一类核型算子的约化原理
Reduction principle for a certain class of kernel-type operators
AI总结：
该研究针对一类核型算子，证明了尽管远离零点的类似Hardy–Littlewood不等式不成立，但受限与非受限不等式的有界性等价性仍成立，与积分区域测度无关。
摘要：
经典的Hardy–Littlewood不等式表明，两个函数乘积的积分始终被其非递增重排的乘积的积分所控制。该结果的核心应用之一是：在零的某右邻域上积分的积分算子，其有界性等价于该算子在正非递增函数锥上的有界性。众所周知，针对远离零点积分的类似不等式并不成立。但本文证明，对于一类核型算子而言，受限不等式与非受限不等式的等价性依然成立，且与积分区域的测度无关。
英文摘要：
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.
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