Gevrey hypoellipticity of a class of pseudodifferential operators
Let a(x,D) be a pseudo-differential operator of type \(S^ m_{\rho,\delta (\Omega)}\). Let its symbol a(x,\(\xi)\) satisfy the conditions: 1) \(| a(x,\xi)| \geq c| \xi |^{m'}\), \(| \xi | \geq B;\) 2) \(| a^{(\alpha)}_{(\beta)}(x,\xi)| \leq C_ 0C_ 1^{| \alpha +\beta |}B!^{\sigma}| a(x,\xi)| (1+| \xi |)^{-\rho | \alpha | +\delta | \beta |}\), \(x\in \Omega.\) The author proves that the operator a(x,D) is Gevrey hypoelliptic of order \(\theta\), where \(\theta =\max (1/\rho,\sigma /(1-\delta))\), that is if \(u\in {\mathcal E}'(\Omega)\) and \(a(x,D)u\in G^ s\) in \(\Omega ',\Omega '\subset \Omega\), then u is also in \(G^ s\) in \(\Omega '\) for \(s\geq \theta\).
scientific article; zbMATH DE number 4061819