The Sobolev-Il’in theorem for the B-Riesz potential

Abstract

We study the Riesz potentials that are generated by the generalized shift operator associated to the Laplace-Bessel operator. We obtain an analog of the Sobolev-Il in theorem for the B-Riesz potential.

On the limiting case for boundedness of the B-Riesz potential in B-Morrey spaces

“…D e f i n i t i o n 3 (see [3]). We denote by B-BMO space the BMO γ (R n k,+ ) the set of all locally integrable functions f (x), x ∈ R n k,+ , with the finite norm…”

“…. , γ k > 0 have been investigated by many researchers, see Muckenhoupt and Stein [13], Kipriyanov [10], Trimeche [18], Lyakhov [12], Stempak [17], Gadjiev and Aliev [2], Guliyev [3,4,5,6], Guliyev and Hasanov [7], Guliyev, Serbetci and Ekincioglu [9], and others. In this paper we consider the generalized shift operator generated by the Laplace-Bessel differential operator Δ B in terms of which we study the boundedness of the modified B-Riesz potential e I α,γ in the limiting case.…”

Inversion of the Mixed Riesz Hyperbolic B-Potentials

“…Such operators in the case of negative powers are analogues of the Riesz potentials with Euclidean distance, and for them the term elliptic B-potentials is accepted. Elliptic B-potentials and its inverses have been studied in papers [12]- [26] and this list is not complete. It is interesting that the Bessel operator and the hyper-Bessel operator could be considered as generalized fractional derivatives too (see [27], p. 97).…”

B L m p , ν estimates for the Riesz transforms associated with Laplace-Bessel operator

“…In the next section, we collect some notation and preliminaries. In Section 3, we recall the definition of the higher order Riesz-Bessel transforms and Riesz potentials, generalized translation operator and prove derivatives of the Riesz potentials that have been previously used in the paper [2,3,8,9]. The Riesz potentials for the Laplace-Bessel differential operator will be studied and to obtain an estimate of the higher order Riesz-Bessel transform is discussed in Section 4.…”