Singular pseudodifferential operators defined using the mixed Fourier–Bessel transform are usually called Kipriyanov singular pseudodifferential operators (SPDO). In the paper, we provide an overview of three types of such operators. The Kipriyanov SPDOs are adapted to work with singular Bessel operators \(B_{\gamma _i}=\dfrac{\partial ^2}{\partial x_i^2}+\dfrac{\gamma _i}{x_i}~\dfrac{\partial }{\partial x_i},\) \( \gamma _i>-1.\) In this work, we focus on two modifications arising on the base of the “even Bessel \(\mathbb {J}\) -transforms” (i.e., for \(\gamma \in (-1,0)\) ) and the “even-odd Bessel–Kipriyanov–Katrakhov \(\mathbb {J}\) -transforms”. The latter ones were introduced to study differential equations containing singular differential operators \(\dfrac{\partial }{\partial x_i}B_{\gamma _i}\) with a negative parameter of the Bessel operator \(\gamma _i\in (-1,0).\)
