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A note on estimates for elliptic systems with L1 data
Comptes Rendus Mathematique ( IF 0.8 ) Pub Date : 2019-11-01 , DOI: 10.1016/j.crma.2019.11.007
Bogdan Raita , Daniel Spector

In this paper we give necessary and sufficient conditions on the compatibility of a $k$th order homogeneous linear elliptic differential operator $\mathbb{A}$ and differential constraint $\mathcal{C}$ for solutions of \begin{align*} \mathbb{A} u=f\quad\text{subject to}\quad \mathcal{C} f=0\quad\text{ in }\mathbb{R}^n \end{align*} to satisfy the estimates \begin{align*} \|D^{k-j}u\|_{L^{\frac{n}{n-j}}(\mathbb{R}^n)}\leq c\|f\|_{L^1(\mathbb{R}^n)} \end{align*} for $j\in \{1,\ldots,\min\{k,n-1\}\}$ and \begin{align*} \|D^{k-n}u\|_{L^{\infty}(\mathbb{R}^n)}\leq c\|f\|_{L^1(\mathbb{R}^n)} \end{align*} when $k\geq n$.

中文翻译:

关于具有 L1 数据的椭圆系统估计的说明

在本文中,我们给出了\begin{align*} 的解的$k$th 阶齐次线性椭圆微分算子$\mathbb{A}$ 和微分约束$\mathcal{C}$ 的兼容性的充要条件\mathbb{A} u=f\quad\text{subject to}\quad \mathcal{C} f=0\quad\text{ in }\mathbb{R}^n \end{align*} 满足估计\begin{align*} \|D^{kj}u\|_{L^{\frac{n}{nj}}(\mathbb{R}^n)}\leq c\|f\|_{ L^1(\mathbb{R}^n)} \end{align*} 用于 $j\in \{1,\ldots,\min\{k,n-1\}\}$ 和 \begin{align *} \|D^{kn}u\|_{L^{\infty}(\mathbb{R}^n)}\leq c\|f\|_{L^1(\mathbb{R}^ n)} \end{align*} 当 $k\geq n$。
更新日期:2019-11-01
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