It is known that

the fractional integral $I_{\alpha}$ 

is bounded from $L^p(\mathbb R^n)$ to $L^q(\mathbb R^n)$

when $0<\alpha<n$, $1<p<n/\alpha$ and $n/q=n/p-\alpha$

as the Hardy-Littlewood-Sobolev theorem.

In [10] %\cite{Nakai2001Taiwan} 

the author 

introduced generalized fractional integrals 

and extended this theorem to the Orlicz spaces.

%on the $n$-dimensional Euclidean space $\mathbb R^n$. 

The purpose of this paper is twofold.

First, we extend this to spaces of homogeneous type.

Secondly, we give several examples and compare with known results.

For example, 

we show the boundedness 

from $\exp L^p$ to $\exp L^q$,

from $\exp\exp L^p$ to $\exp\exp L^q$,

from $L(\log L)^{\alpha}$ to $L(\log L)^{\beta}_{weak}$,

from $L(\log L)^{\alpha}$ to $\exp L^q_{weak}$,

etc.