The boundedness of the generalized anisotropic potentials with rough kernels in the Lorentz spaces

Abstract

In this paper, we study the generalized anisotropic potential integral K _{α, γ}⊗ f and anisotropic fractional integral I _{Ω,α, γ} f with rough kernels, associated with the Laplace–Bessel differential operator Δ _B . We prove that the operator f→K _{α, γ}⊗ f is bounded from the Lorentz spaces to for 1≤p<q≤∞, 1≤r≤s≤∞. As a result of this, we get the necessary and sufficient conditions for the boundedness of I _{Ω,α, γ} from the Lorentz spaces to , 1<p<q<∞, 1≤r≤s≤∞ and from to , 1<q<∞, 1≤r≤∞. Furthermore, for the limiting case p=Q/α, we give an analogue of Adams’ theorem on the exponential integrability of I _{Ω,α, γ} in .

Keywords:

• Laplace–Bessel differential operator
• generalized anisotropic potential integral
• rough anisotropic fractional integral
• Lorentz spaces

2000 Mathematics Subject Classifications :

• 42B20
• 42B25
• 42B35

Acknowledgements

The authors express their thanks to the referee for his/her carefully reading, helpful comments and suggestions on the manuscript of this paper. V.S. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan Project No-01/023 and by Ahi Evran University Scientific Research Projects (BAP FBA-10-05).