Necessary and sufficient conditions for the boundedness of B-Riesz potential in the B-Morrey spaces

We consider the generalized shift operator, associated with the Laplace–Bessel differential operator B = ni =1 ∂2 ∂x2i + ki =1 γi xi ∂ ∂xi . The maximal operator Mγ (B-maximal operator) and the Riesz potential Iα γ (B-Riesz potential), associated with the generalized shift operator are investigated. At first, we prove that the B-maximal operator Mγ is bounded from the B-Morrey space Lp,λ,γ to Lp,λ,γ for all 1 < p <∞ and 0 λ