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

In this paper, we first introduce L.1(logL).2 conditions satisfied by the variable kernels .(x, z) for 0 . . 1 . 1 and . 2 . 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T ., fractional integrals T .,. and parametric Marcinkiewicz integrals . .. with variable kernels on the Hardy spaces H p(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 H p,q(Rn) for all p < q < ..