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

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 } .

scholarly journals Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Emin V. Guliyev ◽  
Yagub Y. Mammadov
Keyword(s):  
Maximal Operator ◽  
Morrey Space ◽  
Shift Operator ◽  
Morrey Spaces ◽  
Dunkl Operator ◽  
Generalized Shift Operator ◽  
Fractional Maximal Operator ◽  
Generalized Shift

We consider the generalized shift operator, associated with the Dunkl operatorΛα(f)(x)=(d/dx)f(x)+((2α+1)/x)((f(x)-f(-x))/2),α>-1/2. We study some embeddings into the Morrey space (D-Morrey space)Lp,λ,α,0≤λ<2α+2and modified Morrey space (modifiedD-Morrey space)L̃p,λ,αassociated with the Dunkl operator onℝ. As applications we get boundedness of the fractional maximal operatorMβ,0≤β<2α+2, associated with the Dunkl operator (fractionalD-maximal operator) from the spacesLp,λ,αtoL∞(ℝ)forp=(2α+2-λ)/βand from the spacesL̃p,λ,α(ℝ)toL∞(ℝ)for(2α+2-λ)/β≤p≤(2α+2)/β.

On Estimating the Approximation of Locally Summable Functions by Gegenbauer Singular Integrals

2008 ◽  
Vol 15 (2) ◽  
pp. 251-262
Author(s):  
Vagif S. Guliyev ◽  
Elman J. Ibrahimov
Keyword(s):  
Differential Operator ◽  
Singular Integrals ◽  
Shift Operator ◽  
Summable Function ◽  
Generalized Shift Operator ◽  
Order Estimation ◽  
Generalized Shift ◽  
Almost All

Abstract Using the generalized shift operator (GSO) generated by the Gegenbauer differential operator we introduce the notion of a Lebesgue–Gegenbauer (L-G)-point of a summable function 𝑓 on the interval [1,∞) and prove that almost all points of this interval are (L-G)-points of 𝑓. Furthermore, we give an exact (by order) estimation of the approximation of locally summable functions by singular integrals generated by GSO (Gegenbauer singular integrals).

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