Let \(G\) be a bounded convex subset of \(\mathbb C^n\) with non-empty interior such that the intersection of \(Q\) and each supporting hyperplane to \(Q\) is compact. Let \(P(z) = \sum_{\alpha \in \mathbb Z_+^n} a_{\alpha}z^{\alpha} \) be an entire function of order one and type zero. The authors give necessary and sufficient conditions in order that each non-zero partial differential operator \(P(D):A(Q) \to A(Q)\) admitted a continuous linear right inverse \(R\) (i.e. \(P(D)\circ Q= R\)). Here \(A(Q)\) is the space of holomorphic functions on \(Q\). The conditions are expressed in terms of functions \(C^{\infty}_H\), \(C^0_H\) (without loss of generality we may assume that \(0\in \int Q\)) defined on the unit sphere \(S \subset \mathbb C^n\) and which are known (due to the second author) to describe the boundary behaviour of the pluricomplex Green functions of the interior of \(Q\) and of the complement of the closure of \(Q\), respectively.
The results of the present paper on the existence of right inverses extend results of K. Schwerdtfeger [Faltungsoperatoren auf Räumen holomorpher aund beliebig oft differenziebarer Funktionen, Thesis, Düsseldorf (1982; Zbl 0536.46014)], B. A. Taylor [Mich. Math. J. 29, 185-197 (1982; Zbl 0479.30018)], R. Meise and B. A. Taylor [Math. Z. 197, 139-152 (1988; Zbl 0618.32014)], for \(Q=\mathbb C^n\), of S. Momm [Bull. Sci. Math., II. Sér. 118, No. 3, 259-270 (1994; Zbl 0819.46039)], S. Melikhov and S. Momm [Stud. Math. 117, 79-99 (1995; Zbl 0842.46051)], Yu. F. Korobeinik and S. N. Melikhov [Sib. Math. J. 34, No. 1, 59-72 (1993); translation from Sib. Mat. Zh. 34, No. 1, 70-84 (1993; Zbl 0833.47044)] for open or compact convex sets in \(\mathbb C^n\), and M. Langenbruch [Stud. Math. 110, 65-82 (1994; Zbl 0824.35147)] and Yu. F. Korobeinik [Sb. Math. 187, No. 1, 53-80 (19996); translation from Mat. Sb. 187, No. 1, 55-82 (1996; Zbl 0873.47022)] for intervals on \(\mathbb R\) and for polygons in \(\mathbb C\), respectively.