Calculus of Variations and Geometric Measure Theory

A. Lorent

Differential inclusions, non-absolutely convergent integrals and the first theorem of complex analysis

created by lorent on 20 Aug 2012
modified on 18 Feb 2014


Accepted Paper

Inserted: 20 aug 2012
Last Updated: 18 feb 2014

Journal: Quarterly Journal of Mathematics
Year: 2014


In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of studies proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide a far reaching generalization.