Commutators of singular \(\varvec{\mathbb {K}}\)-pseudodifferential operators in \(\varvec{\mathbb {R}_n}\)

To study problems with the Kipriyanov singular differential operator \(\Delta _{B_{-\gamma }}\) ${\mathrm{\Delta }}_{B_{-\gamma }}$ with a negative parameter \(-\gamma \in (-1,0)\) $-\gamma \in (-1,0)$ , an integral transformation is introduced in the work, based on the solution \(u= \mathbb {J}_\mu \) $u={\mathbb{J}}_{\mu }$ of the singular Bessel equation \(B_{-\gamma }u{+}u=0\) $B_{-\gamma }u+u=0$ . This solution is expressed through a Bessel function of the first kind with a positive parameter \(\mu =\frac{\gamma {+}1}{2}\) $\mu =\frac{\gamma +1}{2}$ . An even, odd and complete FBK transform (Fourier—Bessel—Kipriyanov—Katrakhov transform) and a class of singular \(\mathbb {K}\) $\mathbb{K}$ -pseudodifferential operators are constructed. Theorems on the order and commutator of singular \(\mathbb {K}\) $\mathbb{K}$ -pseudodifferential operators are obtained.