A note on Marcinkiewicz integrals supported by submanifolds

Research Open Access A note on Marcinkiewicz integrals supported by submanifolds Feng Liu1Email author Journal of Inequalities and Applications20182018:228 https://doi.org/10.1186/s13660-018-1822-8 ©  The Author(s) 2018 Received: 27 April 2018Accepted: 19 August 2018Published: 4 September 2018 Abstract In the present paper, we establish the boundedness and continuity of the parametric Marcinkiewicz integrals with rough kernels associated to polynomial mapping \(\mathcal{P}\) as well as the corresponding compound submanifolds, which is defined by $$ \mathcal{M}_{h,\Omega ,\mathcal{P}}^{\rho }f(x)= \biggl( \int_{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int_{ \vert y \vert \leq t}\frac{\Omega (y)h( \vert y \vert )}{ \vert y \vert ^{n- \rho }}f \bigl(x-\mathcal{P}(y) \bigr)\,dy \biggr\vert ^{2}\frac{dt}{t} \biggr)^{1/2}, $$ on the Triebel–Lizorkin spaces and Besov spaces when \(\Omega \in H ^{1}(\mathrm{S}^{n-1})\) and \(h\in \Delta_{\gamma }(\mathbb{R}_{+})\) for some \(\gamma >1\). Our main results represent significant improvements and natural extensions of what was known previously. Keywords Polynomial compound mappings \(H^{1}(\mathrm{S}^{n-1})\) Triebel–Lizorkin spacesBesov spaces MSC 42B2042B2547G10 1 Introduction As is well known, the Triebel–Lizorkin spaces and Besov spaces contain many important function spaces, such as Lebesgue spaces, Hardy spaces, Sobolev spaces and so on. During the last several years, a considerable amount of attention has been given to investigate the boundedness for several integral operators on the Triebel–Lizorkin spaces and Besov spaces. For examples, see [1–6] for singular integrals, [7–13] for Marcinkiewicz integrals, [14] for the Littlewood–Paley functions, [15–18] for maximal functions. In this paper we continue to focus on this topic. More precisely, we aim to establish the boundedness and continuity of parametric Marcinkiewicz integral operators associated to polynomial compound mappings with rough kernels in Hardy spaces \(H^{1}(\mathrm{S} ^{n-1})\) on the Triebel–Lizorkin spaces and Besov spaces. We now recall the definitions of Triebel–Lizorkin spaces and Besov spaces. Definition 1.1 Let \(d\geq 2\) and \(\mathcal{S}'( \mathbb{R}^{d})\) be the tempered distribution class on \(\mathbb{R} ^{d}\). For \(\alpha \in \mathbb{R}\) and \(0< p\), \(q\le \infty \) (\(p\neq\infty\)), the homogeneous Triebel–Lizorkin spaces \(\dot{F}_{\alpha }^{p,q}(\mathbb{R}^{d})\) and Besov spaces \(\dot{B}_{\alpha }^{p,q}( \mathbb{R}^{d})\) are defined by F ˙ α p , q ( R d ) : = { f ∈ S ′ ( R d ) : ∥ f ∥ F ˙ α p , q ( R d ) = ∥ ( ∑ i ∈ Z 2 − i α q | Ψ i ∗ f | q ) 1 / q ∥ L p ( R d ) < ∞ } ; (1.1) B ˙ α p , q ( R d ) : = { f ∈ S ′ ( R d ) : ∥ f ∥ B ˙ α p , q ( R d ) = ( ∑ i ∈ Z 2 − i α q ∥ Ψ i ∗ f ∥ L p ( R d ) q ) 1 / q < ∞ } , (1.2) where \(\widehat{\Psi _{i}}(\xi )=\phi (2^{i}\xi )\) for \(i\in \mathbb{Z}\) and \(\phi \in \mathcal{C}_{c}^{\infty } (\mathbb{R}^{d})\) satisfies the conditions: \(0\leq \phi (x)\leq 1\); \(\operatorname{supp}(\phi ) \subset \{x: 1/2\leq \vert x \vert \leq 2\}\); \(\phi (x)>c>0\) if \(3/5\leq \vert x \vert \leq 5/3\). The inhomogeneous versions of Triebel–Lizorkin spaces and Besov spaces, which are denoted by \(F_{\alpha }^{p,q} (\mathbb{R}^{d})\) and \(B_{\alpha }^{p,q}(\mathbb{R}^{d})\), respectively, are obtained by adding the term ∥ Θ ∗ f ∥ L p ( R d ) to the right hand side of (1.1) or (1.2) with \(\sum_{i\in \mathbb{Z}}\) replaced by \(\sum_{i\geq 1}\), where \(\Theta \in \mathcal{S}(\mathbb{R}^{d})\) (the Schwartz class), \(\operatorname{supp}(\hat{\Theta })\subset \{\xi : \vert \xi \vert \leq 2\}\), \(\hat{\Theta }(x)>c>0\) if \(\vert x \vert \leq 5/3\). The following properties of the above spaces are well known (see [19–21] for more details): $$\begin{aligned}& \dot{F}_{0}^{p,2} \bigl(\mathbb{R}^{d} \bigr)=L^{p} \bigl(\mathbb{R}^{d} \bigr) \quad \mbox{for } 1< p< \infty ; \end{aligned}$$ (1.3) $$\begin{aligned}& \dot{F}_{\alpha }^{p,p} \bigl(\mathbb{R}^{d} \bigr)= \dot{B}_{\alpha }^{p,p} \bigl( \mathbb{R}^{d} \bigr)\quad \mbox{for } \alpha \in \mathbb{R} \mbox{ and } 1< p< \infty ; \end{aligned}$$ (1.4) F α p , q ( R d ) ∼ F ˙ α p , q ( R d ) ∩ L p ( R d ) and ∥ f ∥ F α p , q ( R d ) ∼ ∥ f ∥ F ˙ α p , q ( R d ) + ∥ f ∥ L p ( R d ) for  α > 0 ; (1.5) B α p , q ( R d ) ∼ B ˙ α p , q ( R d ) ∩ L p ( R d ) and ∥ f ∥ B α p , q ( R d ) ∼ ∥ f ∥ B ˙ α p , q ( R d ) + ∥ f ∥ L p ( R d ) for  α > 0 . (1.6) Let \(n\geq 2\) and \(\mathrm{S}^{n-1}\) be the unit sphere in \(\mathbb{R} ^{n}\) equipped with the normalized Lebesgue measure dσ. Assume that \(\Omega \in L^{1}(\mathrm{S}^{n-1})\) is a function of homogeneous of degree zero and satisfies the cancelation condition $$ \int_{\mathrm{S}^{n-1}}\Omega (u)\,d\sigma (u)=0. $$ (1.7) We denote by \(\Delta_{\gamma }(\mathbb{R}_{+})\) (\(\gamma \geq 1\)) the set of all measurable functions h defined on \(\mathbb{R}_{+}:=(0,\infty )\) satisfying ∥ h ∥ Δ γ ( R + ) : = sup R > 0 ( R − 1 ∫ 0 R | h ( t ) | γ d t ) 1 / γ < ∞ . In 1986, Stein [22] first introduced the singular Radon transforms \(T_{h,\Omega ,\mathcal{P}}\) by $$ T_{h,\Omega ,\mathcal{P}}f(x)=\mathrm{p.v.} \int_{\mathbb{R}^{n}}f \bigl(x- \mathcal{P}(y) \bigr)\frac{\Omega (y)h( \vert y \vert )}{ \vert y \vert ^{n}} \,dy. $$ (1.8) where \(\mathcal{P}=(P_{1},P_{2},\ldots ,P_{d})\) is a polynomial mapping from \(\mathbb{R}^{n}\) into \(\mathbb{R}^{d}\) and \(h\in \Delta_{1}( \mathbb{R}_{+})\). Later on, the bounds of \(T_{h,\Omega ,\mathcal{P}}\) on \(L^{p}\) spaces and other function spaces have been studied by a large number of scholars (see [4, 23, 24] for example). In particular, Chen et al. [4] established the bounds for \(T_{h, \Omega , \mathcal{P}}\) on Triebel–Lizorkin spaces and Besov spaces under the condition that \(\Omega \in H^{1} (\mathrm{S}^{n-1})\) and \(h\in \Delta_{\gamma }(\mathbb{R}_{+})\) for some \(\gamma >1\). It should be pointed out that the class of singular Radon transforms \(T_{h,\Omega ,\mathcal{P}}\) is closely related to the class of Marcinkiewicz integral operators $$ \mathcal{M}_{h,\Omega ,\mathcal{P}}^{\rho }f(x)= \biggl( \int_{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int_{ \vert y \vert \leq t}\frac{\Omega (y)h( \vert y \vert )}{ \vert y \vert ^{n- \rho }}f \bigl(x-\mathcal{P}(y) \bigr)\,dy \biggr\vert ^{2}\frac{dt}{t} \biggr)^{1/2}, $$ (1.9) where h, Ω, \(\mathcal{P}\) are given as in (1.8) and \(\rho =\sigma +i\tau \) (\(\sigma ,\tau \in \mathbb{R}\) and \(\sigma >0\)). The operators defined in (1.9) have their roots in the classical Marcinkiewicz integral operator \ (...truncated)