal functions $M_δf$, $0<δ<1$, by averaging over $δ$-neighborhoods of horizontal unit line segments in the Heisenberg group $\mathbb{H}^1$ equipped with the Korányi distance $d_{\mathbb{H}}$. We show that $$ \|M_δf\|_{L^3(S^1)}\leq C(\varepsilon)δ^{-1/3-\varepsilon}\|f\|_{L^3(\mathbb{H}^1)},\quad f\in L^3(\mathbb{H}^1),$$ for all $\varepsilon>0$. The proof is based on a recent variant, due to Pramanik, Yang, and Zahl, of Wolff's circular maximal function theorem for a class of planar curves related to Sogge's cinematic curvature condition. As an application of our Kakeya maximal inequality, we recover the sharp lower bound for the Hausdorff dimension of Heisenberg Kakeya sets of horizontal unit line segments in $(\mathbb{H}^1,d_{\mathbb{H}})$, first proven by Liu.
