On the Riesz potential in Morrey spaces, associated with the Laplace–Bessel differential operator

2008 ◽  
Vol 19 (8) ◽  
pp. 607-612
Author(s):  
Yusuf Zeren
Keyword(s):  
Differential Operator ◽  
Riesz Potential ◽  
Morrey Spaces ◽  
Bessel Differential Operator

scholarly journals B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 715-730
Author(s):  
Javanshir J. Hasanov ◽  
Rabil Ayazoglu ◽  
Simten Bayrakci
Keyword(s):  
Differential Operator ◽  
Singular Integral ◽  
Riesz Potential ◽  
Type Theorem ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Singular Integral Operators ◽  
Sobolev Type ◽  
Riesz Potentials ◽  
Bessel Differential Operator

Abstract In this article, we consider the Laplace-Bessel differential operator {\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,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .

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

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Rabil Ayazoglu (Mashiyev) ◽  
Javanshir J. Hasanov
Keyword(s):  
Differential Operator ◽  
Shift Operator ◽  
Riesz Potential ◽  
Morrey Spaces ◽  
Generalized Shift Operator ◽  
Bessel Differential Operator ◽  
Generalized Shift

AbstractWe consider the generalized shift operator associated with the Laplace–Bessel differential operator

Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces

2018 ◽  
Vol 25 (2) ◽  
pp. 235-248
Author(s):  
Vagif S. Guliyev ◽  
Elman J. Ibrahimov
Keyword(s):  
Differential Operator ◽  
Morrey Space ◽  
Riesz Potential ◽  
Morrey Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Abstract In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator G_{\lambda}=(x^{2}-1)^{\frac{1}{2}-\lambda}\frac{d}{dx}(x^{2}-1)^{\lambda+% \frac{1}{2}}\frac{d}{dx},\quad x\in(1,\infty),\,\lambda\in\Bigl{(}0,\frac{1}{2% }\Bigr{)}. We prove that the G-Riesz potential {I_{G}^{\alpha}} , {0<\alpha<2\lambda+1} , is bounded from the G-Morrey space {L_{p,\lambda,\gamma}} to {L_{q,\lambda,\gamma}} if and only if \frac{1}{p}-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma},\quad 1<p<\frac{2% \lambda+1-\gamma}{\alpha}. Also, we prove that the G-Riesz potential {I_{G}^{\alpha}} is bounded from the G-Morrey space {L_{1,\lambda,\gamma}} to the weak G-Morrey space {WL_{q,\lambda,\gamma}} if and only if 1-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma}.

scholarly journals Riesz potential on the Heisenberg group and modified Morrey spaces

2012 ◽  
Vol 20 (1) ◽  
pp. 189-212
Author(s):  
Vagif S. Guliyev ◽  
Yagub Y. Mammadov
Keyword(s):  
Heisenberg Group ◽  
Fractional Integral ◽  
Morrey Space ◽  
Riesz Potential ◽  
Morrey Spaces ◽  
Potential Operator ◽  
Group Setting ◽  
Riesz Potential Operator ◽  
Homogeneous Dimension ◽  
Limiting Case

Abstract In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator ℑα, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn).

scholarly journals An observation problem for the Bessel differential operator

1984 ◽  
Vol 26 (1) ◽  
pp. 92-107 ◽  
Author(s):  
K.-D. Werner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Parabolic Partial Differential Equation ◽  
Final State ◽  
Bessel Differential Operator ◽  
State Observability

AbstractIn this paper, the parabolic partial differential equation ut = urr + (1/r)ur − (v2/r2)u, where v ≥ 0 is a parameter, with Dirichlet, Neumann, and mixed boundary conditions is considered. The final state observability for such problems is investigated.

scholarly journals Applications of Littlewood-Paley Theory forB˙σ-Morrey Spaces to the Boundedness of Integral Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Yasuo Komori-Furuya ◽  
Katsuo Matsuoka ◽  
Eiichi Nakai ◽  
Yoshihiro Sawano
Keyword(s):  
Riesz Potential ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Lipschitz Spaces ◽  
Potential Operators

The boundedness of the various operators onB˙σ-Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization ofB˙σ-Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization ofB˙σ-Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators onB˙σ-Lipschitz spaces is discussed.

Sign in / Sign up

Export Citation Format

Share Document