Partial differential equations without solution operators in weighted spaces of (generalized) functions (Q1083603)

Let P(D) be a hypoelliptic partial differential operator with constant coefficients. For a positive convex weight function M let E be one of the classical weighted spaces \[ E(M):=\{f\in C^{\infty}({\mathbb{R}}^ n)| \quad \sup_{| k| \leq n} | D^ kf(x)e^{-M(| x| /n)}| <\infty \quad for\quad any\quad n\in {\mathbb{N}}\} \] (introduced by V. P. Palamodov) or \[ (W_{M,\infty})_ b':=(\{f\in C^{\infty}({\mathbb{R}}^ n)| \quad \sup_{| k| \leq n} | D^ kf(x)e^{M(nx)}| <\infty \quad for\quad any\quad n\in {\mathbb{N}}\})_ b' \] (introduced by I. M. Gelfand/G. E. Shilov). The paper is concerned with the question, if P(D) admits a (continuous linear) right inverse in E. It is shown, that P(D) never has a right inverse in E(M), if the Young conjugate \(M^*\) is stable, while elliptic operators have a right inverse in \((W_{M,\infty})_ b'\), if M is stable. However if (roughly speaking) the restriction of P(x) to two variables is a semielliptic nonelliptic polynomial, then again no right inverse for P(D) exists in \((W_{M,\infty})_ b'\), if M satisfies some mild technical conditions. Especially, if P(D) is a semielliptic nonelliptic operator, then P(D) has no right inverse in the spaces \((S_{\alpha,\infty})_ b'\), \(0<\alpha <1\) (except for at most one value of \(\alpha)\). In the forthcoming papers by the author [see ''Solution operators for partial differential equations in weighted Gevrey spaces'', preprint and ''Complemented kernels of partial differential operators in weighted spaces of (generalized) functions'', preprint] it is shown, that any partial differential operator with constant coefficients has a right inverse in E, if the weighted space E of (ultra) distributions is defined suitably.