Complex Analysis and Operator Theory ( IF 0.8 ) Pub Date : 2020-10-13 , DOI: 10.1007/s11785-020-01044-9 Hichame Amal , Saïd Asserda , Ayoub El Gasmi
We study the complex equations of Hessian type
$$\begin{aligned} (dd^c u)^m\wedge \beta ^{n-m}=F(u,.)d\mu , \end{aligned}$$where \(\mu \) is a positive Borel measure defined on an m-hyperconvex domain of \({\mathbb {C}}^{n}\), m is an integer such that \(1\le m\le n\) and \(\beta :=dd^{c}\vert z\vert ^{2}\) is the standard kähler form in \({\mathbb {C}}^{n}. \) We show that, under some regularity conditions on the density F, if this equation admits a (weak) subsolution in \(\Omega \), then it admits a (weak) solution with a prescribed least maximal m-subharmonic majorant in \(\Omega \).
中文翻译:
任意度量的复数Hessian型方程的弱解
我们研究了Hessian型的复方程
$$ \ begin {aligned}(dd ^ cu)^ m \ wedge \ beta ^ {nm} = F(u,。)d \ mu,\ end {aligned} $$其中\(\ mu \)是在\({\ mathbb {C}} ^ {n} \)的m-超凸域上定义的正Borel度量,m是一个整数,使得\(1 \ le m \ le ñ\)和\(\测试:= DD ^ {C} \ VERTž\ VERT ^ {2} \)是在标准凯勒形式\({\ mathbb {C}} ^ {N} \)。我们表明即,在密度一定的规律性条件下˚F,如果该方程承认一个(弱)下解在\(\欧米茄\)一个(弱)与规定的至少最大解,然后将其承认米-subharmonic majorant在\(\欧米茄\)。