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

In this paper, we first introduce $L^{σ_1}$-$(\log L)^{σ_2}$ conditions satisfied by the variable kernels $Ω(x,z)$ for $0\leqσ_1\leq1$ and $σ_2\geq0$. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators $T_Ω$, fractional integrals $T_{Ω,α}$ and parametric Marcinkiewicz integrals $μ^ρ_Ω$ 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 $p<q<\infty$.

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