Uniform sparse domination and quantitative weighted boundedness for singular integrals and application to the dissipative quasi-geostrophic equation

In this paper, we consider a kind of singular integrals [Formula presented] for any f∈L^q(Rⁿ),1<q<∞ and 0<λ<n, which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂_tθ+u⋅∇θ+κΛ^2βθ=0,(x,t)∈R²×R⁺,κ>0, where u=−∇^⊥Λ^−2+2αθ, [Formula presented] and β∈(0,1]. Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator T_λ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [b,T_λ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation.