room: A1-33, godz. 10:30–12:00
Michael Langenbruch (University of Oldenburg)
Surjective Euler type operators on spaces of smooth functions

Summary. This is a report on joint work with P. Domański (AMU Poznań). I will discuss new developments concerning the solvability of finite order Euler type partial differential operators \(P(\theta)\) on \(C^\infty(\Omega)\) where \(\Omega\subset\mathbb{R}^d\) is open. Recall that these operators are partial differential operators with polynomial coefficients defined by
\[
P(\theta):=\sum_{|\alpha|\leq m}c_\alpha\theta^\alpha\text{ where }\theta^\alpha:=\prod_{j\leq d}\theta_j^{\alpha_j}\text{ for }\theta_j:=x_j\partial_j.
\] Specifically, I will present a complete answer to the following two questions:

1. When is \(P(\theta)\colon C^\infty(\Omega)\to C^\infty_{I(P)}(\Omega)\) surjective?
2. When is \(P(k+\theta)\colon C^\infty(\Omega)\to C^\infty_{I(P(k+\cdot))}(\Omega)\) surjective for any \(k\in\mathbb{R}^d\)?

Recall that \(C^\infty_{I(P)}(\Omega)\) is the canonical upper bound for the range of \(P(\theta)\). Inheritance properties and the relation between questions 1. and 2. are also discussed.