A critical growth rate for the pluricomplex Green function (Q1320600)

It is proved that, for each bounded convex domain \(G\) of \(\mathbb{C}^n\) which contains the origin, the following assertions are equivalent: (i) each nonzero differential operator \(P(D)\): \(A(G) \to A(G)\) admits a solution operator, (ii) there is a neighbourhood of the origin on which the corresponding largest plurisubharmonic function on \(\mathbb{C}^N\) with \(v_H \leq H\) and \(v_H - \log (1 + |z |)\) bounded from above, \(H\) being the support function of \(G\), satisfies \(v_H = H\), (iii) there is a constant \(C > 0\) such that \(G \subset G_x + C |x |U\) for all \(x < 0\), where \(U\) denotes the unit ball in \(\mathbb{C}^N\) and \(G_x : = \{z \in G : g(x) < x\}\), \(x < 0\), denote the level sets of the pluricomplex Green function \(g\).