A note on Marcinkiewicz integrals supported by submanifolds

Abstract In the present paper, we establish the boundedness and continuity of the parametric Marcinkiewicz integrals with rough kernels associated to polynomial mapping P $\mathcal{P}$ as well as the corresponding compound submanifolds, which is defined by Mh,Ω,Pρf(x)=(∫0∞|1tρ∫|y|≤tΩ(y)h(|y|)|y|n−ρf(x−P(y))dy|2dtt)1/2, $$ \mathcal{M}_{h,\Omega ,\mathcal{P}}^{\rho }f(x)= \biggl( \int_{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int_{ \vert y \vert \leq t}\frac{\Omega (y)h( \vert y \vert )}{ \vert y \vert ^{n- \rho }}f \bigl(x-\mathcal{P}(y) \bigr)\,dy \biggr\vert ^{2}\frac{dt}{t} \biggr)^{1/2}, $$ on the Triebel–Lizorkin spaces and Besov spaces when Ω∈H1(Sn−1) $\Omega \in H ^{1}(\mathrm{S}^{n-1})$ and h∈Δγ(R+) $h\in \Delta_{\gamma }(\mathbb{R}_{+})$ for some γ>1 $\gamma >1$. Our main results represent significant improvements and natural extensions of what was known previously