Abstract
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.
Weighted Morrey estimates for Hausdorff operator and its commutator on the Heisenberg group
