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 .