Sławomir Kołodziej

On regularization of plurisubharmonic functions on manifolds

We study the question of when a γ-plurisubharmonic function on a complex manifold, where γ is a fixed (1, 1)-form, can be approximated by a decreasing sequence of smooth 7-plurisubharmonic functions. We show in particular that it is always possible in the compact Kahler case.

Separately harmonic and subharmonic functions

Let u(x, y) be defined in B1×B2 where B1⫅ℝm and B2⫅ℝn, and assume that u(x, ·) harmonic for every fixed x and u(·, y) is subharmonic for every fixed y. We show that if u(·, y) is, in addition, C2 for each y then u is subharmonic in B1×B2 in both variables jointly.