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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 smoothnessconditions, we will prove the boundedness properties of singular integraloperators $T_{\Omega}$, fractional integrals $T_{\Omega,\alpha}$ and parametricMarcinkiewicz integrals $\mu^{\rho}_{\Omega}$ with variable kernels on theHardy 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 correspondingresults for the above integral operators with variable kernels onHardy--Lorentz spaces $H^{p,q}(\mathbb R^n)$ for all $p<q<\infty$.

Estimates of some integral operators with bounded variable kernels on the Hardy and weak Hardy spaces over $\mathbb R^n$