Discrete Calderón's identity, atomic decomposition and boundedness criterion of operators on multiparameter Hardy spaces
The authors consider discrete Calderón's identity, a new atomic decomposition and give an application for this atomic decomposition in the setting of product Hardy space. Suppose \(0<p\leq 1\) and \(1<q<\infty\). Let \(\psi^{(1)}(x_1)\), \(\psi^{(2)}(x_2)\) be Schwartz functions with support on the unit ball satisfying the conditions: for a fixed large integer \(M\) depending on \(p\), \(\sum_j|\hat{\psi}^{(1)}(2^{-j}\xi_1)|^2=1\) for all \(\xi_1\in \mathbb R^n\setminus \{0\}\), and \(\int_{\mathbb R^n}\psi^{(1)}(x_1)x^{\alpha}_1dx_1=0\) for all \(0\leq|\alpha|\leq 2M\); and \(\sum_k|\hat{\psi}^{(2)}(2^{-k}\xi_2)|^2=1\) for all \(\xi_2\in \mathbb R^n\setminus \{0\}\), and \(\int_{\mathbb R^m}\psi^{(2)}(x_2)x^{\beta}_2dx_2=0\) for all \(0\leq|\beta|\leq 2M\). Then for every function \(f\in H^p(\mathbb R^n\times\mathbb R^m)\cap L^q(\mathbb R^{n+m})\), there exists a function \(g\in H^p(\mathbb R^n\times\mathbb R^m)\cap L^q(\mathbb R^{n+m}) \) with \(\| f\|_q\sim \| g\|_q\) and \(\| f\|_{H^p}\sim \| g\|_{H^p}\) such that the following discrete Calderón's identity \[ f(x_1,x_2)=\sum_j\sum_k\sum_I\sum_J|I\times J|\psi_{j,k}(x_1-x_I,x_2-y_J)\psi_{j,k}\ast g(x_I,y_J) \] holds true and the above series converges in both the norms of \(L^q(\mathbb R^{n+m})\) and \(H^p(\mathbb R^n\times\mathbb R^m)\), where \(I\subseteq \mathbb R^n\) and \(J\subseteq\mathbb R^m \) are dyadic cubes, \(x_I, y_J\) are any fixed points in \(I\) and \(J\), respectively. By using the above discrete Calderón's identity the authors give a new atomic decomposition in terms of \((p,q)\)-atoms for the multiparameter product Hardy space. Let \(f\in H^p(\mathbb R^n\times\mathbb R^m)\cap L^q(\mathbb R^{n+m})\). Then there exists a sequence of \((p,q)\)-product atoms \(\{a_j\}\) of \(H^p(\mathbb R^n\times\mathbb R^m)\) and a sequence of scalars \(\{\lambda_j\}\) with \((\sum_j|\lambda_j|^p)^{\frac{1}{p}}\leq C\| f\|_{H^p}\) such that \(f=\sum_j\lambda_ja_j\), where the series converges to \(f\) in both the \(H^p(\mathbb R^n\times\mathbb R^m)\) and \(L^q(\mathbb R^{n+m})\) norms. Furthermore, they give an application for the above atomic decomposition. Let \(T\) be a linear operator which is bounded on \(L^q(\mathbb R^{n+m})\). Then (1) \(T\) is bounded from \(H^p(\mathbb R^n\times\mathbb R^m)\) to \(L^p(\mathbb R^{n+m})\) if and only if \(\| Ta\|_{L^p}\leq C\) for all \((p,q)\)-product atoms; (2) \(T\) is bounded on \(H^p(\mathbb R^n\times\mathbb R^m)\) if and only if \(\| Ta\|_{H^p}\leq C\) for all \((p,q)\)-product atoms, where the constant \(C\) is independent of \(a\).
scientific article; zbMATH DE number 5728277
scientific article