On the boundedness of a B-Riesz potential in the generalized weighted B-Morrey spaces

We consider the generalized shift operator associated with the Laplace-Bessel differential operator Δ B = i=1 n 2 x i 2 +i=1 k γ i x ix i .Δ -B=Σ -i=1ⁿ(∂ ²)/(∂ x-i²) +Σ -i=1^k (γ -i)/(x-i)(∂)/(∂ x-i). The maximal operator M γ M-γ (B-maximal operator) and the Riesz potential I α,γ I-α ,γ (B-Riesz potential), associated with the generalized shift operator are investigated. We prove that the B-maximal operator M γ M-γ and the B-singular integral operator are bounded from the generalized weighted B-Morrey space p,ω 1 ,φ,γ (k,+ n )cal M-p,ω -1,φ ,γ (ℝ-k,+ⁿ) to p,ω 2 ,φ,γ (k,+ n )cal M-p,ω -2,φ ,γ (ℝ-k,+ⁿ) for all 1<p<1 &lt; p &lt; ∞ , φA p,γ (k,+ n )φ ∈ A-p,γ (ℝ-k,+ⁿ) . Furthermore, we prove that the B-Riesz potential I α,γ I-α ,γ , 0<α<n+|γ|0&lt;α &lt;n+|γ | , is bounded from the generalized weighted B-Morrey space p,ω 1 ,φ,γ (k,+ n )cal M-p,ω -1,φ ,γ (ℝ-k,+ⁿ) to q,ω 2 ,φ,γ (k,+ n )cal M-q,ω -2,φ ,γ (ℝ-k,+ⁿ) , where α/(n+|γ|)=1/p-1/qα /(n+|γ |)=1/p-1/q , 1<p<(n+|γ|)/α1&lt;p&lt;(n+|γ |)/α , φA 1+q/p ' ,γ (k,+ n )φ ∈ A-1+q/p^′,γ (ℝ-k,+ⁿ) and 1/p+1/p ' =11/p+1/p^′=1 .