# Reduction principle for a certain class of kernel-type operators

## TL;DR

This paper establishes a reduction principle for a class of kernel-type operators, showing that certain restricted inequalities are equivalent to non-restricted ones, extending classical rearrangement inequalities.

## Contribution

It introduces a new reduction principle for kernel-type operators, demonstrating the equivalence of restricted and non-restricted inequalities for these operators.

## Key findings

- Equivalence of restricted and non-restricted inequalities for certain kernel operators
- Extension of classical rearrangement inequalities beyond zero neighborhood
- Applicability regardless of the measure of the integration domain

## Abstract

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.