Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ⩽ KrN, N ∈ (0, ∞). 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 Lp(⋅)(X, μ) into a certain Musielak-Orlicz space Lp(X, μ) with the N-function Φ(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. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim