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

Abstract

In this paper, we consider a kind of singular integralsTλf(x)= p.v. ∫RnΩ(y)|y|n−λf(x−y)dy for any f∈Lq(Rn),1<q<∞ and 0<λ<n, which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation(0.1)∂tθ+u⋅∇θ+κΛ2βθ=0,(x,t)∈R2×R+,κ>0, where u=−∇⊥Λ−2+2αθ, α∈[0,12] 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 Tλ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [b,Tλ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation.