Singular Solutions to the Complex Monge-Amp\`ere Equation

Abstract

We present an explicit pluripotential and viscosity solution to the complex Monge-Amp\`ere equation with constant right-hand side on $\mathbb D\times\mathbb C^{n-1}\,(n\geq 2)$, which lies merely in $W^{1,2}_{loc}\cap W^{2,1}_{loc}$ and is not even Dini continuous. Additionally, we exhibit two families of explicit entire toric solutions on $\mathbb C^n$ with continuous H\"older exponent $\alpha\in(0,1)$ and $W^{2,p}_{loc}$ exponent $p\in (1,2)$.