1 ISSN 1064–5624, Doklady Mathematics, 2009, Vol. 79, No. 3, pp. 1–5. © Pleiades Publishing, Ltd., 2009. Original Russian Text © L.N. Lyakhov, E.L. Shishkina, 2009, published in Doklady Akademii Nauk, 2009, Vol. 426, No. 4, pp. 443–447. An application of hypersingular integrals to study classical spaces of Riesz potentials was suggested by Samko [1] (see also [2]). Axially symmetric potentials were studied by Weinstein (see book [3] and the refer- ences therein). At present, the potentials introduced by Weinstein are known as (Riesz) B-potentials, or (Riesz) weight potentials. A distinguishing feature of these potentials is that they involve generalized shift, which arises in axially symmetric theory. In [4], the inversion of B-potentials with constant characteristic was stud- ied. This paper presents results on the inversion of more general B-potentials (general B-potentials). The gener- alization consists in that, first, B-potentials with non- constant (homogeneous) characteristic and, secondly, the so-called mixed B-potentials (in which generalized shifts act on a part of variables, while on the remaining variables ordinary shifts act) are allowed. The theory uses general B-hypersingular integrals introduced in [5] and [6] and results of [7] on inversion of similar opera- tors with density from the test function space S ev or Φ γ . Let denote the part of Euclidean space  N con- sisting of points x = ( x ', x ''), where x ' = ( x 1 , x 2 , …, x n ) and x '' = ( x n + 1 , x n + 2 , …, x N ) , for which x 1 > 0, x 2 > 0, …, x n > 0. By (  N ) we denote the set of infinitely differentiable functions even with respect to each of the variables x 1 , x 2 , …, x n , where 0 ≤ n ≤ N . The test func- tion space S ev consists of all functions from ( ) belonging to the space of Schwarz test functions. By we denote the class of distributions generated by the weight linear form ( f , g ) γ = ( x ) g ( x )( x ') γ dx ,  N + C ev ∞ C ev ∞  N + S ev ' f  N + ∫ where x') γ = and γ = (γ 1 , γ 2 , …, γ n ) is a multi- index consisting of fixed positive numbers. The generalized shift acting on a variable x i with i = 1, 2, …, n is defined by where x 1, i = (x 1 , x 2 , …, x i – 1 ) and x i, n = (x i + 1 , x i + 2 , …, x n ). The mixed generalized shift is defined by (1) Shift (1) generates a generalized convolution of functions [3] of the form In this paper, we use a special integral transform which acts as the Hankel transform 1 on each of the first variables x i with i = 1, 2, …, n and as the Fourier trans- form on the remaining variables x '' . Such a mixed Fou- rier–Bessel transform (see [1]) has the form 1 This transform is also known as the Bessel transform or the one- dimensional Fourier–Bessel transform. The latter term is used in [1]; however, it is shown in [1] that this transform is an identical mod- ification of the Hankel transform. x i γ i i 1 = n ∏ T x i y i fx () Γ γ i 1 + 2 ------------     Γ γ i 2 ---     π ---------------------- fx 1 i , , ( 0 π ∫ = x i 2 2 x i y i β i y i 2 + cos – x in , x'' ) β i β i , d sin γ i 1 – , , T y f ( ) x () T x' y' fx' x'' y'' – , ( ) = = T x 1 y 1 …T x n y n fx' x'' y'' – , ( ) . f * g ( ) γ fy () T y g ( ) x () y' ( ) γ y . d  N + ∫ = F B f [ ]ξ () f ˆ ξ () j γ x' ξ' , ( ) e i x'' ξ'' , 〈 〉 – fx () x' ( ) γ x . d  N + ∫ = = MATHEMATICS Inversion of General Riesz B-Potentials with Homogeneous Characteristic in Weight Classes of Functions L. N. Lyakhov a and E. L. Shishkina b Presented by Academician E.I. Moiseev November 20, 2008 Received January 26, 2009 DOI: 10.1134/S106456240903@@@@ a Voronezh State Technological Academy, pr. Revolyutsii 19, Voronezh, 394017 Russia; e-mail: ilina_dico@mail.ru b Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006 Russia; e-mail: lyakhov@box.vsi.ru