sala A1-33, godz. 10:00–11:30

Michael Langenbruch (University of Oldenburg)
Surjectivity of Euler type dierential operators on spaces of smooth functions

Summary. This is a report on joint work with P. Domański (AMU Poznań). We discuss surjectivity of finite order Euler type partial differential operators \(P (\theta)\) on \(C^\infty(\Omega)\) where \(\Omega \subset \mathbb{R}^d\) is open and \(0\in \Omega\). Recall that these operators are partial differential operators with polynomial coefficients defined by
\[
P (\theta) :=\sum_{|\alpha|\leq m} c_\alpha
\] where
\[
\theta^\alpha := \prod_{j\leq d} \theta^{\alpha_j}_j\quad \text{for}\quad \theta_j := x_j \partial_j.
\] Our characterization is different from the one obtained by D. Vogt (Wuppertal) in case \(\Omega \subset ]0, \infty[^d\). Also, the proofs and the characterization differ very much from the surjectivity theory for Euler type differential operators on spaces \(A (\Omega)\) of real analytic functions developed jointly with P. Domański. In fact, the surjectivity problem for smooth functions can be reduced to the solvability theory for smooth functions supported in the canonical quadrants, using also a suitably defined Mellin transform on the corresponding dual spaces.

Tea and coffee at 9:30.