论文标题
椭圆系统的分数整合和最佳估计
Fractional Integration and Optimal Estimates for Elliptic Systems
论文作者
论文摘要
在本文中，我们对div-curl系统的最佳Lorentz估算的Bourgain和Brezis问题的欧几里得类似物给出了肯定的答案：功能$ z = \ permatorAtorname*{curl} {curl}（curl}（-Δ） \ operatorName*{div} z = 0 \ end {align*}，并且存在常数$ c> 0 $，这样\ begin \ begin {align*} \ | z \ | _ {l^{3/2,1}（\ mathbb {r}^3; \ m athbb {r}^3）} \ leq c \ | f \ | _ {l^{1}（\ mathbb {r}^3; \ Mathbb {r}^3）}。 \ end {Align*}我们的证明依赖于新的端点hardy-littlewood-sobolev不平等，用于无差异措施，我们通过独立兴趣而获得的，这是这种对象的原子分解。
In this paper we give an affirmative answer to the Euclidean analogue of a question of Bourgain and Brezis concerning the optimal Lorentz estimate for a Div-Curl system: The function $Z=\operatorname*{curl} (-Δ)^{-1} F$ satisfies \begin{align*} \operatorname*{curl} Z = F \newline \operatorname*{div} Z = 0 \end{align*} and there exists a constant $C>0$ such that \begin{align*} \| Z\|_{L^{3/2,1}(\mathbb{R}^3;\mathbb{R}^3)} \leq C\| F\|_{L^{1}(\mathbb{R}^3;\mathbb{R}^3)}. \end{align*} Our proof relies on a new endpoint Hardy-Littlewood-Sobolev inequality for divergence free measures which we obtain via a result of independent interest, an atomic decomposition of such objects.
