Kipriyanov-Katrakhov singular pseudodifferential operators

Singular pseudodifferential operators created on the base of the mixed Fourier–Bessel transform are usually called Kipriyanov singular pseudodifferential operators (SPDO). The paper provides an overview of three types of such operators. The Kipriyanov SPDOs are adapted to work with singular Bessel operators \(B_{\gamma_i}=\dfrac{\partial^2}{\partial
x_i^2}+\dfrac{\gamma_i}{x_i}~\dfrac{\partial}{\partial x_i},\) \(\gamma_i>-1.\) The main attention in our work is paid to two modifications that arose on the base of the “even \(\mathbb{J}\)-Bessel transforms” (i.e., for \(\gamma\in(-1,0)\)) and the “even-odd \(\mathbb{J}\)-Bessel–Kipriyanov–Katrakhov transforms”. The latter are introduced to study differential equations with singular differential operators \(\dfrac{\partial}{\partial
x_i}B_{\gamma_i}\) with a negative parameter of the Bessel operator \(\gamma_i\in(-1,0).\)