Some results on holomorphic generalized functions

Abstract

In this note, we present three independent results within generalized complex analysis (in the Colombeau sense). The first of them deals with non-removable singularities; we construct a generalized function u on an open subset Ω of C ⁿ , which is not a holomorphic generalized function on Ω but it is a holomorphic generalized function on Ω \ S, where S is a hypersurface contained in Ω. The second result shows the existence of a holomorphic generalized function with prescribed values in the zero-set of a classical holomorphic function. The last result states the existence of a compactly supported solution to the ∂¯ operator.

Keywords:

• generalized functions
• holomorphic generalized functions
• ∂¯ operator
• singularities
• germs of holomorphic generalized functions

AMS Subject Classification :

• 33E99
• 33C55
• 44A99