Growth properties of plurisubharmonic functions related to Fourier-Laplace transforms

Abstract

The purpose of the paper is to study the behavior at infinity of Fourier-Laplace transforms of distributions or more generally plurisubharmonic functions u in Cⁿ with bounds of the form

$$u(\zeta ) \leqslant C + A|\zeta |, \zeta \in C^n , \int_{R^n } {u^ + (\xi )(1 + |\xi |)^{ - n - 1} d\xi< \infty .} $$

The set L∞(u) of limits of T_tu = u(t·)/t as t → +∞ is a compact T invariant subset of the set P_H of plurisubharmonic functions in Cⁿ with v(ξ) ≤H(Im ξ), ξ ∈ Cⁿ, and equality on CRⁿ. Here H is a supporting function associated with u, and T is chain recurrent on L∞(u). The behavior of functions in P_H at CRⁿ is studied in detail, which leads to conditions on a set M ⊂P_H which guarantee that M = L∞(u) for some u as above. One can then choose u = log ¦ F ¦ where F is the Fourier-Laplace transform of a distribution with compact support.