Regular Wavelets, Heat Semigroup and Application to the Magneto-hydrodynamic Equations with Data in Critical Triebel-Lizorkin Type Oscillation Spaces

Abstract

In this paper, based on a Poisson type extension of Triebel-Lizorkin type oscillation spaces $\dot{F}^{\gamma_1, \gamma_2}_{p, q}(\mathbb{R}^{n})$, we establish a bilinear estimate on some new tent spaces $\mathbb{F}^{\gamma_1, \gamma_2}_{p, q, m, m'}$ associated with $\dot{F}^{\gamma_1, \gamma_2}_{p, q}(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the fractional magneto-hydrodynamic equations and quasi-geostrophic equations with initial data in the critical $\dot{F}^{\gamma_1, \gamma_2}_{p, q}(\mathbb{R}^{n})$.