The Cauchy integral operator on weighted Hardy space (Q606359)

The Cauchy integral operator is defined by \[ C_A f(x) = \text{p.v.} \int_{\mathbb{R}^1} \frac{f(y)}{ x-y + i(A(x) - A(y))}dy, \] where \(A(x)\) is a real valued function. It is well-known that \(C_A\) is \(L^p\) bounded if \(A' \in L^{\infty}\). \textit{Y. Komori} [Hokkaido Math. J. 37, No. 2, 389--398 (2008; Zbl 1214.42020)] proved the boundedness of \(C_A\) from Hardy spaces \(H^p\) to local Hardy spaces \(h^p\) under suitable conditions about \(A\) and \(p\). The authors prove the following boundedness on weighted Hardy spaces introduced by \textit{J.-O. Strömberg} and \textit{A. Torchinsky} [Weighted Hardy spaces. Lecture Notes in Mathematics, 1381. Berlin etc.: Springer-Verlag (1989; Zbl 0676.42021)]. Let \(0 < \alpha \leq 1 \leq q, q/2 <p \leq 1\) and \(q/(1 + \alpha) \leq p < q\). If \(w \in A_{q}\) and \(A' \in L^{\infty} \cap \text{Lip}_{loc}^{\alpha}\), then \(C_A\) is bounded from \(H^p_w(\mathbb{R}^1)\) to \(h^p_w(\mathbb{R}^1)\). When \(p=1\) and \(w \equiv 1\), they prove \[ \| C_A f \|_{h^1} \leq C \| f \|_{H^1}\quad \text{where } A' \in L^{\infty}. \] This improves the result by Komori.