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Non-smooth decomposition of homogeneous Triebel-Lizorkin spaces with applications to the Marcinkiewicz integral (The deepening of function spaces and its environment)

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(1)11. Non‐smooth decomposition of homogeneous Triebel‐Lizorkin spaces with applications to the Marcinkiewicz integral Keisuke Asami ( Tokyo Metropolitan Univ.) Abstract. Thc aim of this paper is to develop a theory of non‐smooth decompo‐ sition in homogeneous Triebel‐Lizorkin spaces. As a byproduct, we can recover the decomposition results for Hardy spaces as a special case The result extends what Frazicr and Jawerth obtained in 1990. The result by Frazier and Jawcrth covcrs only thc limited range of the parameters but the result in this paper is valid for all admissible parameters for Triebel‐ Lizorkin spaces. As an application of the main results, we prove that the Marcinkiewicz operator is boundcd. What is new in this paper is to reconstruct sequence spaces other than classical lp spaces.. 1. Preparation and Main result. Definition 1. Let 0<p<\infty, 0<q\leq\infty and s\in R. Let \varphi\in C_{C}^{\infty}(R^{7L}) satisfy \chi_{B(4)\backslash B(2)}\leq\varphi\leq\chi_{B(8)\backslash B(1)} . The homogeneous Triebel‐Lizorkin space \dot{F}_{p,q}^{s}(R^{n}) is defined to be the set of all f\in S'(R^{n})/\mathcal{P}(R^{n}) for which thc quantity. \Vert f\Vert_{F_{p^{s}q} \equiv\Vert\{2^{js}\varphi_{j}(D)f\}_{j\in Z}\Vert_{L^ {p}(l^{q})} is finite, where \varphi_{j}(x)\equiv\varphi(2^{-j}x), \mathcal{P}(R^{n}) denotes the set of all polynomials on R^{n} ,. and. \psi(D)f(x)\equiv \mathcal{F}^{-1}\psi*f(x) (x\in R^{n}) for \psi\in S(R^{7l}) and f\in S'(R^{n}) and \Vert\{f_{j}\cdot\}_{j\in Z}\Vert_{Lp(\iota q} ) stands for the vector‐norm of a sequence \{f_{\dot{j} \}_{j=-\infty}^{\infty} of mesurable functions: For 0<p, q\leq\infty. \Vert\{f_j}\_{j\inZ}\Vert_{L^{p}(l^{q})\equiv(\int_{R^{n}(.\sum_{j=- \infty}^{\infty}|f_{j}(x)|^{q})^{\frac{p}{q}dx)^{\frac{1}{p}.

(2) 2 Remark 2. The space. \dot{F}_{p,q}^{s}(R^{n}). \dot{F}_{p,2}^{0}(R^{n})=L^{p}(R^{rb}). if. realizes many function spaces. Indeed,. 1<p<\infty,\dot{F}_{p,2}^{0}(R^{n})=H^{p}(R^{n}). if 0<p\leq 1. with cquivalence of quasi‐norms, where H^{p}(R^{n}) stands for the Hardy Space. See. [3, Theorem 6.1.2] for the first equivalence and [4, Theorem 2.2.9] for the second. equivalence.. Definition 3. For \nu\in Z and m=(m_{1}, m_{2}, \ldots, m_{n})\in Z^{n} , we define. Q_{\nu,m}\equiv\prod_{j=1}^{r\iota}[\frac{m_{j} {2^{\nu} ,\frac{m_{j}+1} {2^{\nu} ) Denote by \mathcal{D}=\mathcal{D}(R^{n}) thc sct of such cubes. The elements in \mathcal{D}(R^{n}) are callcd dyadic cubes.. We adopt the definition by Grafakos; see [4, Definition 2.3.5]. Definition 4. Let. 0<p<\infty, 0<q\leq\infty and sequences \{r_{Q}\}_{Q\in \mathcal{D}}\subset C such that the function. s\in. R. We consider the set of. g_{q}^{6}( \{r_{Q}\}_{Q\in \mathcal{D} ;x)\equiv(\sum_{Q\in D}(|Q|^{-\frac{s} {n} |r_{Q}|\chi_{Q}(x) ^{q})^{\frac{1}{q} (x\in R^{n}) is in L^{p}(R^{n}) . For such sequences r=\{r_{Q}\}_{Q\in D} sct. r=\{r_{Q}\}_{Q\in \mathcal{D}} is said to belong to. \dot{f}_{p,q}^{6}(R^{n}). if. \Vert r\Vert_{f_{p^{\mathcal{S} q} \equiv\Vert g_{q}^{s}(r)\Vert_{L^{p} . A scqucnce \Vert r\Vert_{f_{p^{8}.q} <\infty.. Definition 5. Let 0<p.<\infty, 0<q\leq\infty and s\in R . A sequence r=\{r_{Q}\}_{Q\in D} is called an \infty ‐atom for f_{p,q}^{s}(R^{n}) with cube Q_{0} if there exists a dyadic cubc Q_{0} such that. g_{q}^{s}(\{r_{Q}\ _{Q\inD}:\cdot)\equiv(\sum_{Q\inD}(|Q|^{-\frac{s}{n} |r_{Q}|\chi_{Q})^{q})^{\frac{1}{q} \leq\chi_{Q_{0} . Our first theorem is as follows:. Theorem 6. Suppose that we are given parameters. p, q, s,. u. satisfying. 0<p< \infty, 0<q\leq\infty, s\in R, 0<u\leq\min(1, q) 1. For any. t\in\dot{f}_{p,q}^{6}(R^{n}) ,. there exists a decomposition. t=\sum_{j=1}^{\infty}\lambda_{j}r_{j},. ..

(3) 3 where each. r_{j}. is an. \infty. ‐atom for. \dot{f}_{p,q}^{s}(R^{TL}). with cube Q_{j} and. \{\lambda_{j}\}_{j=1}^{\infty} satisfies. \Vert(\sum_{j=1}^{\infty}|\lambda_{j}|^{u}\chi_{Q_{j})^{\frac{1}v}\Vert_{L^ {p}\leqC\Vert \Vert_{f p.q}^{s} 2. If a sequence satisfy. \{Q_{j}\}_{j=1}^{\infty}. of cubes and a sequence. \{\lambda_{j}\}_{j=1}^{\infty}. of complex numbers. \Vert(\sum_{j=1}^{\infty}|\lambda_{j}|^{u}\chi_{Q_{j})^{7l}\underline{1} \Vert_{Lp}<\infty, then for any belongs to. \infty. ‐atoms r_{j} for. \dot{f}_{p_{:}q}^{s}(R^{n}). with cube Q_{j} , the series. \dot{f}_{p.q}^{s}(R^{n}) .. t=\sum_{j=1}^{\infty}\lambda_{j}r_{j}. The case of s\in R, 0<p=u\leq 1 and p\leq q\leq\infty is proved in [2, Theorem 7.2]. In this casc there is no condition on thc position of the cubes since. \Vert(\sum_{j=1}^{\infty}|\lambda_{j}|^{u}\chi_{Q_{j})^{\frac{1}?}\Vert= (\sum_{j=1}^{\infty}|\lambda_{j}|^{p}|Q_{j}|)^{\frac{1}p} Definition 7. Lct 0<p<\infty, 0<q\leq\infty, s\in R and 0<v<\infty . One says that a sequence r=\{r_{Q}\}_{Q\in D} is called a v ‐atom for \dot{f}_{p,q}^{s}(R^{n}) with cube Q_{0} if there exists a dyadic cube Q_{0} such that. supp ( g_{q}^{s}(\{r_{Q}\}_{Q\in \mathcal{D} ;\cdot))\subset Q_{0},. \Vert g_{q}^{s}(\{r_{Q}\}_{Q\in D};\cdot)\Vert_{L^{v} \leq|Q_{0}|^{\frac{1}{?)} }.. We can refine the latter half of Theorem 6 as follows:. Theorem 8. In addition to the assumption in Theorem 6, let v \in(\max(1,p), \infty) . If a sequence \{Q_{j}\}_{j=1}^{\infty} of cubes and a sequence \{\lambda_{j}\}_{j=1}^{\infty} of complex numbers satisfy. !??) , then for any ‐atoms f_{p,q}^{s}(R^{n}) v. r_{j}. with cube Q_{j} , the series. t. given by (1) belongs to. .. Thc above rcsults cover the ones in [2, Section 7]. What is new about this. \dot{ \imath} sthc asewhercp>m\dot{ \imath} n(q, 1).Thc ascwhenp>1andq- ,2ises. pecial yintcrcst\dot{ \imath} ngbc auscth\dot{ \imath} sy\dot{ \imath} cldsthcdc ompos\dot{ \imath} t\dot{ \imath} onforL^{p}(R^{n})=\dot{F}_{p2}^{0} (R^{n}). papcr. Definition 9. Let. 0<p<\infty, 0<q\leq\infty, s\in R. Let \nu\in Z and m\in Z^{n}. Suppose that the integers K, L\in Z satisfy K\geq 0 and L\geq-1 . A function a\in C^{K}(R^{n}) is said to be a smooth (K, L) ‐atom centered at Q_{0,m} for \dot{f}_{p_{:}q}^{s}(R^{n}) ,.

(4) 4 if it is supported on 3 Q_{0,\tau n} and if it satisfies the differential inequality and the moment condition:. \Vert\partial^{\alpha}a\Vert_{L^{\infty} \leq 2^{\nu|\alpha|}, |\alpha|\leq K,. \int_{R^{n} x^{\beta}a(x)dx=0. |\beta|\leq L . The case. L=-1. (1). is excluded in (1).. Definition 10. Lct 0<p<\infty_{;}0<q\leq\infty, non‐smooth atom for. \dot{F}_{p,q}^{s}(R^{r\iota}). with cube. \tilde{Q}. s\in. R.. We say that. if there exists a cube. \tilde{Q}. A. is a. such that. A= \sum_{Q\subset Q^{-} r_{Q}a_{Q} where. r=\{r_{Q}\}_{Q\in D}. ccntered at. is an. \infty. ‐atom for. Q.. \dot{f}_{p,q}^{s}. and each a_{Q} is a smooth. (K, L) ‐atom. The following theorem extends [4, Corollary 2.3.9]. Theorem 11. Let 0<p<\infty, 0<q\leq\infty , s\in R, 0<u \leq\min(1, q) , and let. Z\ni L\geq\max(-1, [\sigma_{p,q}-s]) where [\cdot] denotes the Gauss sign,. Then we have the following. 1. Let. f\in\dot{F}_{p,q}^{s}(R^{n}) .. \sigma_{p}\equiv n(\frac{1}{\min(1,p)}-1). and \sigma_{p,q}\equiv\max(\sigma_{p}, \sigma_{q}) .. Then we can write. f= \sum_{j=1}^{\infty}\lambda_{j}A_{j} in S'(R^{n})/\mathcal{P}(R^{n}) , where \{A_{j}\}_{j=1}^{\infty}iS a sequence of non‐smooth atoms and \{\lambda_{j}\}_{j=1}^{\infty} and \{Q_{j}\}_{j=1}^{\infty} satisfy the following condition: The estimate. \Vert(\sum_{j=1}^{\infty}|\lambda_{j}|^{u}\chi_{Q_{2})^{\frac{1}?l} \Vert_{L^{p}\leqC\Vertf\Vert_{F_{p.q}^{s}. holds and. suppA_{j}\subset 3Q_{j}.. 2. Suppose that each A_{j} is a non‐smooth atom with cube Q_{j} and the complex sequence \{\lambda_{j}\}_{j=1}^{\infty} satisfies. \Vert(\sum_{j=1}^{\infty}|\lambda_{j}|^{u}\chi_{Q_{j})^{?}\underline{1}\Vert_ {L^{p}<\infty..

(5) 5 Then by letting satisfies. f\equiv\sum_{j=1}^{\infty}\lambda_{j}A_{j}. , the sum converges in \mathcal{S}'(R^{n})/\mathcal{P}(R^{n}) and. \Vertf\Vert_{F_{p.q}^{s}\leqC\Vert(\sum_{j=1}^{\infty}|\lambda_{j}|^{u} \chi_{Q_{j})^{\frac{1}u}\Vert_{L^{p} In Thcorem 11 the case of s\in R, 0<p=u\leq 1 and p\leq q\leq\infty is [2, Theorem 7.4(ii)].. 2. Application. Definition 12. Let 0<\rho<n, defined by. 1<q<\infty .. The Marcinkiewicz operator is. \mu_{\Omega,\rho,q}f(x)\equiv(\int_{0}^{\infty}|\frac{1}{t^{\rho} \int_{B(t)}f (x-y)\frac{\Omega(y/|y)}{|y^{n-\rho} dy|^{q}\frac{dt}{t})^{\frac{1}{q} where we write. B(r)=\{|x|<r\}\subset R^{r\iota}. for. r>0 here and below.. We suppose. \int_{S^{n-1} \Omega(\omega)d\sigma(\omega)=0, \Omega\in C^{1}(S^{r\iota-1}). ,. where S^{r\iota-1}=\{|x|=1\} . According to [9, Theorem 1], we have. \Vert\mu_{\Omega,\rho,q}f\Vert_{L^{p} \leq C\Vert f\Vert_{F_{p.q}^{0} if. 1<p<\infty.. Thc following is an application of Theorem 11, and extends [9, Theorem 1]. Theorem 13. The estimate. \Vert\mu_{\Omega,\rho,q}f\Vert_{L^{p} \leq C\Vert f\Vert_{F_{p.q}^{0}. for all. f\in\dot{F}_{p,q}^{0}. if. \frac{nq}{nq+1}<p<\infty, 1<q<\infty. References [1] C. Fefferman and E. Stein, Some maximal inequalities, Amer. J. Math, 93 (1971), 107‐115..

(6) 6 [2] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34‐170.. [3] L. Grafakos, Classical Fourier Analysis, Graduate texts in mathematics; 249, New York, Springer, 2014.. [4] L. Grafakos, Modern Fourier Analysis, Graduate texts in mathematics; 250, NGW York, Springcr, 2014.. [5] L. Liu and D.Yang, Boundedness of sublinear opcrators in Triebel‐Lizorkin spaces via atoms, Studia Math. 190 (2009), 164‐183 [6] Y. Han, M. Paluszyn’ski and G. Wciss, A new atomic decomposition for the Triebel‐Lizorkin spaces, in: Harmonic Analysis and Operator Theory. (Caracas, 1994), Contemp. Math. 189, Amer. Math. Soc., Providence, RI, 1995, 235‐249.. [7] G. E. Hu and Y. Meng, Multilinear Calderón‐Zygmund operator on products of Hardy spaces, Acta Math. Sinica 28 (2012), no. 2, 281‐294. [8] Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to boundcd lincar opcrator, Intcgr. Eq. Oper. Theory. 77 (2013), 123‐148.. [9] Y. Sawano and K. Yabuta, Fractional type Marcinkiewicz integral operators associated to surfaces, J. Inequal. Appl. 2014, 2014:232, 29 pp.. [10] H. Tricbel, Fractal and Spectra, Birkhäuser Basel, 1997..

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