The classical Hardy--Littlewood inequality asserts that the integral of a product of two functions is always majorized by that of their non-increasing rearrangements. One of the pivotal applications of this result is the fact that the boundedness of an integral operator which integrates over some right neighbourhood of zero is equivalent to the boundedness of the same operator on the cone of positive non-increasing functions. It is well known that an analogous inequality for integration away from zero is not true. However, as we show in this paper, the equivalence of the restricted inequality for the non-restricted one is still true for certain class of kernel-type operators, regardless of the measure of the integration domain.

中文翻译：

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某类核型算子的约简原理

经典的 Hardy-Littlewood 不等式断言，两个函数的乘积的积分总是被它们的非递增重排的积分所占据。这个结果的一个关键应用是这样一个事实，即在零的某个右邻域上积分的积分算子的有界性等价于同一个算子在正非增函数锥上的有界性。众所周知，远离零积分的类似不等式是不正确的。然而，正如我们在本文中所展示的，无论积分域的度量如何，非受限不等式的受限不等式的等价性对于某些核类型算子仍然适用。