An extension of Marcinkiewicz integrals with rough kernels

This paper considers the following Marcinkiewicz type integrals [inline-graphic not available: see fulltext] which can be regarded as an extension of the classical Marcinkiewicz integral μΩ introduced by Stein in [Trans. Amer. Math. Soc. 88(1958), 159–172], where Ω is a homogeneous function of degree zero on ℝn with mean value zero in the unit sphere Sn−1. Under the assumption that Ω ∈ L∞ (Sn−1), the authors establish the Lq-estimate and weak (1, 1) type estimate as well as the corresponding weighted estimates for μΩ,β with 1 < q < ∞ and 0 < β < (q − 1)n/q. Moreover, the bounds do not depend on β and the strong (q, q) type and weak (1, 1) type estimates for the classical Marcinkiewicz integral μΩ can be recovered from the above estimates of μΩ,β when β → 0.