On regularization of plurisubharmonic functions near boundary points

We prove in an elementary way that for a Lipschitz domain \(D\subset \mathbb {C}^n\), all plurisubharmonic functions on D can be regularized near any boundary point.     

Characterizations of plurisubharmonic functions

Science China Mathematics - We give characterizations of (quasi-)plurisubharmonic functions in terms of Lp-estimates of $overline{partial}$ and Lp-extensions of holomorphic functions.     

Local indicators for plurisubharmonic functions

The notion of index, classical in number theory and its calculation by P. Lelong (1997) for plurisubharmonic functions, allows to define an indicator which is applied to the study of the Monge–Ampère operator and a pluricomplex Green function.     

Distributional boundary values of harmonic and analytic functions

A theory for distributional boundary values of harmonic and analytic functions is presented. In this analysis there arise several indicators that measure the growth of these functions near the boundaries. An extension of the Phragmén-Lindelöf maximum principle is derived. Furthermore, the algebraic properties of the space of real periodic distributions are studied. By introducing a new product, the harmonic product, the boundary conditions involving harmonic functions are transformed into ordinary differential equations.     

Smooth regularization of plurisubharmonic functions

We consider the problem of approximating a given plurisubharmonic function by smooth plurisubharmonic functions. We propose a new constructive approximation method that permits one to obtain more detailed information about the approximating functions. Thus a functionu ∈ PSH(ℂ ⁿ ) having finite growth order can be approximated by smooth functionsv ∈ PSH(ℂ ⁿ ) so that the difference |v−u| has almost logarithmic growth (Theorem 2). It can also be approximated so that the difference |v−u| has a power-law growth; in this case, however, power-law estimates on |gradv| appear (Theorem 3). Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 312–320, August, 1997. Translated by I. P. Zvyagin     

Extreme Jensen measures

Let Ω be an open subset of R ^d , d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u  dμ≤u(x) for every superharmonic function u on Ω. Denote by J _x (Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J _x (Ω)), the set of extreme elements of J _x (Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains. This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53. As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α _n } _n=1 ^∞ and a continuous function , there exists an entire function f≢0 satisfying f(α _n )=0 for all n, and |f(z)|≤M(z) for all z∈C.     

Jensen measures and analytic multifunctions

This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowski's theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions. The author was partially supported by an NSF Grant.     

On moduli of boundary values of holomorphic functions

This work was supported in part by the Republic of Slovenia Science Foundation     

Valuative analysis of planar plurisubharmonic functions

We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set of valuations on R and – by way of a natural Laplace operator defined in terms of the tree structure on – a positive measure on . This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampère mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed (1,1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers. Mathematics Subject Classification (1991) 32U25, 13A18, 13H05     

Acharacterization of Newtonian functions with zero boundary values

We give an intrinsic characterization of the property that the zero extension of a Newtonian function, defined on an open set in a doubling metric measure space supporting a strong relative isoperimetric inequality, belongs to the Newtonian space on the entire metric space. The theory of functions of bounded variation is used extensively in the argument and we also provide a structure theorem for sets of finite perimeter under the assumption of a strong relative isoperimetric inequality. The characterization is used to prove a strong version of quasicontinuity of Newtonian functions.