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Weighted Estimates of Generalized Potentials in Variable Exponent Lebesgue Spaces on Homogeneous Spaces

  • Mubariz G. Hajibayov
  • Stefan G. Samko
Part of the Operator Theory: Advances and Applications book series (OT, volume 210)

Abstract

For generalized potential operators with the kernel \( \frac{{a\left[ {\varrho \left( {x,y} \right)} \right]}} {{\left[ {\varrho \left( {x,y} \right)} \right]^N }} \) on bounded measure metric space (X, μ, ϱ) with doubling measure μ satisfying the upper growth condition μB(x, r) ≤ Cr N ∈ (0,∞), we prove weighted estimates in the case of radial type power weight w = [ϱ(x, x 0)] v . Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the weighted variable exponent Lebesgue space L p(·)(X, w, μ) into a certain weighted Musielak-Orlicz space \( L^\Phi \left( {X,w^{\frac{1} {{p\left( {x_0 } \right)}}} ,\mu } \right) \) with the N-function ϕ(x, r) defined by the exponent p(x) and the function a (r).

Mathematics Subject Classification (2000)

Primary 43A85 46E30 Secondary 47B38 

Keywords

Weighted estimates generalized potential variable exponent variable Lebesgue space metric measure space space of homogeneous type Musielak-Orlicz space Matuszewska-Orlicz indices 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Mubariz G. Hajibayov
    • 1
  • Stefan G. Samko
    • 2
  1. 1.Institute of Mathematics and Mechanics of NAS of AzerbaijanBakuAzerbaijan
  2. 2.Department of MathematicsUniversity of AlgarveFaroPortugal

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