  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.
