Algebras of pseudo-differential operators acting on holomorphic Sobolev spaces

TL;DR

This paper investigates pseudo-differential operators on holomorphic Sobolev spaces over real-analytic manifolds and Lie groups, characterizing their algebraic structure and mapping properties in relation to the Laplacian and other operators.

Contribution

It identifies and characterizes algebras of pseudo-differential operators acting on holomorphic Sobolev spaces, extending classical results to complexified settings and Lie groups.

Findings

Operators in the commutant of the Laplacian are of the desired type.

Larger operator algebras are characterized by global matrix-valued symbols.

Elliptic elements within these algebras are explicitly characterized.

Abstract

We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The operators should mirror the standard Sobolev mapping property in the holomorphic analogues. The setting is a closed real-analytic Riemannian manifold, or Lie group with a bi-invariant metric, and the holomorphic Sobolev spaces are defined on a fixed Grauert tube about the core manifold. We find that every pseudo-differential operator in the commutant of the Laplacian is of this kind. Moreover, so are all the operators in the commutant of certain analytic pseudo-differential operators, but for more general tubes, provided that an old statement of Boutet de Monvel holds true generally. In the Lie group setting, we find even larger algebras, and characterize all their elliptic elements. These latter algebras are determined by global matrix-valued symbols.