CR$CR$ analysis via local uniform completion, a sharp maximum modulus principle and holomorphic extension

Abstract Using iterated uniform local completion, we introduce a notion of continuous functions on locally closed subsets of reduced complex spaces, generalising both holomorphic functions and functions on submanifolds. Under additional assumptions of set‐theoretical weak pseudo‐concavity , we prove optimal maximum modulus principles for these functions, extending classical results for holomorphic functions and ordinary functions. Restricting to real submanifolds (possibly with singularities) of complex manifolds, we generalise results on holomorphic extension to full neighbourhoods known before only for submanifolds. The article is concluded by a study of singularities and explicit constructions of submanifolds on which the extension results are valid.