The molecular characterization of weighted Hardy spaces
Let \(0<p\leq 1\leq q\leq\infty\), \(p\neq q\) and \(\omega\in A_q\) (the Muckenhoupt's class). The authors introduce the weighted molecules which characterize the weighted Hardy spaces \(H^p_\omega(\mathbb R^n)\). As an application, the authors give some sufficient conditions on the kernel \(k\) such that the convolution operator \(Tf=k\ast f\) is bounded on weighted Hardy spaces \(H^p_\omega(\mathbb R^n)\) for some \(p\leq 1\) and \(\omega\in A_1\). The authors also prove that the Hilbert transform is bounded on \(H^p_\omega(\mathbb R)\) for \(1/2<p\leq 1\) and \(\omega\in A_1\) and that the Riesz transforms are bounded on \(H^p_\omega(\mathbb R^n)\) for \(n/(n+1)<p\leq 1\) and \(\omega\in A_1\).
scientific article; zbMATH DE number 1763778
scientific article