A Removability Result for Holomorphic Functions of Several Complex Variables Pages 50-52

A Removability Result for Holomorphic Functions of Several Complex Variables
Pages 50-5288×31
Juhani Riihentaus

DOI: http://dx.doi.org/10.6000/1927-5129.2016.12.07

Published: 03 February 2016

Abstract: Suppose that Ω is a domain of Cnn≥1, E⊂Ω closed in Ω, the Hausdorff measure H2n-1 (E) = 0 , and ƒis holomorphic in Ω\E . It is a classical result of Besicovitch that if n=1 and ƒ is bounded, then ƒ has a unique holomorphic extension to Ω. Using an important result of Federer, Shiffman extended Besicovitch’s result to the general case of arbitrary number of several complex variables, that is, for  n≥1 . Now we give a related result, replacing the boundedness condition of ƒ by certain integrability conditions of ƒ and of ∂2ƒ/∂Ζ2jj=1,2,K ,n.

Keywords: Holomorphic function, subharmonic function, Hausdorff measure, exceptional sets.