In the present paper, we consider a kind of singular integral  which can be viewed as an extension of the classical Calder/'{o}n-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 $/Omega.$ Moreover, the bounds   does not depend on $/beta$ and the strong $(q, q)$ type estimate and weak $(1, 1)$ type estimate of the Calder/'{o}n-Zygmund type singular integral can be recovered when $/beta /rightarrow 0$ from our obtained estimates.

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