2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106218Let $\T (0\leq α<n)$ be the singular and fractional integrals with variable kernel $Ω(x,z)$, and $[b,\T]$ be the commutator generated by $\T$ and a Lipschitz function $b$. In this paper, the authors study the boundedness of $[b,\T]$ on the Hardy spaces, under some assumptions such as the $L^r$-Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators $\tT (0\leq α<n)$. The smoothness conditions imposed on $\tOmega$ are weaker than the corresponding known results.12 pagesClassical Analysis and ODEs42B20; 42B30text