Wavelets and Triebel type oscillation spaces

We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $dot{F}^{gamma_1,gamma_2}_{p,q}(mathbb{R}^{n})$. 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 Calderon-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations.