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:
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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] ,
![]()
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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 ![]()
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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).