The boundedness of Calderón-Zygmund operators on Hardy spaces \(HA^ p\) and its applications (Q1200061)

The authors show that for an \(A_ 1\)-weight \(\omega\) (Muckenhoupt class) and \(1<p<\infty\), if \(T\) is a \(\delta\)-Calderón-Zygmund operator and \(T^ \ast(1)=0\), then \(T\) can be extended to a bounded operator on the weighted \`\` Beurling-Hardy'' space \(HA_ \omega^ p(\mathbb{R}^ n)\). Here \(H A_ \omega^ p\) is defined by using central atoms. A function \(a(x)\) is said to be a weighted \((1,p)\)-central atom, if \(\operatorname{supp} a\subset B(0,r)\) \(r\geq1\), \(\bigl(\int| a(x)|^ p\omega(x) dx\bigr)^{1/p} \leq \bigl(\omega(B(0,r))\bigr)^{1/p-1}\), and \(\int a(x) dx=0\). \(HA _ \omega^ p\) is the set of all \(f\in L_ \omega^ 1\) such that \(f=\sum_ k \lambda_ k a_ k\), each \(a_ k\) is a central atom, and \(\sum_ k|\lambda_ k| <\infty\). Its norm is \(\inf\{\sum_ k|\lambda_ k|\}\), where the infimum is taken over all such decompositions. A necessary condition is given for a wavelet series to be in \(HA^ p\).