On weighted \(H^ p\) boundedness of C-Z type singular integral operators (Q1916756)

The author considers the problem of boundedness of the Calderón-Zygmund type singular integral operator \(T\) acting from the weighted Hardy space \(H^p_\omega(\mathbb{R}^n)\) to \(L^p_\omega\) by the formula \(T(f,x)=\text{p.v.}(H*f)(x)\), where the kernel \(H(x)=h(x)K(x)\), \(K(x)=\Omega(x)/|x|^n\) is a Calderón-Zygmund kernel and \(h\) is a bounded radial function. As it has been proved in the article the operator \(T\) is of type \((H^p_\omega,L^p_\omega)\) \((0<p<1)\) under conditions on \(H\) weaker than \(|H(x-y)-H(x)|\leq C|y|/|x|^{n+1}\), \(|x|>2|y|>0\). The proof makes use of some atomic decomposition on \(H^p_\omega(\mathbb{R}^n)\).