L p $L^{p}$ and BMO bounds for weighted Hardy operators on the Heisenberg group

Abstract

Abstract In the setting of the Heisenberg group H n $\mathbb{H}^{n}$ , we characterize those nonnegative functions w defined on [ 0 , 1 ] $[0,1]$ for which the weighted Hardy operator H w $\mathsf{H}_{w}$ is bounded on L p ( H n ) $L^{p}(\mathbb{H}^{n})$ , 1 ≤ p ≤ ∞ $1\leq p\leq\infty$ , and on BMO ( H n ) $\operatorname{BMO}(\mathbb{H}^{n})$ . Meanwhile, the corresponding operator norm in each case is derived. Furthermore, we introduce a type of weighted multilinear Hardy operators and obtain the characterizations of their weights for which the weighted multilinear Hardy operators are bounded on the product of Lebesgue spaces in terms of Heisenberg group. In addition, the corresponding norms are worked out.

Keywords

• Heisenberg group
• weighted Hardy operator
• BMO
• weighted multilinear Hardy operator