Describing the space of the Riesz \(B\)-potentials \(\mathbb{U}_ \alpha^ \gamma (L_ p^ \gamma)\) using \(B\)-derivatives of order \(2[\alpha/2]\) (Q1363903)

The author continues his studies of the spaces of Riesz \(B\)-potentials, see for example his paper in Dokl. Akad. Nauk, Ross. Akad. Nauk 334, No. 3, 278-280 (1994; Zbl 0864.46019) and others. The Riesz \(B\)-potential is organized in a fashion similar to the usual Riesz potential but with respect to the generalized translation operator in \( R_+^n = \{x \in R^n: x_1 > 0,\dots,x_n >0 \}: \) \[ T^y f(x) = c \int _0 ^{\pi} \dots\int _0 ^{\pi} \sin ^{\gamma -1 } t \;f(A_{x,y} t) dt \] where \( t = (t_1,\dots,t_n), dt = dt_1 \dots dt_n , \sin ^{\gamma -1} t = \prod _{k=1}^{n} \sin ^{\gamma _k-1 }t_k \) and \[ A_{x,y} t = (\sqrt{x_1^2+ y_1^2-2x_1y_1\cos t_1},\dots,\sqrt{x_n^2+ y_n^2-2x_ny_n\cos t_n}), \] \(c\) being some normalizing constant. This ``translation'' is well suited to differential operators which are products of powers of Bessel type operators \( \frac{\partial ^2}{\partial x_k ^2} + \frac{\gamma _k}{x_k} \frac{\partial }{\partial x_k }. \) In 1980 the reviewer and \textit{S. M. Umarkhadzhiev} [Izv. Vyssh. Uchebn. Zaved., Mat. 1980, No. 11(222), 79-81 (1980; Zbl 0461.46030)] have shown that the space \(I^{\alpha}(L_p)\) of usual Riesz potentials can be described in terms of the higher derivatives of order \(2\left[\frac{\alpha}{2}\right]\). The author extends this characterization to the case of \(B\)-potentials. To this end, the corresponding apparatus is developed familiar from the case of usual Riesz potentials: simultaneous approximation of functions and their Riesz derivatives in different norms, existence of intermediate weak derivatives and so on. One of the main difficulties arised, is in the extension of the simultaneous approximation mentioned above, which is aggravated by the complicated constructions due to the generalized shift.