Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel a([rho(x,y)])/[rho(x,y)](N) on bounded quasimetrie measure space (X, mu, d) with doubling measure mu satisfying the upper growth condition mu B(x, r) <= Kr-N, N is an element of (0, infinity). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space L-p(.) (X, mu) into a certain Musielak-Orlicz space L-Phi (X, mu) with the N-function Phi(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska-Orlicz indices of the function a(r) is also given. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim