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.

令P ( x ,  y ) 是\({\mathbb {R}}^n\times {\mathbb {R}}^n\)上的实值多项式。我们用\(\deg _x(P)\) (resp., \(\deg _y(P)\) ) 表示P在x (resp., y ) 中的度数。在本文中，我们研究了由\(T_{P,K}f(x)=\mathrm{pv}\int _{{\mathbb {R}}^n}\mathrm{e }^{iP(x,y)}K(x,y)f(y)\mathrm{d}y,\)其中K是 Calderón–Zygmund 非卷积类型核。如果核K ( x ,  y ) 满足 Hölder 条件且P( x ,  y ) 满足条件\(\deg _x(P)\le 1\)或\(\deg _y(P)\le 1\)，我们证明\(T_{P,K}\)和它的换向器\(T_{b,P,K}\)在\(L_w^p({\mathbb {R}}^n)\)上有界，对于\(1<p<\infty \) , \( b\in \mathrm{BMO}({\mathbb {R}}^n)\)和\(w\in A_p({\mathbb {R}}^n)\)。我们还证明了交换子\(T_{b,P,K}\)是\(L_w^p({\mathbb {R}}^n)\)上的紧算子，如果\(b\in \mathrm{ CMO}({\mathbb {R}}^n)\)对于所有\(1<p<\infty \)和\(w\in A_p({\mathbb {R}}^n)\)。这里\(\mathrm{CMO}({\mathbb {R}}^n)\)表示\({\mathcal {C}}_c^\infty ({\mathbb {R}}^n)\ )在\(\mathrm{BMO}({\mathbb {R}}^n)\)拓扑中。