On commutativity of the second-order cross partial derivatives

Abstract

In 1956 J. Geffroy has given the following theorem: Let f be a real function defined at each point of a plane domain G. If f =(f') and fvx= (fg,) ' exist and are finite in G, and fJ' is bounded in G then fv =fv in G if and only if fx, considered as a function of x, is a derivative function [2]. To establish this theorem, J. Geffroy uses as a main tool Lebesgue's integral. But it is known that there are derivative functions which are not integrable in Lebesgue sense. It is more natural to use in this problem the Denjoy-Perron integral, which permits us to integrate each finite derivative function. But some general theorems concerning the Denjoy-Perron integral are too complicated to be used for practical purposes. The passage to limit under the integral sign, which is essential in Geifroy's proof, is a very difficult problem for the Denjoy-Perron integral. By modifications in Geffroy's proof, I have replaced the boundedness of f,' by the continuity of f-' with respect to x. I think that these conditions are not essential, being required only by the method of proof. I make the following CONJECTURE. If f tv and fv exist and are finite in G, then fx =fv in G if and only if f,,, is, for each y, a derivative function with respect to x and f{, is, for each x, a derivative function with respect to y. I am unable to prove or to contradict this conjecture, but I can establish the following: