论文标题
海森伯格集团的平面事件和几何不平等现象
Planar incidences and geometric inequalities in the Heisenberg group
论文作者
论文摘要
我们证明，如果$ p，\ mathcal {l} $是$δ$ - 分离的点和$ \ mathbb {r}^{2} $的有限集，则$Δ$ - incessences的数量，$Δ$ - incences $ p $和$ \ m m ianctcal {l} $不超过不变的时间。 $ | p |^{2/3} | \ MATHCAL {l} |^{2/3} \cdotΔ^{ - 1/3}。$$我们应用界限以获取Heisenberg Group中Loomis-Whitney不平等的以下变体\ Lessim |π_{x}（k）|^{2/3} \ cdot |π_{y}（k）|^{2/3}，\ qquad k \ subset \ subset \ subset \ mathbb {h h}。 $$这里$π_{x} $和$π_{y} $分别是$ xt $ - 和$ yt $ - 平面的垂直预测，而$ | \ cdot | $是指在$ \ mathbb {h} $上的自然haar量度，或者是指飞机中的一个。最后，作为Loomis-Whitney不平等的推论，我们推断出$$ \ | f \ | _ {4/3} \ Lessim \ sqrt {\ | xf \ | \ | yf \ | }，\ qQuad f \ in Bv（\ Mathbb {h}），$$其中$ x，y $是$ \ mathbb {h} $中的标准水平向量字段。这是经典的几何Sobolev不等式的敏锐版本$ \ | f \ | _ {4/3} \ sillesim \ | \ | \ nabla _ {\ Mathbb {h} h}} f \ | $ for $ f \ for $ f \ in Bv（\ sathbb {h}）$。
We prove that if $P,\mathcal{L}$ are finite sets of $δ$-separated points and lines in $\mathbb{R}^{2}$, the number of $δ$-incidences between $P$ and $\mathcal{L}$ is no larger than a constant times $$|P|^{2/3}|\mathcal{L}|^{2/3} \cdot δ^{-1/3}.$$ We apply the bound to obtain the following variant of the Loomis-Whitney inequality in the Heisenberg group: $$ |K| \lesssim |π_{x}(K)|^{2/3} \cdot |π_{y}(K)|^{2/3}, \qquad K \subset \mathbb{H}. $$ Here $π_{x}$ and $π_{y}$ are the vertical projections to the $xt$- and $yt$-planes, respectively, and $|\cdot|$ refers to natural Haar measure on either $\mathbb{H}$, or one of the planes. Finally, as a corollary of the Loomis-Whitney inequality, we deduce that $$ \|f\|_{4/3} \lesssim \sqrt{\|Xf\| \|Yf\| }, \qquad f \in BV(\mathbb{H}), $$ where $X,Y$ are the standard horizontal vector fields in $\mathbb{H}$. This is a sharper version of the classical geometric Sobolev inequality $\|f\|_{4/3} \lesssim \|\nabla_{\mathbb{H}}f\|$ for $f \in BV(\mathbb{H})$.
