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arXiv 1604.04744math.CV
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紧支集下$\bar\partial$ -方程的Hörmander解

Hörmander's solution of the $\bar\partial$ -equation with compact support

Eric Amar

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摘要：

本文的工作是对H.赫登马尔姆设$φ$是Cn中C2类严格多重次调和函数，$c_φ(z)$是$i\partial\bar\partialφ$的最小特征值，则$\forall z\in\mathbb{C}^n$，$c_φ(z)>0$ .我们用$L^2_{p,q}(\mathbb{C}^n, e^φ)$表示系数在$L^2_{p,q}(\mathbb{C}^n, e^φ)$中的$(p, q)$电流。我们证明了如果$ω\in L^2_{p,q}(\mathbb{C}^n,e^φ)$，$\bar\partial$ $ω$ = 0，q <n，则存在$\bar\partial$ u = $ω$的解u $\in L ^2_{p,q-1}(\mathbb{C}^n,c_φe^φ)$.这是通过一个定理来完成的，如果数据具有紧支持，则给出具有紧支持的解决方案。

英文摘要：

This work is a complement of the study on Hörmander's solution of the $\bar\partial$ equation initialised by H. Hedenmalm. Let $φ$ be a strictly plurisubharmonic function of class C 2 in C n, let $c_φ(z)$ be the smallest eigenvalue of $i\partial\bar\partialφ$ then $\forall z\in\mathbb{C}^n$, $c_φ(z)>0$. We denote by $L^2_{p,q}(\mathbb{C}^n, e^φ)$ the $(p, q)$ currents with coefficients in $L^2_{p,q}(\mathbb{C}^n, e^φ)$. We prove that if $ω\in L^2_{p,q}(\mathbb{C}^n,e^φ)$, $\bar\partial$$ω$ = 0 for q <n then there is a solution u $\in L ^2_{p,q-1}(\mathbb{C}^n,c_φe^φ)$ of $\bar\partial$u = $ω$. This is done via a theorem giving a solution with compact support if the data has compact support.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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