We prove variational inequalities for the truncated rough singular integrals \(\mathcal {T}_{\Omega ,\gamma }=\{T_{\Omega ,\gamma ,\varepsilon }\}_{\varepsilon >0}\) , where \(\Omega \in L^{1}(\mathbb {S}^{n-1})\) is homogeneous and cancellative. For \(n\ge 2\) , \(1\le p<q<\infty \) , \(s>1\) , \(\frac{1}{q}+1=\frac{1}{p}+\frac{1}{s}\) , and \(0<\gamma <\frac{n(s-1)}{s}\) , we show that the \(r\) -variation operator \(V_{r}(\mathcal {T}_{\Omega ,\gamma })\) is bounded on \(L^{q}\) uniformly in \(\gamma \) as \(\gamma \rightarrow 0\) , provided \(\Omega \in L^{s}(\mathbb {S}^{n-1})\) . This extends earlier \(L^{q}\) bounds of Chen and Guo (J Funct Anal 281:109196, 2021) and Lin and Xie (Arch Math 120:631–642, 2023) to the variational setting, offering a more refined characterization of the oscillation behavior of rough singular integral families. The findings contribute to the framework of variational analysis for rough singular integrals and hold potential applications in the study of geophysical flows modeled by the surface quasi-geostrophic (SQG) equation.
