Convexity with Respect to Differential Operators

Abstract

In Section 6.1 we sum up with brief indications of the proofs some facts on the open sets in Rⁿ where a differential equation P(D)u = f with constant coefficients can always be solved. Depending on whether f is allowed to be an arbitrary distribution or a C^∞ function (or a distribution of finite order), we get two classes of admissible open sets depending on P. Those which are admissible for every P are precisely the genuinely convex sets. However, more general domains are admissible for individual operators P. In Section 6.2 we prove by methods close to those used in Section 4.2 that in a pseudo-convex open set in Cⁿ all equations of the form $P(\partial /\partial \bar z_1 , \ldots ,\partial /\partial \bar z_n )u = f$ can be solved. In fact, we prove more general results for operators in a product space Rⁿ × C^N which have this structure with respect to the complex variables. In Section 6.3 we pass to the existence of analytic solutions of equations of the form P(∂/∂z₁,..., ∂/∂z_n)u = f in a pseudo-convex open set ⊂ Cⁿ where f is analytic. We show that it is precisely in the C convex sets that a solution exists for arbitrary P and f.