  In this paper, we first introduce $L^{\sigma_1}$-$(\log L)^{\sigma_2}$
conditions satisfied by the variable kernels $\Omega(x,z)$ for
$0\leq\sigma_1\leq1$ and $\sigma_2\geq0$. Under these new smoothness
conditions, we will prove the boundedness properties of singular integral
operators $T_{\Omega}$, fractional integrals $T_{\Omega,\alpha}$ and parametric
Marcinkiewicz integrals $\mu^{\rho}_{\Omega}$ with variable kernels on the
Hardy spaces $H^p(\mathbb R^n)$ and weak Hardy spaces $WH^p(\mathbb R^n)$.
Moreover, by using the interpolation arguments, we can get some corresponding
results for the above integral operators with variable kernels on
Hardy--Lorentz spaces $H^{p,q}(\mathbb R^n)$ for all $p<q<\infty$.
