A note on Marcinkiewicz integrals supported by submanifolds.

In the present paper, we establish the boundedness and continuity of the parametric Marcinkiewicz integrals with rough kernels associated to polynomial mapping P as well as the corresponding compound submanifolds, which is defined by M h , Ω , P ρ f ( x ) = ( ∫ 0 ∞ | 1 t ρ ∫ | y | ≤ t Ω ( y ) h ( | y | ) | y | n - ρ f ( x - P ( y ) ) d y | 2 d t t ) 1 / 2 , on the Triebel-Lizorkin spaces and Besov spaces when Ω ∈ H 1 ( S n - 1 ) and h ∈ Δ γ ( R + ) for some γ > 1 . Our main results represent significant improvements and natural extensions of what was known previously.