On the state of the second part of Hilbert's fifth problem (Q6185270)

In this interesting paper the author gives a survey of several of his results related to the second part of Hilbert's fifth problem. Consider the following general functional equation: \[ f(x)=h(x,y,f(g_1(x,y)),\cdots,f(g_n(x,y))) \] where \(f\) is the unknown function. The problem can be roughly formulated in this way: it is true that every solution \(f\) which is either Lebesgue or Baire measurable is \(C^{\infty}\)? The paper presents results for various types of functional equations under different conditions on the sets and functions involved, along with open problems.