Hypersingular integrals and Riesz potential spaces (Q674590)

Let \( I^{\alpha}\varphi = k_{\alpha} \ast \varphi , \alpha > 0 , \) be the well known Riesz potential operator. Its range \(I^{\alpha}(L^p), 1 < p < \infty , \) was characterized by the reviewer in terms of convergence of difference hypersingular integrals. This range was treated in the usual sense if \( \alpha < \frac{n}{p}\) and in the sense of distributions (over the Lizorkin test function space \(\Phi\)) if \( \alpha \geq \frac{n}{p}\), see e.g. \textit{S. Samko, A. Kilbas} and \textit{O. Marichev} [Fractional integrals and derivatives. Theory and applications, Gordon \& Breach (1993; Zbl 0818.26003)]. The author obtains similar results for the modification of the Riesz potential operator defined in the usual sense for all \(\alpha \) as \[ U^f_{\alpha , m} \;= \;\int_{\mathbb{R}^n} k_{\alpha,m}(x,y)f(y)dy , \] where \(k_{\alpha,m}\) is the Riesz kernel ``improved'' at infinity: \[ k_{\alpha,m} = k_{\alpha}(x-y) \;- \;\sum_{{}j{}\leq m} \frac{D^jk_{\alpha}(-y)}{j!}x^j \;, \] the sum being omitted if \( m \leq -1, \) using the corresponding type of hypersingular integrals -- with subtraction of the Taylor sum instead of taking finite differences. This approach allows also to obtain more information about the polynomial behaviour at infinity of functions which have a convergent hypersingular integral.