2012/08/22 by Andrew Lorent, Lorent, Andrew
Mathematics · #26A39 #30A99 #35J47 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical and Theoretical Analysis #math.AP #math.CV #msc:26A39 #msc:30A99 #msc:35J47
paper · pdf · doi:10.48550/arxiv.1208.4659
openalex publication_date 2012/08/22 · arxiv created 2014/02/18 · arxiv updated 2014/02/19 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies C∞ 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.