Atomic decomposition and boundedness criterion of operators on multi-parameter Hardy spaces of homogeneous type
The authors consider the Hardy spaces on a space of homogeneous type \((X,\rho,\mu)\) satisfying the following conditions: \(X\) is a topological space endowed with a Borel measure \(\mu\) and a quasi-metric \(\rho\), satisfying the conditions below: (a) \(\rho(x,y)=\rho(y,x)\), (b) \(\rho(x,y)>0\) if and only if \(x\neq y\), (c) there exists a constant \(K\) such that \(\rho(x,y)\leq K[\rho(x,z)+\rho(z,y)]\) for all \(x,y,z\in X\), (d) the balls \(B_r(x)=\{y\in X;\,\rho(x,y)<r\}\) centered at \(x\) and of radius \(r>0\) form a basis of open neighborhoods of the point \(x\), (e) there exists a positive constant \(d\) such that \(\mu(B_{r}(x))\sim r^d\), (f)\( |\rho(x,y)-\rho(z,y)|\leq C_0\,\rho(x,z)^\theta|\rho(x,y)+\rho(z,y)|^{1-\theta}\). (It was shown by \textit{R. A. Macias} and \textit{C. Segovia} [Adv. Math. 33, 271--309 (1979; Zbl 0431.46019)] that there exists such a quasi-metric which gives the same topology as the space of homogeneous type in the sense of Coifman and Meyer.) Let \((X_1,\rho_1,\mu_1)_{d_1,\theta_1}, (X_2,\rho_2,\mu_2)_{d_2,\theta_2}\) be two spaces of homogeneous type in the above sense. On the product space \(X_1\times X_2\) there is a Littlewood-Paley theorem, and there can be built the multi-parameter Hardy spaces \(H^p(X_1\times X_2)\) in terms of the discrete Littlewood-Paley function \(g_2\). In this paper, the authors define \((p,q)\)-atoms of \(H^p(X_1\times X_2)\), and give the following: Let \(0<\varepsilon_i\leq\theta_i\), and \(0<\max(d_1/(d_1+\varepsilon_1), d_2/(d_2+\varepsilon_2))<p\leq 1<q<\infty\). Then if \(f\in H^p(X_1\times X_2)\cap L^q(X_1\times X_2)\), there exist a sequence of numbers \(\{\lambda_k\}_{k\in\mathbb Z}\) and a sequence of \((p,q)\)-atoms \(\{a_k\}_{k\in\mathbb Z}\) such that \((\sum_{k\in\mathbb Z}|\lambda_k|^p)\) is equivalent to \(\|f\|_{H^p(X_1\times X_2)}\) and \(f=\sum_{k\in\mathbb Z}\lambda_k a_k\) both in \(H^p(X_1\times X_2)\) and \(L^q(X_1\times X_2)\). Using this, one has that every \(L^q(X_1\times X_2)\) bounded operator is bounded from \(H^p(X_1\times X_2)\) to \(L^p(X_1\times X_2)\) if and only if it is bounded for \((p,q)\)-atoms.
scientific article; zbMATH DE number 6102792
scientific article