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Weighted anisotropic product Hardy spaces and boundedness of sublinear operators. (English) Zbl 1205.42021
Let $A_1$ and $A_2$ be expansive dilations, respectively, on $\mathbb{R}^n$ and $\mathbb{R}^m$. Let $\vec{A}=(A_1,A_2)$ and $\mathcal {A}_p(\vec{A})$ be the class of product Muckenhoupt weights on $\mathbb{R}^n\times\mathbb{R}^m$ for $p\in(1,\infty]$. Suppose that $\varphi^{(1)}\in \mathcal {S}(\mathbb{R}^n)$ and $\varphi^{(2)}\in \mathcal{S}(\mathbb{R}^m)$ with $\widehat{\varphi^{i}}(0)=0$ for $i=1,2.$ Set $\varphi(x)=\varphi^{(1)}(x_1)\varphi^{(2)}(x_2)$ and $\varphi_{k_1,k_2}(x)=b^{-k_1}_1b^{-k_2}_2\varphi(A^{-k_1}_1x_1,A^{-k_2}_2x_2)$ for all $x=(x_1,x_2)\in \mathbb{R}^n\times\mathbb{R}^m$. For all $f\in\mathcal {S}'(\mathbb{R}^n\times\mathbb{R}^m)$, the anisotropic product Lusin area function of $f$ is defined as follows: $$\vec{S}_{\varphi}(f)(x)=\left\{\sum_{k_1,k_2\in \mathbb{Z}}b^{-k_1}_1b^{-k_2}_2 \int_{B^{(1)}_{k_1}\times B^{(2)}_{k_2}}|\varphi_{k_1,k_2}\ast f(x-y)|^2dy\right\}^{\frac{1}{2}},$$ where $B^{(i)}_{k_i}=A^{k_i}\Delta_i$ and $\Delta_i$ is an open and symmetric convex ellipsoid, respectively, in $\mathbb{R}^n$ and $\mathbb{R}^m$ for $i=1,2$. The authors give the first main result in this paper. Let $1<p<\infty$ and $\omega\in \mathcal {A}_p(\vec{A})$. Then $f\in L^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m)$ if and only $f\in \mathcal {S}'_{\infty,\omega}(\mathbb{R}^n\times\mathbb{R}^m)$ and $\vec{S}_{\varphi}(f)\in L^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m)$. Moreover, for all $f\in L^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m)$, $\Vert f\Vert_{L^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m)}\sim \Vert\vec{S}_{\varphi}(f)\Vert_{L^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m)}$, where $\mathcal {S}'_{\infty,\omega}(\mathbb{R}^n\times\mathbb{R}^m)$ denotes the set of all $f\in \mathcal {S}'(\mathbb{R}^n\times\mathbb{R}^m)$ vanishing weakly at infinity. Let $0<p\le 1$ and $\omega\in \mathcal {A}_{\infty}(\vec{A})$. The weighted anisotropic product Hardy space is defined by $$H^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m; \vec{A})=\left\{f\in\mathcal {S}'_{\infty,\omega}(\mathbb{R}^n\times\mathbb{R}^m): \Vert f\Vert_{H^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m; \vec{A})}=\Vert \vec{S}_{\psi}(f)\Vert_{L^p_{\omega} (\mathbb{R}^n\times\mathbb{R}^m)}<\infty\right\},$$ where $\psi(x)=\psi^{(1)}(x_1)\psi^{(2)}(x_2)$ is a Schwartz function satisfying other extra conditions. For the above Hardy space, the authors give an atomic decomposition. They show that if $(p,q,\vec{s})_{\omega}$ is an admissible triplet, then $f\in H^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m; \vec{A})$ if and only if $f=\sum_{j\in \mathbb{N}}\lambda_ja_j$ in $\mathcal {S}'(\mathbb{R}^n\times\mathbb{R}^m)$, where $\sum_{j\in \mathbb{N}}|\lambda_j|^p<\infty$ and $\{a_j\}_{j\in \mathbb{N}}$ are $(p,q,\vec{s})_{\omega}$-atoms. Furthermore, they prove that all finite linear combinations of $(p,q,\vec{s})^{*}_{\omega}$-atoms is dense in $H^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m; \vec{A}).$ As an application, they prove that if $T$ is a sublinear operator and maps all $(p,q,\vec{s})^{*}_{\omega}$-atoms into uniformly bounded elements of a quasi-Banach space $\mathcal {B}$, then $T$ uniquely extends to a bounded sublinear operator from $H^p_{\omega}(\mathbb{R}^n\times\mathbb{R}^m; \vec{A})$ to $\mathcal {B}$. The results of this paper improve the existing results for weighted product Hardy spaces and are new in the unweighted anisotropic setting.
Reviewer: Liu Yu (Beijing)

42B30$H^p$-spaces (Fourier analysis)
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
42B35Function spaces arising in harmonic analysis
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