On the behavior of strictly plurisubharmonic functions near real hypersurfaces

We describe the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in ℂ ⁿ , n≥3. Given a hypersurface we study continuous plurisubharmonic functions which are zero on the hypersurface and have Monge–Ampère mass greater than one in a one-sided neighborhood of the hypersurface. If we can find complex curves which have sufficiently high contact order with the hypersurface then the plurisubharmonic functions we study cannot be globally Lipschitz in the one-sided neighborhood.     

Approximation of plurisubharmonic functions

We extend a result by Fornaæss and Wiegerinck [Ark. Mat. 1989;27:257–272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.     

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.     

Langlands classification for real Lie groups with reductive Lie algebra

LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call admissible but we call them of Harish-Chandra type. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, ) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations U^V, with imaginary and withV from the quasi-discrete series of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.We show that irreducible continuous unitary representations of really reductive groups are of Harish-Chandra type. Then the results above yield the canonical decomposition of the unitary spectrum>G for any really reductiveG. In particular, this holds ifG/G ₀ is finite, so the center of the connected semi-simple subgroup with Lie algebra [g, g] may be infinite!Research supported, in part, by the Hungarian National Fund for Scientific Research (grant Nos. 1900 and 2648).     

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     

Extension of functions to plurisubharmonic functions in a topological vector space

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 11, pp. 1443–1449, November, 1989.     

A class of maximal plurisubharmonic functions

In this note, we introduce a class of maximal plurisubharmonic functions and use that class to prove some properties of maximal plurisubharmonics functions.