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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 thestudy of the continuity of the fractional maximal operator of Hardy-Littlewoodand 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 ofhomogeneous groups. We generalize it here to spaces of homogeneous type and weprove that the results obtained in \cite{FFK} such as relations between thesespaces and Lebesgue spaces, weak Lebesgue and Morrey spaces, remain true.

Integrable fractional mean functions on spaces of homogeneous type