Pseudo-differential operators in algebras of generalized functions and global hypoellipticity
Claudia Garetto
TL;DR
This work develops a comprehensive global pseudodifferential calculus on Colombeau-type algebras, introducing the algebras $\mathcal{G}_{\tau,\mathcal{S}}(\mathbb{R}^n)$ and $\mathcal{G}^\infty_{\mathcal{S}}(\mathbb{R}^n)$ to study PDEs globally on $\mathbb{R}^n$. It defines weighted symbol and amplitude classes $\mathcal{S}^m_{\Lambda,\rho}$ and related constructions, and builds a robust theory of pseudodifferential operators acting on these generalized function spaces, including composition, kernel regularity, and mollifier-invariant definitions. A key contribution is the global hypoellipticity framework, via hypoelliptic/elliptic symbols and a parametrix, yielding regularity results: if $Au$ is $\mathcal{S}$-regular, then $u$ is $\mathcal{S}$-regular. The framework unifies and extends classical global calculus to Colombeau settings, enabling analysis of PDEs globally defined on $\mathbb{R}^n$ with generalized coefficients.
Abstract
The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on $\mathbb{R}^n$.