Abstract
In the setting of the Heisenberg group $\mathbb{H}^{n}$ , we characterize those nonnegative functions w defined on $[0,1]$ for which the weighted Hardy operator $\mathsf{H}_{w}$ is bounded on $L^{p}(\mathbb{H}^{n})$ , $1\leq p\leq\infty$ , and on $\operatorname{BMO}(\mathbb{H}^{n})$ . 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.
Highlights
The history of weighted Hardy operators can be traced back to the end of the th century when Hadamard [ ] used the idea of fractional differentiation of an analytic function via differentiation of its Taylor series
Corresponding to fractional differentiation, we note that Hadamard dealt with fractional integration in the form of
The Heisenberg group Hn is a homogeneous group with dilations δr(x, x, . . . , x n, x n+ ) := rx, rx, . . . , rx n, r x n+, r >
Summary
The history of weighted Hardy operators can be traced back to the end of the th century when Hadamard [ ] used the idea of fractional differentiation of an analytic function via differentiation of its Taylor series. (α) which led him further to consider generalized fractional integrals of the form g(xξ )v(ξ ) dξ. In , Carton-Lebrun and Fosset [ ] defined the weighted Hardy operators Hψ as follows. Hψ f (x) := f (tx)ψ(t) dt, x ∈ Rn. Sometimes Hψ is called the generalized Hardy operator [ ]. For other results of the weighted Hardy operators on the Euclidean space one can refer to [ ] and references therein. We will consider the weighted Hardy operators on the Heisenberg group. Form a natural basis for the Lie algebra of left-invariant vector fields. The Heisenberg group Hn is a homogeneous group with dilations δr(x , x , . . . , x n, x n+ ) := rx , rx , . . . , rx n, r x n+ , r >
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