We search for pseudo-differential operators acting on holomor-phic Sobolev spaces. The operators should mirror the stan-dard 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, pro-vided that an old statement of Boutet de Monvel holds true generally. In the Lie group setting, we find even larger alge-bras, and characterize all their elliptic elements. These latter algebras are determined by global matrix-valued symbols. (c) 2023 Elsevier Inc. All rights reserved.
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