  We apply wavelets to identify the Triebel type oscillation spaces with the
known Triebel-Lizorkin-Morrey spaces
$\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$. Then we establish a
characterization of $\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$ via the
fractional heat semigroup. Moreover, we prove the continuity of
Calder\'on-Zygmund operators on these spaces. The results of this paper also
provide necessary tools for the study of well-posedness of Navier-Stokes
equations.
