Open Journal of Applied Sciences
Vol.04 No.09(2014), Article ID:48591,3 pages

Some Rearrangement Inequalities on Space of Homogeneous Type

Tiejun Chen

Yiyang Medical College Hunan Pro of China, Yiyang, China


Copyright © 2014 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 9 June 2014; revised 22 July 2014; accepted 2 August 2014


Let be a Muckenhoupt weight. In this paper we get the estimate of rearrangement in homogeneous space that is. The similar estimate is obtained only on space of.


Rearrangement, Homogeneous Space, Weight

1. Introduction

We first recall some basic notions about the homogeneous space and the weights we are going to use.

Definition 1 [1] . (Homogeneous space X). Let X be a set. A function d: is called a quasi- distance on X if the following conditions are satisfied:

1) for every x and y in X, , and if and only if x = y,

2) for every x and y in X, ,

3) there exists a constant K such that for every x, y and z in X.

Let μ be a positive measure on the -algebra of subsets of X generated by the d-balls, with and r > 0. Then a structure (X, d, μ), with d and μ as above, is called a space of homogeneous type.

We say that (X, d, μ) is a space of homogeneous type regular in measure if μ is regular, that is for every measurable set E, given, there exists an open set G such that and. In what follows we always assume that the space (X, d, μ) is regular in measure.

A non-negative locally integrable on homogeneous space X function is called a weight. With any

weight function we call the measure. Given a measurable function f on homogeneous space

X, define its non-increasing rearrangement with respect to a weight similar to (see [1] , p. 32).


Definition 2 (weight) [2] . A weight is in Muckenhoupt’s class respect to μ if there are positive constants C and such that the inequality:

holds for every ball B and every measurable set. The infimum of such C will be denoted by.

2. Basic Lemmas

Denote doubling condition D, a weight if and only if for any ball holds. Clearly if then.

Lemma 1 [3] . Let (X, d, μ) be a space of homogeneous type. Let be a family of balls in X such that is measurable and. Then there exists a disjoint sequence, possibly finite, such that for some constant C. Moreover, every is contained in


Lemma 2. (C-Z decomposition) [4] [5] . Let (X, d, μ) be a space of homogeneous type such that the open balls are open sets. Let f be a nonnegative integrable function defined on X, then for every (if), there exist a sequence of disjoint balls such that if, C is the constant in Lemma [1] then


2) for every ball B centered at, holds.

Lemma 3. and, If X is a ball and is an arbitrary measurable set of positive measure with , there exist mutually disjoint balls such that

Bi cover E and

Proof: If

Letting, then


For every ball B centered at



If there exist and, now exists such that, then


this is a contradiction.

Then and


3. Inequalities Conclusion

Theorem 1. then .

Proof: The proof is similar to Lerner [5] - [7] ,

From [6] , We get two collections of balls, then

Fix X, with, for all E, there is, then exist dis-

joint balls, hold

Which contains


Select from the balls, which are not contained in,

. That is for all. There exist then

Note that






We have

Taking supremum over all with, we get the argument .


A project supported by scientific research fund of Hunan provincial education department in China (NO: 13C 955).


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