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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{P}$\end{document}P as well as the corresponding compound submanifolds, which is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{M}_{h,\Omega ,\mathcal{P}}^{\rho }f(x)= \biggl( \int_{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int_{ \vert y \vert \leq t}\frac{\Omega (y)h( \vert y \vert )}{ \vert y \vert ^{n- \rho }}f \bigl(x-\mathcal{P}(y) \bigr)\,dy \biggr\vert ^{2}\frac{dt}{t} \biggr)^{1/2}, $$\end{document}Mh,Ω,Pρf(x)=(∫0∞|1tρ∫|y|≤tΩ(y)h(|y|)|y|n−ρf(x−P(y))dy|2dtt)1/2, on the Triebel–Lizorkin spaces and Besov spaces when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \in H ^{1}(\mathrm{S}^{n-1})$\end{document}Ω∈H1(Sn−1) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h\in \Delta_{\gamma }(\mathbb{R}_{+})$\end{document}h∈Δγ(R+) for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma >1$\end{document}γ>1. Our main results represent significant improvements and natural extensions of what was known previously.


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][2][3][4][5][6] for singular integrals, [7][8][9][10][11][12][13] for Marcinkiewicz integrals, [14] for the Littlewood-Paley functions, [15][16][17][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 (S n-1 ) on the Triebel-Lizorkin spaces and Besov spaces.
Based on the above, a natural question, which arises from the above results, is the following.
Question A Is the operator M ρ h, ,P bounded on Triebel-Lizorkin spaces and Besov spaces under the condition that ∈ H 1 (S n-1 ) and h ∈ γ (R + )?
Question A is the main motivation for this work. The main purpose of this paper will not only be to address the above question by treating a more general class of operators but also to establish the corresponding continuity of Marcinkiewicz integral operators on Triebel-Lizorkin spaces and Besov spaces. More precisely, let h, , ρ, P be given as in (1.9) and ϕ : R + → R be a suitable function, we define the parametric Marcinkiewicz integral operator M ρ h, ,P,ϕ on R d by Our main result can be listed as follows.
Remark 1.2 We remark that the set R γ was originally given by Yabuta [10] in the study of the boundedness for Marcinkiewicz integrals associated to surfaces {ϕ(|y|)y : y ∈ R n } with ϕ ∈ F on Triebel-Lizorkin spaces. Actually, Theorem 1.1 extends the partial result of [10, Theorem 1.1], which corresponds to the case n = d, ρ > 0 and P(y) = y. Clearly, R γ 1 R γ 2 for any 1 < γ 1 < γ 2 ≤ ∞ and R ∞ = (0, 1) × (0, 1). There are some model examples for the class F, such as t α (α > 0), t β ln(1 + t) (β ≥ 1), t ln ln(e + t), real-valued polynomials P on R with positive coefficients and P(0) = 0 and so on. Note that there exists B ϕ > 1 such that ϕ(2t) ≥ B ϕ ϕ(t) for any ϕ ∈ F (see [7] γ (R + ) for any 1 < γ < ∞ and L r (S n-1 ) H 1 (S n-1 ) for any r > 1, the boundedness part in Theorem 1.2 improves and generalizes greatly the results of [12,13]. It should be pointed out that our main results are new even in the special case: The paper is organized as follows. Section 2 contains two vector-valued inequalities on maximal functions, which are the main ingredients of our proofs. Section 3 is devoted to presenting some preliminary lemmas. The proof of Theorem 1.2 will be given in Sect. 4. We would like to remark that some ideas in our proofs are taken from [7,10,17,23,34] and the main novelty in this paper is to give the continuity for Marcinkiewicz integral operators on Triebel-Lizorkin spaces and Besov spaces.
Throughout this note, we denote by p the conjugate index of p, i.e. 1/p + 1/p = 1. The letter C or c, sometimes with certain parameters, will stand for positive constants not necessarily the same one at each occurrence, but are independent of the essential variables. If f ≤ Cg, we then write f g or g f ; and if f g f , we then write f ∼ g. In what follows, we denote by J -1 and J t the inverse transform and the transpose of the linear transformation J, respectively. We also denote the Dirac delta function on We also use the conventions i∈∅ a i = 0 and i∈∅ a i = 1.
Comments on conclusions and methods. This aim of this paper is to investigate the boundedness and continuity for the parametric Marcinkiewicz integral operators supported by polynomial compound mappings M ρ h, ,P,ϕ on the Triebel-Lizorkin spaces and Besov spaces. This is motivated by some recent results (see [4,10,11,25,31]). In [4], the authors established the bounds for the singular integral operators supported by polynomial mappings on the Triebel-Lizorkin spaces and Besov spaces; In [10,11] the authors proved the boundedness for Marcinkiewicz integral operators M ρ h, on the Triebel-Lizorkin spaces; In [25,31] the authors gave the L p bounds for the Marcinkiewicz integral operators supported by polynomial mappings M h, ,P . The main purpose of this paper will not only address the residual problems with respect to exponents [25,31] but also establish the corresponding continuity of Marcinkiewicz integral operators on Triebel-Lizorkin spaces and Besov spaces. Although the methods and idea used in proofs of main results are motivated by some previous work [7,10,16,22,31], the methods and techniques are more delicate and difficult than those in the above references. Moreover, the main results are new and the proofs are highly non-trivial. On the other hand, the main results greatly extended and generalized some previous work [10][11][12].

Two vector-valued inequalities on maximal functions
The following lemma can be seen as a general case of [ is well-defined as an element of B 2 for all L ∞ (B 1 , R d ) functions F with compact supported provided x lies outside the support of F. Assume that the kernel K satisfies Hörmander condition Then, for any 1 < p, q < ∞ and all B 1 -valued functions F j , there exists C > 0, such that .
We now establish the following vector-valued inequality of a Hardy-Littlewood maximal function, which is of interest in its own right.

Lemma 2.2 Let M (d) be the Hardy-Littlewood maximal operator defined on
for all 1 < p, q, r, s < ∞.
Proof Let be a positive radial symmetrically decreasing Schwartz function on (2.1) together with the L r ( s , R d )-boundedness of the Hardy-Littlewood maximal functions and Fubini's theorem shows that On the other hand, for any x, y ∈ R d , This together with (2.2) yields for any 1 < p, q < ∞. This proves Lemma 2.2.
We end this section by presenting the following lemma, which plays a key role in the proof of Theorem 1.1. Then, for any 1 < p, q, r < ∞, there exists a constant C > 0 independent of the coefficients of .
Let {b k } be a lacunary sequence such that 1 < δ 1 ≤ b k+1 b k ≤ δ 2 for all k ∈ Z. Let {λ k } k∈Z be a collection of C ∞ 0 (R + ) with the following properties: , 0 ≤ λ k (t) ≤ 1 and k∈Z λ k (t) = 1. We have the following result. Then, for 1 < p, q, r < ∞, there exists a constant C > 0 depending only on δ 2 and d such that .
Proof Define the operator Tf : for any 1 < p, q, r < ∞. One can easily check that k∈Z | k (ξ )| 2 ≤ 1 for all ξ = 0. By Plancherel's theorem we see that T is bounded from L 2 (R d ) to L 2 ( 2 , R d ). Next we shall prove that |x|≥2|y| k∈Z It is clear that ix·ξ dξ for any multi-index α.
This completes the proof of Lemma 3.3.