Fractional integral operators between Banach function lattices

We study the generalized fractional integral transforms associated to a measure on a quasi-metric space. We give a characterization of those measures for which these operators are bounded between LpLp-spaces defined on nonhomogeneous spaces. The key in the proof of one of the main theorems is the boundedness of the modified sublinear Hardy–Littlewood maximal operator in the classical Lebesgue space with general measure. We also provide necessary and sufficient conditions for some classes of integral operators to be bounded from Lorentz to Marcinkiewicz spaces.