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arXiv 1312.7136math.CV
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严格伪凸域上PDMATHAAB方程解的$ L^{r}-L^{s}$估计

Estimates $ L^{r}-L^{s}$ for solutions of the $\bar \partial $ equation in strictly pseudo convex domains in ${\mathbb{C}}^{n}.$

Eric Amar

备注：
The result in the case Lr-Ls were known, thank to the referee, even in the case of q-convex domains. But here the method is different and we get the BMO and Lipschitz estimates
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摘要：

本文证明了$\bar \partial u=ω$方程在严格伪凸区域$ Ω$中解的估计。例如，如果$ (p,q)$当前的$ω$的系数在$L^{r}(Ω)$中有$1\leq r<2(n+1)$，则在$L^{s}(Ω)$中有解$u$，$\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$。Krantz在$(0,1)$形式的情况下，仅用于$L^{r}-L^{s}$部分。Ma和S. Vassiliadou一般$(p,q)$形式。为了得到完整的结果，我们提出了另一种方法，基于Carleson措施的顺序$α$和从属引理。

英文摘要：

We prove estimates for solutions of the $\bar \partial u=ω$ equation in a strictly pseudo convex domain $ Ω$ in ${\mathbb{C}}^{n}.$ For instance if the $ (p,q)$ current $ω$ has its coefficients in $L^{r}(Ω)$ with $1\leq r<2(n+1)$ then there is a solution $u$ in $L^{s}(Ω)$ with $\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$ We also have $BMO$ and Lipschitz estimates for $r\geq 2(n+1).$ These results were already done by S. Krantz in the case of $(0,1)$ forms and just for the $L^{r}-L^{s}$ part by L. Ma and S. Vassiliadou for general $(p,q)$ forms. To get the complete result we propose another approach, based on Carleson measures of order $α$ and on the subordination lemma.

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