A note on Marcinkiewicz integrals supported by submanifolds

Abstract

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.

Keywords

• Polynomial compound mappings
• H 1 ( S n − 1 ) $H^{1}(\mathrm{S}^{n-1})$
• Triebel–Lizorkin spaces
• Besov spaces