Notions of Convexity pp 328-352 | Cite as

# 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 **R**^{n} 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 C^{n} 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 **R**^{n} × **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*(*∂/∂z*_{1},..., *∂/∂z*_{n})*u* = *f* in a pseudo-convex open set ⊂ **C**^{n} where *f* is analytic. We show that it is precisely in the C convex sets that a solution exists for arbitrary *P* and *f*.

## Keywords

Differential Operator Existence Theorem Finite Order Complex Line Boundary Distance## Preview

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