Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform L^p−L^q estimates with respect to a parameter β. From this one can cover the L^p-boundedness of the Calderón-Zygmund operator with rough kernel by letting β→0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space B_p,q^s and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021.