On an extension of singular integrals along manifolds of finite type

for all test functions f , where y′ = y/|y| ∈ Sn−1. We denote SIΩ,h( f ) by SIΩ( f ) if h= 1. The operator SIΩ was first studied by Calderón and Zygmund in their well-known papers (see [1, 2]). They proved that SIΩ is Lp(Rn) bounded, 1 < p < ∞, provided that Ω ∈ LLog+L(Sn−1) satisfying (1.1). They also showed that the space LLog+L(Sn−1) cannot be replaced by any Orlicz space Lφ(Sn−1) with a monotonically increasing function φ satisfying φ(t) = o(t log t), t → ∞, that is, L(Log+L)1−ε(Sn−1), 0 < ε ≤ 1. The idea of their proof was as follows. Suppose that Ω∈ L1(Sn−1) is an odd function, then one can easily show that