Wavelets and Triebel type oscillation spaces

We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces ${\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n)$. Then we establish a characterization of ${\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n)$ 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.