On restricting Cauchy-Pexider functional equations to submanifolds (Q387613)

The author considers the restricted Cauchy-Pexider functional equation \(f(x)g(y)=h(x+y)\) for complex-valued functions on subsets of Euclidean spaces. The functions are assumed to be measurable and non-vanishing almost everywhere. The main result is that if the functional equation is satisfied almost everywhere on a hypersurface which is nowhere flat and contains no cylinder, then \(f\) and \(g\) are exponential affine functions almost everywhere. Related results are proved on curves and for similar equations with more than two unknown functions.