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

Abstract

This paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group Hn{{\mathbb{H}}}^{n}. A sharp strong estimate for TΦm{T}_{\Phi }^{m} is obtained. As an application, we derive the sharp constant for the product Hardy operator on Hn{{\mathbb{H}}}^{n}. Some weak-type (p,q)\left(p,q) (1≤p≤∞)\left(1\le p\le \infty ) estimates for TΦ,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 TΦ,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.

Keywords

• hausdorff operator
• heisenberg group
• multilinear
• sharp bound
• 42b20
• 42b35