Hasanov, Javanshir J.; Ayazoglu, Rabil; Bayrakci, Simten

In this article, we consider the Laplace-Bessel differential operator Δ B k, n = a' i = 1 k a, 2 a, x i 2 + γ i x i a, a, x i + a' i = k + 1 n a, 2 a, x i 2, γ 1 > 0,γ k > 0. {\Delta }{{B}{k,n}}=\mathop{\sum }\limits{i=1}{k}\left(\frac{{\partial }{2}}{\partial {x}{i}{2}}+\frac{{\gamma }{i}}{{x}{i}}\frac{\partial }{\partial {x}{i}}\right)+\mathop{\sum }\limits{i=k+1}{n}\frac{{\partial }{2}}{\partial {x}{i}{2}},{\gamma }{1}\gt 0,\ldots,{\gamma }{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator M b, γ {M}{b,\gamma } and the commutator [ b, A γ[ {[}b,{A}{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator [ b, I α, γ[ {[}b,{I}{\alpha,\gamma }] of the B-Riesz potential on B-Morrey spaces L p, λ, γ {L}{p,\lambda,\gamma }, when b â BMO γ b\in {\text{BMO}}{\gamma }.

AtıfScopus5·WoS5·OpenAlex7

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Ayazoglu, Rabil; Hasanov, Javanshir J.

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 .

AtıfScopus2·WoS2·OpenAlex2

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