Inverting operators of the potential type whose kernels have singularities on a cone (Q1594009)

The paper deals with the invertibility of potential operators \(H_{m}^{\alpha }\) on \( L^{p}(\mathbb{R}^{n})\) \((0\leq m\leq 1,\) \(\text{Re }\alpha >n-2),\) with symbols \(( -r^{2}(\xi)+\frac{1-m^{2}}{4}-i\xi _{1}) ^{-\alpha /2},\) where \(r(y)=( y_{1}^{2}-y_{2}^{2}-\dots-y_{n}^{2}) ^{1/2}\) denotes the Lorentz distance. Also, they are inverted on ``sufficiently good '' smooth functions in terms of hypersingular integrals with generalized weighted differences. For \(0\leq m<1,\) the operators \(H_{m}^{\alpha }\) are hyperbolic analogues of the Bessel potentials over \(\mathbb{R}^{n},\) while for \(m=1,\) they correspond to negative powers of the telegraphic operator \(\square +\partial /\partial x_{1}\).