A note on the harmonic derivative (Q1772914)

The harmonic derivative is understood as in Chapter VIII in [\textit{E. M. Stein}, ``Singular integrals and differentiability properties of functions'' (1970; Zbl 0207.13501)]: if \(f\) has harmonic derivatives \(D^\alpha_h f(a)\) of all orders \(|\alpha|\leq m\) it is said that \(f\) has a harmonic derivative at \(x=a\) of order \(m\) and the harmonic differential of \(h\) at \(a\) of order \(m\) is \(P(x-a)=\sum_{|\alpha|\leq m} d_\alpha\cdot(x-a)^\alpha\). Let \(m-1<\ell\leq m\), where \(m\) is a positive integer. It is said that \(f\) has property \(B_\ell\) at \(x=a\) if there is a polynomial \(Q(x)=\sum_{1\leq|\alpha|\leq m-1}c_\alpha\cdot x^\alpha\) without constant term and of degree at most \(m-1\), \(Q(x)\equiv 0\) for \(0<\ell\leq 1\), such that if \(f_m(x)=f(x)-Q(x)\), then for every \(\varepsilon>0\) there are positive numbers \(t\) and \(\delta\), \(0<t<\min(\varepsilon,1)\), such that \(0<|x-a|<\delta\) and \(|z-x|\leq t\cdot|x-a|\) imply \(|f_m(z)-f_m(x)|\leq\varepsilon\cdot|x-a|^\ell\). The main result is contained in the following theorems: Let \(m\) be a positive integer and \(f\) a function defined in a neighborhood of \(x=a\) in \(\mathbb R^n\). Then \(f\) is differentiable at \(x=a\) of order \(m\) if and only if (1) \(f\) has a harmonic derivative of order \(m\) at \(x=a\) and (2) \(f\) has property \(B_m\) at \(x=a\). This result follows from a more general theorem, obtained by replacing condition (1) with the condition that \(D^\alpha_h f(a)\) exists for all \(|\alpha|=m\).