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

Rabil Ayazoglu (Mashiyev) and Javanshir J. Hasanov

Abstract

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

ΔB=i=1n2xi2+i=1kγixixi.

The maximal operator Mγ (B-maximal operator) and the Riesz potential Iα,γ (B-Riesz potential), associated with the generalized shift operator are investigated. We prove that the B-maximal operator Mγ and the B-singular integral operator are bounded from the generalized weighted B-Morrey space p,ω1,φ,γ(k,+n) to p,ω2,φ,γ(k,+n) for all 1<p<, φAp,γ(k,+n). Furthermore, we prove that the B-Riesz potential Iα,γ, 0<α<n+|γ|, is bounded from the generalized weighted B-Morrey space p,ω1,φ,γ(k,+n) to q,ω2,φ,γ(k,+n), where α/(n+|γ|)=1/p-1/q, 1<p<(n+|γ|)/α, φA1+q/p',γ(k,+n) and 1/p+1/p'=1.

The authors would like to express their gratitude to the referee for very valuable comments and suggestions.

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Received: 2014-5-7
Revised: 2014-6-28
Accepted: 2014-11-19
Published Online: 2016-3-17
Published in Print: 2016-6-1

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