论文标题
有限度量的Riesz电位梯度的精确代表
The precise representative for the gradient of the Riesz potential of a finite measure
论文作者
论文摘要
给定有限的非负Borel测量$ m $ IN $ \ MATHBB {r}^{d} $，我们识别Lebesgue设置$ \ Mathcal {l}（v_ {s}）\ subset \ subset \ mathbb {r}^r}^{d} \ int _ {\ Mathbb {r}^{d}} \ frac {x -y} {| x -y |^{s + 1}} \ s + 1}} \ mathrm {d} m（y），$$，$ 0，对于任何顺序$ 0 <s <d $。我们证明$ a \ in \ mathcal {l}（v_ {s}）$，并且仅当上述积分具有$ a $ a $ a和$ a和$ a和$ a和$ a和$ a和$ a \ lim_ {r \ to 0} {\ frac {m（b_ {b_ {r}（r}（r}（a））} {r^r^{r^{s s}}的$ n in and的castiants的castiants at prectiants and and and的cansepant的主值。 $ v_ {s} $ at $ a $ a $ cys与积分的主要价值一致。我们还研究了与平面康托尔集合相关的内在概率度量的库奇积分的勒布斯格点的存在，这导致了挑战新问题。
Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in \mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.
