Hausdorff operators on weighted Bergman and Hardy spaces
Ha Duy Hung, Luong Dang Ky
TL;DR
This work provides a sharp, unified treatment of Hausdorff operators on weighted spaces of holomorphic functions in the upper half-plane, establishing precise boundedness criteria in Bergman and Hardy-type settings via kernels $\varphi$ and the weight parameter $\alpha$. Central to the approach are carefully crafted test functions ${\Phi_\varepsilon}$ that yield sharp norm estimates and enable necessity arguments, together with derivative identities ${\left( {\mathscr H_\varphi}f \right)'= {\mathscr H_{\widetilde{\varphi}}}(f')}$ and boundary-value transfer results ${({\mathscr H_\varphi}f)^*= {\mathcal H_\varphi}(f^*)}$. The paper also extends these results to the real line, proving analogous boundedness and norm-equality statements for ${\mathcal H}_\varphi$ on ${L^p_{|\cdot|^\alpha}(\mathbb R)}$ and establishing commutation with the Hilbert transform under suitable weight ranges. Altogether, the results connect holomorphic function spaces with their real-variable counterparts, recovering and extending known operators (e.g., Cesàro, Stieltjes) within a cohesive framework.
Abstract
Let $1\leq p<\infty$, $α>-1$, and let $\varphi$ be a measurable function on $(0,\infty)$. The main purpose of this paper is to study the Hausdorff operator \[ \mathscr H_\varphi f(z)=\int_0^\infty f\left(\frac{z}{t}\right) \frac{\varphi(t)}{t} dt, \quad z\in \mathbb C^+, \] on the weighted Bergman space $\mathcal A^p_α(\mathbb C_+)$ and on the power weighted Hardy space $\mathcal H^p_{|\cdot|^α}(\mathbb{C_+})$ of the upper half-plane. Some applications to the real version of $\mathscr H_\varphi$ are also given.