Differential representation of measures and functions (Q678676)

Let \({\mathcal M}\) denote the vector space of all Radon measures \(\mu\) on \(\mathbb{R}^n\) with the property that \(p_m(\mu)= \int| x|^m d|\mu|(x)<\infty\) for every \(m\geq 0\), and let \({\mathcal M}_r\) be the Banach space of those Radon measures \(\mu\) for which \(\|\mu\|_{{\mathcal M}_r}= \int(1+| x|)^{r-1}d|\mu|(x)< \infty\). A Radon measure \(\mu\) is called \(r\)-oscillating if \(\mu\in{\mathcal M}_r\) and \(\int x^\alpha d\mu(x)= 0\) for every multiindex \(\alpha= (\alpha_1,\alpha_2,\dots,\alpha_n)\) with \(|\alpha|< r\). Among other results obtained the authors show that if \(r\) is a positive integer, \(\mu\in{\mathcal M}\) and \(\mu\) is \(r\)-oscillating then \(\mu\) can be represented in the form \(\sum_{|\alpha|= r}\partial^\alpha R_\alpha(\mu)\), where \(R_\alpha:{\mathcal M}\to{\mathcal M}\) is the linear operator defined by the formula \[ \langle R_\alpha(\mu), f\rangle= (-1)^{|\alpha|} {|\alpha|\over\alpha!} \int^1_0 (1-t)^{|\alpha|- 1}\int x^\alpha f(tx)d\mu(x)dt \] valid for every continuous function \(f\) on \(\mathbb{R}^n\) with compact support.