A division problem in the space of entire functions of exponential type
The surjectivity of convolution operators and in particular of linear partial differential operators with constant coefficients is often equivalent to a problem of division in an appropriate space of entire functions of exponential type. In this frame Malgrange citel3 proved that for each convex domain G in R N each partial differential operator on E(G) and each convolution operator on A(C N) is surjectire. Ehrenpreis [4] characterized the surjective convolution operators on E(R N) by means of lower bounds of its symbol. HSrmander [7] extended this result to convolution operators ~(G2)----~(G1) where G~, i--1, 2 are convex domains in R N. Martineau [14] extended Malgrange's result proving that for each convex domain G in C N each partial differential operator of infinite order on A(G) is surjective. After these early papers, in the sequel several authors extended Ehrenpreis' result to convolution operators on spaces of ultradifferentiable functions $~(RN), Chou [2], Cioranescu [3], Meise, Taylor, Vogt [15], and Momm [18]. Others considered convolution operators A(G2)----~A(G1) for convex domains Gi, i=1, 2 in C N, KorobeYnik [9], Epifanov [5], [6], Tka~enko [29], [30], Napalkov [23], [24], Meril, Struppa [16], Morzhakov [21], [22], Sigurdsson [28], and recently Krivosheev [11] with the complete solution of this problem. There are many further results on this subject which cannot be mentioned here.