Boundedness of sublinear operators on product Hardy spaces and its application. (English) Zbl 1195.42060

Chang, Der-Chen; Yang, Dachun; Zhou, Yuan

J. Math. Soc. Japan 62, No. 1, 321-353 (2010).

Let \(0<q\leq1\). A quasi-Banach space \(\mathcal B_q\) with quasi norm \(\|\cdot\| _{\mathcal B_q}\) is said to be a \(q\)-quasi-Banach space if \(\|\cdot\| _{\mathcal B_q}\) satisfies \(\|f+g\|_{\mathcal B_q}^q\leq\|f\|_{\mathcal B_q}^q +\|g\|_{\mathcal B_q}^q\). For any given \(q\)-quasi-Banach space \(\mathcal B_q\) and a vector space \(\mathcal Y\), an operator \(T\) from \(\mathcal Y\) to \(\mathcal B_q\) is said to be \(\mathcal B_q\)-sublinear if for any \(f,g\in \mathcal Y\) and \(\lambda, \mu\in\mathbb C\) one has
\[ \|T(\lambda f+\mu g)\|_{\mathcal B_q}\leq \Bigl(|\lambda|^q\|T(f)\|_{\mathcal B_q}^q +|\mu|^q\|T(g)\|_{\mathcal B_q}^q\Bigr)^{1/q} \]
and
\[ \|T(f)-T(g)\|_{\mathcal B_q}\leq \|T(f-g)\|_{\mathcal B_q}. \]
Obviously, if \(T\) is linear it is \(\mathcal B_q\)-sublinear. And, if \(\mathcal B_q\) is a function space, \(T\) is sublinear in the classical sense and \(T(f)\geq0\) for all \(f\in\mathcal Y\), then \(T\) is also \(\mathcal B_q\)-sublinear. \(\mathcal D_{s_1,s_2}(\mathbb R^n\times\mathbb R^m)\) \((s_1, s_2\in \{0\}\cup\mathbb N)\) denotes the set of all smooth functions on \(\mathbb R^n\times\mathbb R^m\) with compact support and vanishing moments up to order \(s_1\) with respect to the first variable and order \(s_2\) with respect to the second variable. With \(0\leq\sigma_1, \sigma_2<\infty\), \(\mathcal D_{s_1,s_2;\sigma_1,\sigma_2}(\mathbb R^{n_1}\times\mathbb R^{n_2})\) denotes the Banach space \(\mathcal D_{s_1,s_2}(\mathbb R^{n_1}\times\mathbb R^{n_2})\) endowed with the norm
\[ \|f\|_{\mathcal D_{s_1,s_2;\sigma_1,\sigma_2}} =\sup_{x_1\in\mathbb R^{n_1}, x_2\in\mathbb R^{n_2}} (1+|x_1|)^{\sigma_1}(1+|x_2|)^{\sigma_2}|f(x_1,x_2)|. \]
For \(0<p\leq1\), \(H^p(\mathbb R^n\times\mathbb R^m)\) denotes the atomic Hardy space, introduced by Chang and Fefferman, using \((p,2,s_1,s_2)\)-atoms for \(s_1,s_2\in \mathbb N\cup \{0\}\) satisfyig \(s_1\geq [n(1/p-1)]\) and \(s_2\geq [m(1/p-1)]\). One of the main results in this paper is the following Theorem. Let \(0<p\leq q\leq1\) and \(\mathcal B_q\) be a \(q\)-quasi-Banach space. Suppose that \(s_1\geq [n(1/p-1)]\) and \(s_2\geq [m(1/p-1)]\). Let \(T\) be a \(\mathcal B_q\)-sublinear operator from \(\mathcal D_{s_1,s_2}(\mathbb R^n\times\mathbb R^m)\) to \(\mathcal B_q\). Then \(T\) can be extended as a bounded \(\mathcal B_q\)-sublinear operator from \(H^p(\mathbb R^n\times\mathbb R^m)\) to \(\mathcal B_q\) if and only if \(T\) maps all \((p,2,s_1,s_2)\)-atoms in \(\mathcal D_{s_1,s_2}(\mathbb R^n\times\mathbb R^m)\) into uniformly bounded elements of \(\mathcal B_q\).
This generalizes the boundedness criterion on \(H^p(\mathbb R^n)\) to the product Hardy space \(H^p(\mathbb R^n\times\mathbb R^m)\), in the paper [D. Yang and Y. Zhou, Constr. Approx. 29, No. 2, 207–218 (2009; Zbl 1283.42029)]. In the proof of this theorem, the following two lemmas are the key ones: Lemma 1. Let \(0<p\leq1\), \(s_i\geq [n_i(1/p-1)]\) and \(\max\{n_i+s_i, n_i/p\}<\sigma_i<n_i+s_i+1\) for \(i=1,2\). Then for any \(f\in\mathcal D_{s_1,s_2}(\mathbb R^{n_1}\times\mathbb R^{n_2})\), there exist \(\{\lambda_k\}_{k\in\mathbb N}\subset \mathbb C\) and \((p,2,s_1,s_2)\)-atoms \(\{a_k\}_{k\in\mathbb N}\subset \mathcal D_{s_1,s_2}(\mathbb R^{n_1}\times\mathbb R^{n_2})\) such that \(f=\sum_{k\in\mathbb N}\lambda_ka_k\) in \(\mathcal D_{s_1,s_2;\sigma_1,\sigma_2}(\mathbb R^{n_1}\times\mathbb R^{n_2})\) and \(\bigl(\sum_{k\in\mathbb N}|\lambda_k|^p\bigr)^{1/p} \leq C\|f\|_{H^p(\mathbb R^{n_1}\times\mathbb R^{n_2})}\).
Lemma 2. Let \(0<p\leq q\leq1\) and \(\mathcal B_q\) be a \(q\)-quasi-Banach space. Let \(s_1, s_2\in \{0\}\cup\mathbb N\), \(\max\{n_i+s_i, n_i/p\}<\sigma_i<n_i+s_i+1\) for \(i=1,2\), and \(T\) be a \(\mathcal B_q\)-sublinear operator from \(\mathcal D_{s_1,s_2}(\mathbb R^n\times\mathbb R^m)\) to \(\mathcal B_q\). If there exists \(C>0\) such that for any \(f\in\mathcal D_{s_1,s_2}(\mathbb R^{n_1}\times\mathbb R^{n_2})\), \(\|Tf\|_{\mathcal B_q}\leq C\bigl(\sup_{x_2\in \mathbb R^{n_2}}\text{diam} (\text{supp}f(\cdot,x_2))\bigr)^{n_1/p}\times \bigl(\sup_{x_1\in \mathbb R^{n_1}}\text{diam} (\text{supp}f(x_1,\cdot))\bigr)^{n_2/p} \|f\|_{L^\infty(\mathbb R^{n_1}\times\mathbb R^{n_2})}\),
then \(T\) can be extended as a bounded \(\mathcal B_q\)-sublinear operator from \(\mathcal D_{s_1,s_2;\sigma_1,\sigma_2}(\mathbb R^{n_1}\times\mathbb R^{n_2})\) to \(\mathcal B_q\).
Using their theorem and its corollary, the authors give the boundedness of the commutators generated by Calderón-Zygmund operatros and Lipschitz functions from the usual Lebesgue space \(L^p(\mathbb R^n\times\mathbb R^m)\) with some \(p>1\) or the Hardy space \(H^p(\mathbb R^n\times\mathbb R^m)\) with some \(p\leq1\) near \(1\) to the Lebesgue space \(L^q(\mathbb R^n\times\mathbb R^m)\) with \(q>1\) determined by \(p\) and the Lipschitz constants.

Reviewer: Kôzô Yabuta (Nishinomiya)

MSC:

42B30	\(H^p\)-spaces
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25	Maximal functions, Littlewood-Paley theory
47B47	Commutators, derivations, elementary operators, etc.

Keywords:

Hardy space; \(L^p\) space; product space; sublinear operator; commutator; Calderón-Zygmund operator; Lipschitz function

Citations:

Zbl 1283.42029

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