Mårten Nilsson: Quasibounded solutions to the complex Monge–Ampère equation

Abstract:

In potential theory of the complex plane, or more generally on \(\mathbb{R}^n\), we suggest a way to capture Poisson integrals of integrable, almost everywhere continuous functions (with respect to harmonic measure) in terms of superharmonic functions. Specifically, such harmonic functions may be characterized by two properties: (1) having boundary data with b-polar discontinuity set, i.e. a boundary singularity set of a positive superharmonic function, and (2) being quasibounded, i.e. the modulus is suitably majorized by a superharmonic function. We will discuss how these notions naturally extend to the non-linear setting of pluripotential theory, and how they may be applied to allow for discontinuous, unbounded boundary data in the Dirichlet problem for the complex Monge–Ampère operator.