Growth properties of plurisubharmonic functions related to Fourier-Laplace transforms

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 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.