Power series spaces and weighted solution spaces of partial differential equations (Q1074763)

Let \(P(D)\) be a hypoelliptic system of partial differential operators with constant coefficients and let \(M\) be a weight function. The paper is concerned with the following weighted solution spaces of \(P(D)\): \[ N_ r:=\{f\in C^{\infty}(\mathbb R^ N)^ s\mid P(D)f=0; \| fe^{-\delta \cdot M}\|_{\infty}<\infty \quad\text{for some}\;\delta >r\} \] \[ N^{\rho}:=\{f\in C^{\infty}(\mathbb R^ N)^ s\mid P(D)f=0; \| fe^{-\delta \cdot M}\|_{\infty}<\infty \quad\text{for some}\;\delta <\rho \}. \] The spaces are provided with their natural projective (resp. inductive) topology. If \(M\) satisfies some mild technical conditions, then there is a sequence \(\alpha =(\alpha_ n)\), such that \(N_ r\) is (linear topologically) isomorphic to \(\Lambda_ 0(\alpha)\) and \(N^{\rho}\) is isomorphic to \(\Lambda_ 0(\alpha)_ b'\) for \(0<r\), \(\rho <\infty\). So \(N_ r\) and \(N^{\rho}\) have a basis. The power series space of finite type \(\Lambda_ 0(\alpha)\) is independent of \(r\) and \(\rho\). Especially, the dual \((N_ r)_ b'\) is isomorphic to \(N^{\rho}\) for \(0<r\), \(\rho <\infty\) (and fixed \(P(D)\)). \(\Lambda_ 0(\alpha)\) may be calculated for many systems and weight functions.