L2-Boundedness of Marcinkiewicz Integrals along Surfaces with Variable Kernels: Another Sufficient Condition

We give the estimates for the Marcinkiewicz integral with rough variable kernels associated to surfaces. More precisely, we give some other sufficient conditions which are different from the conditions known before to warrant that the -boundedness holds. As corollaries of this result, we show that similar properties still hold for parametric Littlewood-Paley area integral and parametric functions with rough variable kernels. Some of the results are extensions of some known results.