RIESZ TRANSFORMS ON $Q$-TYPE SPACES WITH APPLICATION TO QUASI-GEOSTROPHIC EQUATION

Abstract

By an equivalent characterization of Morrey space associated with the fractional heat semigroup, we establish a relation between the generalized $Q-$type spaces and Morrey spaces. By this relation, in this paper, we prove the boundedness of the singular integral operatoes on the Q-type spaces $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the quasi-geostrophic equation with initial data in $Q_{\alpha}^{\beta, -1}(\mathbb{R}^{n})$.