Wavelets and Triebel type oscillation spaces

We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $\dot{F}^{γ_1,γ_2}_{p,q}(\mathbb{R}^{n})$. Then we establish a characterization of $\dot{F}^{γ_1,γ_2}_{p,q}(\mathbb{R}^{n})$ via the fractional heat semigroup. Moreover, we prove the continuity of Calderón-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations.

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