  The class of Banach spaces $(L^{q},L^{p}) ^{\alpha}(X,d,\mu)$, $1\leq q\leq
\alpha \leq p\leq \infty ,$ introduced in \cite{F1} in connection with the
study of the continuity of the fractional maximal operator of Hardy-Littlewood
and of the Fourier transformation in the case $% X=\mathbb{R}^{n}$ and $\mu $
is the Lebesgue measure, was generalized in \cite{FFK} to the setting of
homogeneous groups. We generalize it here to spaces of homogeneous type and we
prove that the results obtained in \cite{FFK} such as relations between these
spaces and Lebesgue spaces, weak Lebesgue and Morrey spaces, remain true.
