Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group

This paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group H n {{\mathbb{H}}}^{n} . A sharp strong estimate for T Φ m {T}_{\Phi }^{m} is obtained. As an application, we derive the sharp constant for the product Hardy operator on H n {{\mathbb{H}}}^{n} . Some weak-type ( p , q ) \left(p,q) ( 1 ≤ p ≤ ∞ ) \left(1\le p\le \infty ) estimates for T Φ , β {T}_{\Phi ,\beta } are also obtained. As applications, we calculate some sharp weak constants for the fractional Hausdorff operator on the Heisenberg group. Besides, we give an explicit weak estimate for T Φ , β → m {T}_{\Phi ,\overrightarrow{\beta }}^{m} under some mild assumptions on Φ \Phi . We extend the results of Guo et al. [Hausdorff operators on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714] to the fractional setting.