For \(0 \le \beta < n\) , we study a generalized singular integral and its commutator with rough variable kernels, defined as \(\begin{aligned} {T_{\Omega ,\beta }}f(x) =p.v. \int _{\mathbb {R}^n} \frac{\Omega (x, x - y)}{|x - y|^{n - \beta }} f(y)\, dy \end{aligned}\) and \(\begin{aligned} {[}b, T_{\Omega ,\beta }]f(x) =p.v.\int _{\mathbb {R}^n} \frac{\Omega (x, x - y)}{|x - y|^{n - \beta }} (b(x) - b(y)) f(y)\, dy, \end{aligned}\) where \(\Omega \) is defined on \(\mathbb {R}^n \times \mathbb {R}^n\) , \(b \in \text {BMO}(\mathbb {R}^n)\) , and \(\Omega (x, \cdot )\) is homogeneous of degree zero for any fixed \(x \in \mathbb {R}^n\) . Moreover, for some \(r \ge 1\) , \(\Vert \Omega (x, \cdot )\Vert _{L^r(S^{n-1})}\) is uniformly bounded with respect to \(x \in \mathbb {R}^n\) . The authors prove that, if \(0 < \beta \le 1/2\) and \(\Omega (x, \cdot )\) has mean value zero on the unit sphere for any fixed \(x \in \mathbb {R}^n\) , then \(r > 2(n - 1)/(n - 2\beta )\) is sufficient for both \(T_{\Omega ,\beta }\) and \([b, T_{\Omega ,\beta }]\) to be bounded from \(L^{2n/(n + 2\beta )}\) to \(L^2\) , uniformly with respect to \(\beta \) . This result recovers the \(L^2\) -boundedness of \(T_{\Omega ,0}\) and \([b, T_{\Omega ,0}]\) by letting \(\beta \rightarrow 0^+\) , aligning with the known results established by Calderón and Zygmund (Trans Am Math Soc 78:209–224, 1955) and Chen and Ding (Rev Mat Iberoam 24(2):531–547, 2008).
