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(alpha,gamma) circle times f and anisotropic fractional integral I(Omega,alpha,gamma) f with rough kernels, associated with the Laplace-Bessel differential operator Delta(B). We prove that the operator f -> K(alpha,gamma) circle times f is bounded from the Lorentz spaces L(p,r,gamma) (R(k)(n),(+)) to L(q,s,gamma) (R(k)(n),(+)) for 1 <= p < q <= infinity, 1 <= r <= s <= infinity. As a result of this, we get the necessary and sufficient conditions for the boundedness of I(Omega,alpha,gamma) from the Lorentz spaces L(p,s,gamma) (R(k)(n),(+)) to L(q,r,gamma) (R(k)(n),(+)), 1 < p < q < infinity, 1 <= r <= s <= 8 and from L(1,r,gamma) (R(k)(n),(+)) to L(q,infinity,gamma) (R(k)(n),(+)) = WL(q,gamma) (R(k)(n),(+)), 1 < q < infinity, 1 <= r <= 8. Furthermore, for the limiting case p = Q/alpha, we give an analogue of Adams' theorem on the exponential integrability of I(Omega,alpha,gamma) in L(Q/alpha,gamma) (R(k)(n),(+)).