In the present paper, we consider a kind of singular integral$Tf(x)=\mathrm{p}.\mathrm{v}.\underset{{\mathbb{R}}^n}{\int }\frac{\mathrm{\Omega }(y)}{|y{|}^{n-\beta }}f(x-y)dy$ which can be viewed as an extension of the classical Calderón-Zygmund type singular integral. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish an estimate of the singular integral in the $L^q$ space for $1<q<\infty$ and a weak $(1,1)$ type of the singular integral when $0<\beta <\frac{(q-1)n}{q}$ without any smoothness assumed on Ω. Moreover, the bounds do not depend on β and the strong $(q,q)$ type estimate and weak $(1,1)$ type estimate of the Calderón-Zygmund type singular integral can be recovered when $\beta \to 0$ from our obtained estimates.

在本文中，我们考虑一种奇异积分$吨F(X)=\mathrm{磷}.\mathrm{v}.\underset{{\mathbb{电阻}}^n}{\int }\frac{\mathrm{\Omega }(是)}{|是{|}^{n-\beta }}F(X-是)d是$这可以看作是经典 Calderón-Zygmund 型奇异积分的扩展。这种奇异积分出现在从广义 SQG 方程逼近表面准地转 (SQG) 方程中。我们建立了奇异积分的估计${升}^q$ 空间 $1<q<\infty$ 和一个弱 $(1,1)$ 奇异积分的类型，当 $0<\beta <\frac{(q-1)n}{q}$Ω 上没有任何平滑假设。此外，边界不依赖于β和强$(q,q)$ 类型估计和弱 $(1,1)$ 当 Calderón-Zygmund 型奇异积分的类型估计可以恢复时 $\beta \to 0$ 从我们得到的估计。