Generalized hypersingular integrals with application to inversion of operators of potential type
The authors investigate convergence of hypersingular integrals of the form \[ \int_{\mathbb{R}^n} u_\varepsilon(t) (\Delta^\ell_t f)(x) \Omega(t) dt/|t|^{n+ \alpha},\quad x\in \mathbb{R}^n, \] where \(\lim_{\varepsilon\to 0} u_\varepsilon(t)= 1\) and \((\Delta^\ell_t f)(x)\) is the (generalized) finite difference of \(f\) of the Marchaud type. The results are applied to the inversion problem for operators of potential type \[ K^\alpha_\theta \varphi= \int_{\mathbb{R}^n} {\theta(x- t)\over |x- t|^{n- \alpha}} \varphi(t) dt,\quad 0< \text{Re } \alpha< n, \] in \(L^p\)-spaces.
scientific article; zbMATH DE number 815663