Weak solutions to the complex Hessian equation.
Let \(\Omega\subset\mathbb C^n\) be open and let \(u\in{\mathcal C}^2(\Omega)\). For \(m=1,\dots,n\), put \[ H_m(u)(z):=\sum_{1\leq j_1<\dots<j_m\leq n} \lambda_{j_1}\cdots\lambda_{j_m}, \] where \(\lambda_1,\dots,\lambda_n\) are eigenvalues of the complex Hessian \([\frac{\partial^2u}{\partial z_j\partial\overline z_j}(z)]\), \(z\in\Omega\); \(H_1=(1/4)\Delta\), \(H_n=\) the complex Monge-Ampère operator. The author studies various properties of the class \({\mathcal P}_m(\Omega)\) of all \(m\)-subharmonic functions, which in the smooth case are defined by the condition \(H_m(u+A| z| ^2)\geq0\) for all \(A\geq0\); we have \(\mathcal{PSH}(\Omega)={\mathcal P}_n(\Omega)\subset\dots\subset {\mathcal P}_1(\Omega)=\mathcal{SH}(\Omega)\). Let \({\mathcal D}_m(\Omega)\) denote the set of all \(u\in{\mathcal P}_m\) for which there exists a regular Borel measure \(\mu=:H_m(u)\) such that if locally \(u_j\searrow u\), where \((u_j)_{j=1}^\infty\) is a sequence of smooth \(m\)-subharmonic functions, then \(H_m(u_j)\to\mu\) weakly. The main result of the paper are the following theorems: (1) If \(K\subset\subset\Omega\), \(u\in{\mathcal D}_m(\Omega)\), \(v\in{\mathcal P}_m(\Omega)\), \(u\leq v\) on \(\Omega\setminus K\), then \(v\in{\mathcal D}_m(\Omega)\). (2) A function \(u\in{\mathcal D}_m(\Omega)\) is \(m\)-maximal (i.e., for every \(v\in{\mathcal P}_m(\Omega)\), if \(v\leq u\) on \(\Omega\setminus K\), where \(K\subset\subset\Omega\), then \(v\leq u\) on \(\Omega\)) iff \(H_m(u)=0\). (3) \({\mathcal P}_m\subset L^p_{\text{loc}}\), \(p<n/(n-m)\) (it is an open question whether the inclusion is true for all \(p<nm/(n-m)\)).
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