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We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some subelliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest.doi:10.4310/mrl.2013.v20.n6.a15 fatcat:aau3o64obrb5ro7ylnmlq5xidu