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Reduction principle for a certain class of kernel-type operators

Dalimil Peša

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.

Reduction principle for a certain class of kernel-type operators

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.

Paper Structure

This paper contains 8 sections, 10 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-increasing rearrangement
  4. Rearrangement invariant Banach function norms
  5. Operators
  6. The fall of the star
  7. Auxiliary statements
  8. Main result

Key Result

Theorem \oldthetheorem

Let $I:(0,\infty) \rightarrow (0,\infty)$ be a non-decreasing and let $\lVert \cdot \rVert_X$ and $\lVert \cdot \rVert_Y$ be rearrangement invariant Banach function norms on $M((0, \infty), \lambda)$. Then the following statements are equivalent:

Theorems & Definitions (28)

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  • ...and 18 more