Inversion and description of generalized Riesz potentials with quadratic characteristics (Q1902723)

The present article deals with the problem of inversion of generalized Riesz potentials (GRP) \[ K^\alpha_\theta\varphi= \int_{\mathbb{R}^n} {\theta(t')\over|t|^{n-\alpha}} \varphi(x- t)dt,\quad 0<\alpha<n,\quad\varphi\in L_p,\tag{1} \] \(t'=t/|t|\), where the characteristic \(\theta\) is a restriction on the unit sphere of the following quadratic form of non-zero rank: \[ \theta(t')= A(t',t')= \sum^n_{i,j=1} a_{ij}t_i' t_j',\quad a_{ij}\in\mathbb{R}^1.\tag{2} \] In the present article, we effectively construct an inversion formula for GRP (1) with characteristic (2).