Nonlinear conditions for differentiability by almost analytic extension
A remarkable theorem of Joris states that a function f:Rd→R is of class C∞ if two coprime powers of f, e.g., f2 and f3, are of class C∞. Naively dividing f3 by f2 clearly does not prove this result, but there is a way to make this strategy work. This path leads through the complex domain by almost analytic extension and holomorphic approximation. I will explain this approach and show that it can similarly be applied to a wide
variety of smooth regularity classes C, including for instance quasianalytic Denjoy--Carleman classes. Furthermore, I will present a full characterization of the analytic germs Φ:(R,0)→(Rn,0) with the property that Φ∘f∈C implies f∈C, for all continuous function germs f, in terms of a condition on the support of
the Taylor series of Φ.