MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS ON PRODUCT SPACES

"Suppose that ＄￥Omega(x^{￥prime}, y^{￥prime})￥in L^{1}(S^{n-1}￥times S^{m-1})＄ is ahomogeneous function of degree zero ＄satis￥Phi ing＄ the mean zero propeIty (1.1), and that ＄h(s,t)＄ ls a bounded function on ＄Rx＄ R. The Marcinkiewicz integral operator ＄m(f)＄ along a continuous surface ＄￥gamma(u, v)＄ on the product space ＄R^{n}xR^{m}(n￥geq 2, m￥geq 2)＄ is defined by ＄￥iota_{O}f(￥xi, ￥eta, z)=(￥int_{R}￥int_{R}|F_{l,￥epsilon}(x, y, z)|^{2}2^{-2t-2￥epsilon}dtds)^{1/2}＄ where ＄F_{t,e}(￥xi, ￥eta, z)＄ ＄=￥int_{|_{￥nu}^{x}|^{:}}<2h(|x|, |y|)|x|^{-n+1}|y|^{-m+1}￥Omega(x^{￥prime}, y^{￥prime})f(￥xi-x,￥eta-y, z-￥gamma(|x|, |y|))dxdy<2＄ We prove that the operator ＄￥nu_{￥Omega}f＄ is bounded on ＄L^{P}(R^{n} xR^{m}xR),＄ ＄p￥in(1, ￥infty)＄ , provided that ＄￥Omega＄ is a function in certain block space ＄B_{q}^{0,1}(S^{n-1}xS^{m-1})＄ for some ＄q>1＄ and that two lower dimensional maximal functions related to ＄￥gamma＄ are bounded on ＄L^{p}＄ . These two lower dimensional maximal functions are natural extension of a well-known maximal function along curves.