Atomic decomposition characterizations of weighted multiparameter Hardy spaces
Let \(\omega\in A_{\infty}(\mathbb{R}^n \times \mathbb{R}^m)=\bigcup_{1\leq q<\infty}A_q\). Let \(q_{\omega}=\inf\{q: \omega\in A_q\}\) be the critical index of \(\omega\). The author introduces the weighted-\((p, q)\) atomic Hardy spaces \( H^{p,q}_\omega (\mathbb{R}^n \times \mathbb{R}^m)\) for \(0 < p\leq 1\), \(q >q_{\omega}\) and shows that the weighted Hardy space \( H^{p}_\omega (\mathbb{R}^n \times \mathbb{R}^m)\) defined via Littlewood-Paley square functions coincides with \( H^{p,q}_\omega (\mathbb{R}^n \times \mathbb{R}^m)\) for \(0 < p\leq 1\), \(q >q_{\omega}\). As applications, the author proves that a linear operator \(T\) is bounded from \( H^{p}_\omega (\mathbb{R}^n \times \mathbb{R}^m)\) to \( L^{p}_\omega (\mathbb{R}^n \times \mathbb{R}^m)\), when \(T\) is bounded on \( H^{p}_\omega (\mathbb{R}^n \times \mathbb{R}^m)\) and on \( L^{q}_\omega (\mathbb{R}^n \times \mathbb{R}^m)\) for \(q >q_{\omega}\) and obtains a boundedness criterion for two parameter singular integrals on weighted Hardy spaces.
scientific article; zbMATH DE number 6152751
scientific article