On the state of the second part of Hilbert’s fifth problem

Abstract

In the second part of his fifth problem Hilbert asks for functional equations “In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption.” In the case of the general functional equation

$$\begin{aligned} f(x)=h\Bigl (x,y,\bigl (g_1(x,y)\bigr ),\ldots ,\bigl (g_n(x,y)\bigr )\Bigr ) \end{aligned}$$

for the unknown function f under natural condition for the given functions it is proved on compact manifolds that \(f\in C^{-1}\) implies \(f\in C^{\infty }\) and practically the general case can also be treated. The natural conditions imply that the dimension of x cannot be larger than the dimension of y. If we remove this condition, then we have to add another condition. In this survey paper a new problem for this second case is formulated and results are summarised for both cases.