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

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.

### 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.

Zbl 1283.42029
Full Text:

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