On the density of a special class of Lizorkin functions in a weighted Lebesgue space $L^\gamma_p$

We study the class of test functions $\Phi^+_\gamma,$ constructed on the principle of Lizorkin spaces by means of mixed Fourier–Kipriyanov–Katrakhov transform. Initially, such classes of functions, constructed on the basis of a mixed Fourier–Bessel transform, were investigated by L. N. Lyakhov. The spaces introduced by him could not take into account “odd” orders of singular derivatives. But the latter appeared to be fundamentally necessary in the problems of determining the fundamental solutions of differential equations (ordinary and in partial derivatives). The integral Kipriyanov–Katrakhov transform (belonging to the class of Bessel transforms) is adapted to work with singular differential operators of the type $D^{2m+k}_B\frac{\partial^k}{\partial x^k}B^m_x,$ where $k$ takes values 0 or 1, $B^m_x$ is a singular differential Bessel operator and the order of differentiation is 2$m$. The spaces of the basic functions that represent the images of the mixed Fourier–Kipriyanov–Katrakhov transform of functions vanishing at the origin and infinity are considered in this paper. We study the possibility of approximating functions from weighted Lebesgue classes $L^{\gamma}_p$ with power weight $\Pi|x_i|^{\gamma_i},$ namely, the density theorem $\Phi^+_\gamma$ in the Lebesgue function space $L^\gamma_p$.