论文标题
关于加权强加权常数的特性
On properties of weighted Hardy constant for means
论文作者
论文摘要
对于给定的加权平均值$ \ MATHSCR {m} $定义在$ \ Mathbb {r} _+$的子插图和一个权重序列$λ=（λ_n）_ {n = 1}^\ infty $我们定义了一个硬质量的$ \ mathscr h（λ） \ sum_ {n = 1}^\inftyλ_n\ mathscr {m} \ big（（（x_1，\ dots，x_n），（λ_1，\ dots，λ_n）\ le le \ le \ le \ le \ le \ le \ mathscr h（λ） } x \ in \ ell^1（λ）。$$本注释的目的是对映射$ \ Mathscr H $进行全面研究。例如，我们证明它在点式拓扑中是较低的半连续性。此外，我们表明，每当$ \ mathscr {m} $都是单调和詹森 - 孔concave平均值时，它的权重是连续的，而$ \ m rathscr h $就矢量的分区而言是单调的。最后，我们为$λ$提供了一些足够的条件，以验证每个对称和单调的平等$ \ mathscr h（λ）= \ sup \ mathscr h $。
For a given weighted mean $\mathscr{M}$ defined on a subinterval of $\mathbb{R}_+$ and a sequence of weights $λ=(λ_n)_{n=1}^\infty$ we define a Hardy constant $\mathscr H(λ)$ as the smallest extended real number such that $$ \sum_{n=1}^\infty λ_n \mathscr{M}\big((x_1,\dots,x_n),(λ_1,\dots,λ_n)\big) \le \mathscr H(λ) \cdot \sum_{n=1}^\infty λ_n x_n \text{ for all }x \in \ell^1(λ).$$ The aim of this note is to present a comprehensive study of the mapping $\mathscr H$. For example we prove that it is lower semicontinuous in the pointwise topology. Moreover we show that whenever $\mathscr{M}$ is a monotone and Jensen-concave mean which is continuous in its weights then $\mathscr H$ is monotone with respect to the partitioning of the vector. Finally we deliver some sufficient conditions for $λ$ to validate the equality $\mathscr H(λ)=\sup \mathscr H$ for every symmetric and monotone mean.
