In this paper, we consider a kind of singular integral which can be viewed as an extension of the classical Calder$\acute{ \rm o}$n-Zygmund type singular integral. We establish an estimate of the singular integral in the $L^q$ space for $1<q<\infty$. In particular, the Calder$\acute{ \rm o}$n-Zygmund estimate can be recovered from our obtained estimate. The proof of our main result is via the so called "geometric approach", which was applied in \cite{ CP} on the $L^q$ estimate of the elliptic equations and in \cite{ LW,Wang} on a new proof of the the Calder$\acute{ \rm o}$n-Zygmund estimate.
Categories: math.AP