Abstract

The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete variants of monotone operators in ideal spaces and different monotonicity conditions for functions. As applications, we calculate the norms of some integral operators on the cones, associate norms over some cones in Lebesgue spaces, the norms of the dilation operator and embedding operators on weighted Lorentz spaces with general weights. Under some more general conditions, we present order sharp estimates for Hardy-type operators on the cones.

中文翻译：

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单调性函数锥上单调算子范数的计算

摘要

该论文致力于研究具有单调性的函数锥上的单调算子在理想空间中范数的精确计算问题。我们对这个问题实施了一种通用方法，该方法涵盖了理想空间中单调运算符的许多具体变体和函数的不同单调性条件。作为应用，我们计算锥体上一些积分算子的范数，关联勒贝格空间中一些锥体上的范数，膨胀算子的范数和具有一般权重的加权洛伦兹空间上的嵌入算子。在一些更一般的条件下，我们对锥体上的 Hardy 型算子给出了阶次尖锐的估计。