On Generalized Fractional Integrals in the Orlicz Spaces

Abstract

Let X = (X, d, μ) be a space of homogeneous type, i.e. X is a topological space endowed with a quasi-distance d and a positive measure μ such that

$$ \begin{array}{*{20}{c}} {d\left( {x,y} \right)0\;and\;d\left( {x,y} \right) = 0\;if and only if{\text{ }}x = y,} \\ {d\left( {x,y} \right) = d\left( {y,x} \right),} \\ {d\left( {x,y} \right){{K}_{1}}\left( {d\left( {x,z} \right) + d\left( {z,y} \right)} \right),} \\ \end{array} $$

the balls B(x, r) = {y ∈ X: d(x, y) < r},r > 0, form a basis of neighborhoods of the point x, μ is defined on a σ-algebra of subsets of X which contains the balls, and

$$ 0 < \mu \left( {B\left( {x,2r} \right)} \right){{K}_{2}}\mu \left( {B\left( {x,r} \right)} \right) < \infty , $$

where K _i ≥ 1 (i = 1, 2) are constants independent of x, y, z ∈ X and r >0.