On Gevrey Regularity of the Supercritical SQG equation in Critical Besov Spaces

In this paper we show that the solution of the supercrti- cal surface quasi-geostrophic (SQG) equation, starting from initial data in homogeneous critical Besov spaces belong to a subanalytic Gevrey class. In particular, we improve upon the result of Dong and Li in [26], where they showed that the solutions of Chen-Miao-Zhang (cf. [11]) are classical solutions. We extend the approach of Biswas (cf. [7]) to critical, L^p -based Besov spaces, and adapt the point of view of Lemarie- Rieusset (cf. [36]), who treated the operator arising from applying the analytic Gevrey operator to a product of analytic functions as a bilinear multiplier operator. In order to obtain L^p bounds, we prove that our bilinear multiplier operator is of Marcinkiewicz type, and show that due to additional localizations inherited from working in Besov spaces, this condition implies boundedness.