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).
(1)
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
some.
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
1),
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
then
For every ball B centered at
i.e.
,
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
Then
Select from the balls, which are not contained in,
. That is for all. There exist then
Note that
Since
Then
,
i.e.
.
We have
Taking supremum over all with, we get the argument .
Fund
A project supported by scientific research fund of Hunan provincial education department in China (NO:
13C
955).