Hölder estimates and regularity for holomorphic and harmonic functions (Q1424015)

In this paper about analysis on singular Riemannian manifolds, it is assumed that on such a manifold a weak mean value property holds. This property is the existence of constants \(b,\lambda\geq1\) such that \(u^2(q)\leq{\lambda\over| B_r(q)| }\int_{B_{br}(q)}u^2\) holds for all subharmonic functions \(u\). If this condition holds on a Kähler manifold, there is an oscillation bound, implying Hölder continuity, for locally defined holomorphic functions. A typical situation to apply this is the case when the Kähler manifold is a singular algebraic or minimal variety, when the mean value property is implied by the Sobolev inequality of the ambient manifold. In a similar vein, Liouville properties of holomorphic functions can be studied, to the effect that a constant \(\alpha>0\) exists such that every holomorphic function of growth order at most \(\alpha\) must be constant. In the real case, i.e.\ harmonic instead of holomorphic functions, it is suggested that for each singular point \(p\) of the manifold, all harmonic functions up to a finite-dimensional exception should be Hölder continuous in \(p\). This question remains partially open, but is answered for \(C^0\) regularity.