Atomic decomposition and boundedness of operators on weighted Hardy spaces (Q2888783)

The main purpose of this paper is to give a criterion for the boundedness of operators on weighted Hardy spaces \(H_{\omega}^p(\mathbb R^n)\).NEWLINENEWLINEThe authors first establish a new atomic decomposition for \(f\in L_\omega ^2(\mathbb{R}^n) \cap H_{\omega}^p(\mathbb{R}^n)\), where the decomposition converges in \(L_\omega ^2\)-norm instead of in the distribution sense.NEWLINENEWLINELet \(0<p \leq 1\) and let \(\omega\) be a Muckenhoupt \(A_2\) weight function. Set \(N=\lfloor n(2/p-1) \rfloor\), the integer part of \(n(2/p-1)\). For \(f\in L_\omega ^2(\mathbb{R}^n) \cap H_{\omega}^p(\mathbb{R}^n)\), there exist a sequence \(\{a_i \}\) of \(\omega\)-\((p, 2, N)\)-atoms and a sequence \(\{\lambda_i \}\) of real numbers satisfying \(\sum| \lambda_i|^p \leq C \|f\|_{[b]H_{\omega}^p}^p\) such that \(f=\sum \lambda_i a_i\), where the series converges in \(L_\omega ^2(\mathbb{R}^n)\) and hence a subsequence converges almost everywhere.NEWLINENEWLINEAs applications of this decomposition, assuming that \(T\) is a linear operator bounded on \(L_\omega ^2(\mathbb R^n)\) and \(0<p \leq 1\), the authors obtain the following.NEWLINENEWLINE(i) If \(T\) is uniformly bounded in \(L_\omega ^p\)-norm for all \(\omega\)-\(p\)-atoms, then \(T\) can be extended to be bounded from \(H_{\omega}^p(\mathbb R^n)\) to \(L_{\omega}^p(\mathbb R^n)\);NEWLINENEWLINE(ii) If \(T\) is uniformly bounded in \(H_\omega ^p\)-norm for all \(\omega\)-\(p\)-atoms, then \(T\) can be extended to be bounded on \(H_{\omega}^p(\mathbb R^n)\);NEWLINENEWLINE(iii) If \(T\) is bounded on \(H_\omega ^p(\mathbb R^n)\), then \(T\) can be extended to be bounded from \(H_{\omega}^p(\mathbb R^n)\) to \(L_{\omega}^p(\mathbb R^n)\).