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

Abstract

Let P(x, y) be a real-valued polynomial on ${\mathbb {R}}^n\times {\mathbb {R}}^n$. 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 $T_{P,K}f(x)=\mathrm{p.v.}\int _{{\mathbb {R}}^n}\mathrm{e}^{iP(x,y)}K(x,y)f(y)\mathrm{d}y,$ 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 $\deg _x(P)\le 1$ or $\deg _y(P)\le 1$, we show that both $T_{P,K}$ and its commutator $T_{b,P,K}$ are bounded on $L_w^p({\mathbb {R}}^n)$ for $1<p<\infty$ , $b\in \mathrm{BMO}({\mathbb {R}}^n)$ and $w\in A_p({\mathbb {R}}^n)$. We also prove that the commutator $T_{b,P,K}$ is a compact operator on $L_w^p({\mathbb {R}}^n)$ if $b\in \mathrm{CMO}({\mathbb {R}}^n)$ for all $1<p<\infty$ and $w\in A_p({\mathbb {R}}^n)$. Here $\mathrm{CMO}({\mathbb {R}}^n)$ denotes the closure of ${\mathcal {C}}_c^\infty ({\mathbb {R}}^n)$ in the $\mathrm{BMO}({\mathbb {R}}^n)$ topology.