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In this paper we present the Lp mapping properties for a class of multiple singular integral operators along polynomial compound surfaces provided that the integral kernels are given by the radial function h ∈ Δγ (or h ∈ Uγ ) for some γ > 1 and the sphere function Ω ∈ F̃β (Sm−1×Sn−1) for some β > 0 , which is distinct from L(log+ L)2(Sm−1×Sn−1) . In addition, the Lp bounds for the related maximal operators are also established. Some previous results are greatly extended and improved. Mathematics subject classification (2010): 42B20, 42B25.

Multiple singular integrals and maximal operators with mixed homogeneity along compound surfaces