Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ n

Abstract

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ1 ≤ 1 and σ2 ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators TΩ, fractional integrals TΩ,α and parametric Marcinkiewicz integrals μΩρ with variable kernels on the Hardy spaces Hp(Rn) and weak Hardy spaces WHp(Rn). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces Hp,q(Rn) for all p < q < ∞.