On the \(\mathcal L\)-characteristic of some potential-type operators with oscillating symbols and singularities of the kernels on a sphere. (Q1432174)

The authors consider integral convolution operators over \(\mathbb R^n\) defined as pseudo-differential operators with the symbols \(\omega (\xi')| \xi| ^{-\alpha}e^{i\gamma | \xi| }\) where \(\frac{n-1}2<\alpha <n\), \(\gamma >0\), \(\xi'=\xi /| \xi| \), and \(\omega\) is a smooth non-degenerate homogeneous function of degree zero. As integral operators, they have kernels, non-integrable at infinity, with a possible singularity on the sphere \(\{ | x| =\gamma\}\). The \(\mathcal L\)-characteristic of such operators is found, that is, it is determined exactly for which pairs \((p,q)\) the operator acts continuously from \(L_p\) to \(L_q\).