On oscillatory singular integrals and their commutators with non-convolutional Hölder class kernels

Let P(x, y) be a real-valued polynomial on Rn× Rn. We denote by deg x(P) (resp., deg y(P)) the degree of P in x (resp., y). In this paper, we investigate the properties of the oscillatory integral given by TP,Kf(x)=p.v.∫RneiP(x,y)K(x,y)f(y)dy, where K is a Calderón–Zygmund non-convolutional type kernel. If the kernel K(x, y) satisfies a Hölder condition and P(x, y) satisfies the condition deg x(P) ≤ 1 or deg y(P) ≤ 1 , we show that both TP,K and its commutator Tb,P,K are bounded on Lwp(Rn) for 1 < p< ∞ , b∈ BMO (Rn) and w∈ Ap(Rn). We also prove that the commutator Tb,P,K is a compact operator on Lwp(Rn) if b∈ CMO (Rn) for all 1 < p< ∞ and w∈ Ap(Rn). Here CMO (Rn) denotes the closure of Cc∞(Rn) in the BMO (Rn) topology. © 2021, Tusi Mathematical Research Group (TMRG).