Riesz transforms on Q-type spaces with application to quasi-geostrophic equation

TL;DR

This paper establishes the boundedness of Riesz transforms on Q-type spaces and applies this to prove well-posedness and regularity results for the quasi-geostrophic equation with specific initial data.

Contribution

It proves the boundedness of Riesz transforms on Q-type spaces and applies this to analyze the well-posedness of the quasi-geostrophic equation.

Findings

Boundedness of Riesz transforms on Q-type spaces established

Well-posedness of quasi-geostrophic equation proved for initial data in Q-type spaces

Regularity results obtained for solutions to the quasi-geostrophic equation

Abstract

In this paper, we prove the boundedness of Riesz transforms $\partial_{j}(-\Delta)^{-1/2}$ ($j=1,2,...,n$) 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 $Q_{\alpha}^{\beta,-1}(\mathbb{R}^{2}).$