A property of Peano derivatives in several variables (Q2839311)

A function \(f: \mathbb R^d\to \mathbb R \) is \(n\) times Peano differentiable if for each multiindex \( \alpha \) of degree \(|\alpha|\leq n\) there is a number \(f_{(\alpha)}(x)\), called the Peano derivative of \(f\), such that NEWLINE\[NEWLINEf(x+h)=f(x)+\sum_{1\leq |\alpha|\leq n} \frac{f_{(\alpha)}(x)}{\alpha!} h^\alpha +o(\|h\|^n).NEWLINE\]NEWLINE NEWLINEFor \(n>1\) this property of a function is weaker than being \(n\) times differentiable. But the authors show that one-sided boundedness of Peano derivatives implies that the classical derivatives \(f^{(\alpha)}\) exist and agree with \(f_{(\alpha)}\).NEWLINENEWLINESpecifically, suppose that \(f: \mathbb R^d\to \mathbb R \) is \(n\) times Peano differentiable on \(\mathbb R^d\) and that for each multiindex \( \alpha \) of degree \(|\alpha|=n\) the Peano partial derivative \(f_{(\alpha)}\) is either bounded from above or bounded from below on \(\mathbb R^d\). Corollary 3 of the present paper then asserts that \(f\) is \(n\) times differentiable on \(\mathbb R^d\) and its partial derivatives are equal to the Peano partials. In particular, the mixed partials of \(f\) do not depend on the order in which they are taken.NEWLINENEWLINEThe above result was proved by \textit{H. W. Oliver} [Trans. Am. Math. Soc. 76, 444--456 (1954; Zbl 0055.28505)] in one dimension. \textit{A. Fischer} [Proc. Am. Math. Soc. 136, No. 5, 1779--1785 (2008; Zbl 1151.26011)] extended it to several dimensions, but under more restrictive assumptions on the Peano derivatives.