论文标题
关于手段及其耐力财产的整体方法
On the integral approach to means and their Hardy property
论文作者
论文摘要
著名的顽强不等式可以用$ \ int_0^\ int_0^\ mathcal {p} _p \ big（f | _ {[0，x]} \ big）dx \ le（1-p）^{ - 1/p}^{ - int_0^fext_ fexpt（f | _ {[0，x]} } p \ in（0,1）\ text {and} f \ in l^1 \ text {with} f \ ge0，$ $，其中$ \ mathcal {p} _p $代表$ p $ th power的平均值。人们可以向另一个家庭询问该物业可能的概括（取决于平均值）。 适应Riemann积分的概念，对于每个加权平均值，我们都定义了下部和上部的均值。我们证明，每一个对称，单调，$ \ mathbb {r} $ - 加权平均值$ i $，它的条目中是连续的，权重最多具有一个整体不可分割的连续扩展。此外，此扩展可以保留强壮的常数。该结果允许将后者的不平等扩展到同质的凹偏差手段的家族。
The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where $\mathcal{P}_p$ stands for the $p$-th power mean. One can ask about possible generalizations of this property to another families (with sharp constant depending on the mean). Adapting the notion of Riemann integral, for every weighted mean we define the lower and the upper integral mean. We prove that every symmetric, monotone, $\mathbb{R}$-weighted mean on $I$ which is continuous in its entries and weights has at most one continuous extension to the integral one. Moreover this extension preserves the Hardy constant. This result allows to extend the latter inequality to the family of homogeneous, concave deviation means.
