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 $dot{F}^{gamma_1,gamma_2}_{p,q}(mathbb{R}^{n})$ v
ia 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.
In this paper, we consider the Fefferman-Stein decomposition of $Q_{alpha}(mathbb{R}^{n})$ and give an affirmative answer to an open problem posed by M. Essen, S. Janson, L. Peng and J. Xiao in 2000. One of our main methods is to study the structure
of the predual space of $Q_{alpha}(mathbb{R}^{n})$ by the micro-local quantities. This result indicates that the norm of the predual space of $Q_{alpha}(mathbb{R}^{n})$ depends on the micro-local structure in a self-correlation way.
In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M^{t,tau}_{r,p}(mathbb{R}^{n})$ to establish the inclusion relation between Morr
ey spaces and multiplier spaces. By wavelet characterization and fractal skills, we construct a counterexample to show that the scope of the index $tau$ of $M^{t,tau}_{r,p}(mathbb{R}^{n})$ is sharp. As an application, we consider a Schrodinger type operator with potentials in $M^{t,tau}_{r,p}(mathbb{R}^{n})$.
