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.

中文翻译：

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海森堡群上线性和多线性分数阶Hausdorff算子的弱估计和强估计

本文致力于Heisenberg群H n {{\\ mathbb {H}}} ^ {n}上线性和多线性分数Hausdorff算子的弱估计和强估计。获得了对TΦm {T} _ {\ Phi} ^ {m}的精确估计。作为应用程序，我们在H n {{\ mathbb {H}}} ^ {n}上推导产品Hardy运算符的锐常数。TΦ，β{T} _ {\ Phi，\的一些弱类型（p，q）\ left（p，q）（1≤p≤∞）\ left（1 \ le p \ le \ infty）估计也可以得到beta}。作为应用程序，我们为Heisenberg群上的分数Hausdorff算子计算了一些尖锐的弱常数。此外，我们在Φ\ Phi的一些温和假设下，给出了TΦ的显式弱估计，β→m {T} _ {\ Phi，\ overrightarrow {\ beta}} ^ {m}。我们扩展了郭等人的结果。[海森堡集团的Hausdorff运算符Acta Math。罪。（英文）.31（2015），No. 11