arXiv subjects
Publications and source records attributed to Per Ahag.
With inspiration from the K\"ahler geometry, we introduce a metric structure on the energy class, $\mathcal{E}_{1,m}$, of $m$-subharmonic functions with bounded energy and show that it is complete. After studying how the metric convergence relates to the accepted convergences in this Caffarelli-Nirenberg-Spruck model, we end by constructing geodesics in a subspace of our complete metric space.
We construct a family of quasimetric spaces in generalized potential theory containing $m$-subharmonic functions with finite $(p,m)$-energy. These quasimetric spaces will be viewed both in $\mathbb{C}^n$ and in compact K\"ahler manifolds, and their convergence will be used to improve known stability results for the complex Hessian equations.
This note aims to investigate the regularity of a solution to the Dirichlet problem for the complex Hessian equation, which has a density of the $m$-Hessian measure that belongs to $L^q$, for $q\leq\frac nm$.
By proving an estimate of the sublevel sets for $(\omega,m)$-subharmonic functions we obtain a Sobolev type inequality that is then used to characterize the degenerate complex Hessian equations for such functions with bounded $(p,m)$-energy.
By using quasi-Banach techniques as key ingredient we prove Poincar\'e- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of B\l ocki's integrability conjecture for $m$-subharmonic functions.
We characterize those compact sets for which the Dirichlet problem has a solution within the class of continuous $m$-subharmonic functions defined on a compact set, and then within the class of $m$-harmonic functions.
In this paper we prove the pluricomplex counterpart of the Moser-Trudinger and Sobolev inequalities in complex space. We consider these inequalities for plurisubharmonic functions with finite pluricomplex energy, and we estimate the concerned constants.
We study the problem of classifying the holomorphic $(m,n)$-subharmonic morphisms in complex space. This determines which holomorphic mappings preserves $m$-subharmonicity in the sense that the composition of the holomorphic mapping with a $m$-subharmonic functions is $n$-subharmonic. We show that there are three different scenarios depending on the underlying dimensions, and the model itself. Either the holomorphic mappings are just the constant functions, or up to composition with a homotethetic map, canonical orthogonal projections. Finally, there is a more intriguing case when subharmonicity is gained in the sense of the Caffarelli-Nirenberg-Spruck framework.
Let $\Omega\subset \mathbb C^n$ be a bounded domain, and let $f$ be a real-valued function defined on the whole topological boundary $\partial \Omega$. The aim of this paper is to find a characterization of the functions $f$ which can be extended to the inside to a $m$-subharmonic function under suitable assumptions on $\Omega$. We shall do so by using a function algebraic approach with focus on $m$-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of $m$-subharmonic functions.
We study the geometry of $m$-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every $m$-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly $m$-subharmonic, and has bounded $m$-Hessian measure.
In this article we address the question whether the complex Monge-Ampère equation is solvable for measures with large singular part. We prove that under some conditions there are no solution when the right-hand side is carried by a smooth subvariety in $\CEP{n}$ of dimension $k<n$.
The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods.
In this article we solve the complex Monge-Ampere equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kolodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampere measure, then it is a complex Monge-Ampere measure.