On \(\mathcal L\) characteristic of some potential type operators with oscillating symbols and singularities of the kernels on a sphere (Q1857388)

The paper under review is concerned with operators of convolution type \[ A^\alpha\varphi :x \mapsto \int_{\mathbb R^\alpha} \Omega_\alpha(|y|)\varphi(x-y)dy, \] with kernels \(\Omega_\alpha\) having singularities on the unit sphere (acting on sufficiently smooth functions \(\varphi\)). In the Fourier picture these operators act on Schwartz functions vanishing together with all derivatives on the unit sphere. Under additional conditions the authors prove \(L^p-L^q\)-boundedness, representing the operators as \(B^\alpha_1+B^\alpha_2\), where \(B^\alpha_1\) is given by a distribution with Fourier transform \(\xi\mapsto a(|\xi|)\cdot |\xi|^{-\alpha}e^{i|\xi|}\) and \(B^\alpha_2=AI^\alpha\) for some multiplier \(A\) and a Riesz potential \(I^\alpha\).