论文标题
Hölder连续订阅定理的复杂Hessian方程
The Hölder continuous subsolution theorem for complex Hessian equations
论文作者
论文摘要
令$ω\ subset \ mathbb c^n $成为一个有限的$ m $ -m $ -pseudoconvex域（$ 1 \ leq m \ leq n $）和$μ$ $ $ $ $ $ $ $ $ $ $ $ $ $ω$。然后，我们解决了复杂的Hessian方程$（dd^c u）^m \wedgeβ^{n -m} =μ$上的hölder连续订阅问题。也就是说，我们表明，如果给定的Hölder连续边界值在$ω$上承认唯一的Hölder连续解决方案，如果它承认hölder连续订购$ω$。解决问题的主要步骤是建立一个新的容量估计，表明$ m $ - hessian的hölder连续$ m $ m $ $ m $ -subharmonic在$ω$上具有零边界价值的主导，由$ m $ -Hessian的容量在$ $ -Hessian的容量中，$ $ω$带有（explicit）expentonent $ termentent $τ> 1 $。
Let $Ω\Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $μ$ a positive Borel measure with finite mass on $Ω$. Then we solve the Hölder continuous subsolution problem for the complex Hessian equation $(dd^c u)^m \wedge β^{n - m} = μ$ on $Ω$. Namely, we show that this equation admits a unique Hölder continuous solution on $Ω$ with a given Hölder continuous boundary values if it admits a Hölder continuous subsolution on $Ω$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a Hölder continuous $m$-subharmonic function on $Ω$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $Ω$ with an (explicit) exponent $τ> 1$.
