Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel $\frac{a([\varrho (x,y)])}{[\varrho (x,y)]^N}$ on bounded quasimetric measure space ( X , μ, d ) with doubling measure μ satisfying the upper growth condition μ B ( x , r ) ⩽ Kr N , 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 L p (⋅) ( X , μ) into a certain Musielak‐Orlicz space L p ( 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