- Research Article
- Open access
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Potential Operators in Variable Exponent Lebesgue Spaces: Two-Weight Estimates
Journal of Inequalities and Applications volume 2010, Article number: 329571 (2010)
Abstract
Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for corresponding inequalities. Appropriate examples of weights are also given.
1. Introduction
We study the two-weight problem for Hardy-type and potential operators in Lebesgue spaces with nonstandard growth defined on quasimetric measure spaces . In particular, our aim is to derive easily verifiable sufficient conditions for the boundedness of the operators
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ1_HTML.gif)
in weighted spaces which enable us to effectively construct examples of appropriate weights. The conditions are simultaneously necessary and sufficient for corresponding inequalities when the weights are of special type and the exponent
of the space is constant. We assume that the exponent
satisfies the local log-Hölder continuity condition, and if the diameter of
is infinite, then we suppose that
is constant outside some ball. In the framework of variable exponent analysis such a condition first appeared in the paper [1], where the author established the boundedness of the Hardy-Littlewood maximal operator in
. As far as we know, unfortunately, an analog of the log-Hölder decay condition (at infinity) for
is not known even in the unweighted case, which is well-known and natural for the Euclidean spaces (see [2–5]). Local log-Hölder continuity condition for the exponent
, together with the log-Hölder decay condition, guarantees the boundedness of operators of harmonic analysis in
spaces (see, e.g., [6]). The technique developed here enables us to expect that results similar to those of this paper can be obtained also for other integral operators, for instance, for maximal and Calderón-Zygmund singular operators defined on
.
Considerable interest of researchers is focused on the study of mapping properties of integral operators defined on (quasi)metric measure spaces. Such spaces with doubling measure and all their generalities naturally arise when studying boundary value problems for partial differential equations with variable coefficients, for instance, when the quasimetric might be induced by a differential operator or tailored to fit kernels of integral operators. The problem of the boundedness of integral operators naturally arises also in the Lebesgue spaces with nonstandard growth. Historically the boundedness of the maximal and fractional integral operators in spaces was derived in the papers [7–14]. Weighted inequalities for classical operators in
spaces, where
is a power-type weight, were established in the papers [10–12, 15–19], while the same problems with general weights for Hardy, maximal, and fractional integral operators were studied in [10, 20–25]. Moreover, in the latter paper, a complete solution of the one-weight problem for maximal functions defined on Euclidean spaces is given in terms of Muckenhoupt-type conditions.
It should be emphasized that in the classical Lebesgue spaces the two-weight problem for fractional integrals is already solved (see [26, 27]), but it is often useful to construct concrete examples of weights from transparent and easily verifiable conditions.
To derive two-weight estimates for potential operators, we use the appropriate inequalities for Hardy-type transforms on (which are also derived in this paper) and Hardy-Littlewood-Sobolev-type inequalities for
and
in
spaces.
The paper is organized as follows: in Section 1, we give some definitions and prove auxiliary results regarding quasimetric measure spaces and the variable exponent Lebesgue spaces; Section 2 is devoted to the sufficient governing two-weight inequalities for Hardy-type operators defined on quasimetric measure spaces, while in Section 3 we study the two-weight problem for potentials defined on .
Finally we point out that constants (often different constants in the same series of inequalities) will generally be denoted by or
. The symbol
means that there are positive constants
and
independent of
such that the inequality
holds. Throughout the paper is denoted the function
by the symbol
.
2. Preliminaries
Let be a topological space with a complete measure
such that the space of compactly supported continuous functions is dense in
and there exists a nonnegative real-valued function (quasimetric)
on
satisfying the conditions:
(i) if and only if
;
(ii)there exists a constant , such that
for all
;
(iii)there exists a constant , such that
for all
.
We assume that the balls are measurable and
for all
and
; for every neighborhood
of
, there exists
, such that
. Throughout the paper we also suppose that
and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ2_HTML.gif)
for all , positive
and
with
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ3_HTML.gif)
We call the triple a quasimetric measure space. If
satisfies the doubling condition
, where the positive constant
does not depend on
and
, then
is called a space of homogeneous type (SHT). For the definition, examples, and some properties of an SHT see, for example, monographs [28–30].
A quasimetric measure space, where the doubling condition is not assumed, is called a nonhomogeneous space.
Notice that the condition implies that
because we assumed that every ball in
has a finite measure.
We say that the measure is upper Ahlfors
-regular if there is a positive constant
such that
for for all
and
. Further,
is lower Ahlfors
-regular if there is a positive constant
such that
for all
and
. It is easy to check that if
is a quasimetric measure space and
, then
is lower Ahlfors regular (see also, e.g., [8] for the case when
is a metric).
For the boundedness of potential operators in weighted Lebesgue spaces with constant exponents on nonhomogeneous spaces we refer, for example, to the monograph [31, Chapter 6] and references cited therein.
Let be a nonnegative
-measurable function on
. Suppose that
is a
-measurable set in
. We use the following notation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ4_HTML.gif)
Assume that . The variable exponent Lebesgue space
(sometimes it is denoted by
) is the class of all
-measurable functions
on
for which
. The norm in
is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ5_HTML.gif)
It is known (see, e.g., [8, 15, 32, 33]) that is a Banach space. For other properties of
spaces we refer, for example, to [32–34].
We need some definitions for the exponent which will be useful to derive the main results of the paper.
Definition 2.1.
Let be a quasimetric measure space and let
be a constant. Suppose that
satisfies the condition
. We say that
belongs to the class
, where
, if there are positive constants
and
(which might be depended on
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ6_HTML.gif)
holds for all ,
. Further,
if there are positive constants
and
such that (2.5) holds for all
and all
satisfying the condition
.
Definition 2.2.
Let be an SHT. Suppose that
. We say that
(
satisfies the log-Hölder-type condition at a point
) if there are positive constants
and
(which might be depended on
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ7_HTML.gif)
holds for all satisfying the condition
. Further,
(
satisfies the log-Hölder type condition on
) if there are positive constants
and
such that (2.6) holds for all
with
.
We will also need another form of the log-Hölder continuity condition given by the following definition.
Definition 2.3.
Let be a quasimetric measure space, and let
. We say that
if there are positive constants
and
(which might be depended on
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ8_HTML.gif)
for all with
. Further,
if (2.7) holds for all
with
.
It is easy to see that if a measure is upper Ahlfors
-regular and
(resp.,
), then
(resp.,
. Further, if
is lower Ahlfors
-regular and
(resp.,
), then
(resp.,
).
Remark 2.4.
It can be checked easily that if is an SHT, then
.
Remark 2.5.
Let be an SHT with
. It is known (see, e.g., [8, 35]) that if
, then
. Further, if
is upper Ahlfors
-regular, then the condition
implies that
.
Proposition 2.6.
Let be positive and let
and
(resp.,
, then the functions
, and
belong to
resp.,
. Further if
resp.,
then
, and
belong to
resp.,
.
The proof of the latter statement can be checked immediately using the definitions of the classes ,
,
, and
.
Proposition 2.7.
Let be an
and let
. Then
for all
with
, where
is a small constant, and the constant
does not depend on
.
Proof.
Due to the doubling condition for , Remark 1.1, the condition
and the fact that
we have the following estimates:
, which proves the statement.
The proof of the next statement is trivial and follows directly from the definition of the classes and
. Details are omitted.
Proposition 2.8.
Let be a quasimetric measure space and let
. Suppose that
be a constant. Then the following statements hold:
-
(i)
if
(resp.,
, then there are positive constants
,
, and
such that for all
and all
(resp., for all
with
), one has that
.
-
(ii)
Let
, then there are positive constants
,
, and
(in general, depending on
) such that for all
(
) and all
one has
.
-
(iii)
Let
, then there are positive constants
,
, and
such that for all balls
with radius
(
) and all
, one has that
.
It is known that (see, e.g., [32, 33]) if is a measurable function on
and
is a measurable subset of
, then the following inequalities hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ9_HTML.gif)
Further, Hölder's inequality in the variable exponent Lebesgue spaces has the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ10_HTML.gif)
Lemma 2.9.
Let be an SHT.
(i)If is a measurable function on
such that
and if
is a small positive number, then there exists a positive constant
independent of
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ11_HTML.gif)
(ii)Suppose that and
are measurable functions on
satisfying the conditions
and
. Then there exists a positive constant
such that for all
the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ12_HTML.gif)
holds.
Proof.
Part (i) was proved in [35] (see also [31, page 372], for constant ). The proof of Part (ii) is given in [31, (Lemma 6.5.2, page 348)] for constant
and
, but repeating those arguments we can see that it is also true for variable
and
. Details are omitted.
Lemma 2.10.
Let be an
. Suppose that
, then
satisfies the condition
(resp.,
) if and only if
resp.,
.
Proof.
We follow [1].
Necessity.
Let , and let
with
for some positive constant
. Observe that
, where
. By the doubling condition for
, we have that
, where
is a positive constant which is greater than 1. Taking now the logarithm in the last inequality, we have that
. If
, then by the same arguments we find that
.
Sufficiency.
Let . First observe that If
, then
. Consequently, this inequality and the condition
yield
. Further, there exists
such that
and
, where
and
are positive constants. Hence
.
Let, now, and let
where
is a small number. We have that
and
for some positive constant
. Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ13_HTML.gif)
Definition 2.11.
A measure on
is said to satisfy the reverse doubling condition
if there exist constants
and
such that the inequality
holds.
Remark 2.12.
It is known that if all annulus in are not empty (i.e., condition (2.1) holds), then
implies that
(see, e.g., [28, page 11, Lemma 20]).
Lemma 2.13.
Let be an
. Suppose that there is a point
such that
. Let
be the constant defined in Definition 2.11. Then there exist positive constants
and
(which might be depended on
) such that for all
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ14_HTML.gif)
holds, where and the constant
is independent of
.
Proof.
Taking into account condition (2.1) and Remark 2.12, we have that . Let
. By the doubling and reverse doubling conditions, we have that
. Suppose that
, where
is a sufficiently small constant. Then by using Lemma 2.10 we find that
.
In the sequel we will use the notation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ15_HTML.gif)
where the constants and
are taken, respectively, from Definition 2.11 and the triangle inequality for the quasimetric
, and
is a diameter of
.
Lemma 2.14.
Let be an
and let
. Suppose that there is a point
such that
. Assume that if
, then
and
outside some ball
. Then there exists a positive constant C such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ16_HTML.gif)
for all and
.
Proof.
Suppose that . To prove the lemma, first observe that
and
. This holds because
satisfies the reverse doubling condition and, consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ17_HTML.gif)
Moreover, the doubling condition yields , where
. Hence,
.
Further, since we can assume that , we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ18_HTML.gif)
Moreover, using the doubling condition for we have that
. This gives the estimates
.
For simplicity, assume that . Suppose that
is an integer such that
. Let us split the sum as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ19_HTML.gif)
Since outside the ball
, by using Hölder's inequality and the fact that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ20_HTML.gif)
Let us estimate . Suppose that
and
. Also, by Proposition 2.6, we have that
. Therefore, by Lemma 2.13 and the fact that
, we obtain that
and
, where
. Further, observe that these estimates and Hölder's inequality yield the following chain of inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ21_HTML.gif)
Now we claim that , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ22_HTML.gif)
and the positive constant does not depend on
. Indeed, suppose that
. Then taking into account Lemma 2.13 we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ23_HTML.gif)
Consequently, since and
, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ24_HTML.gif)
This implies that . Thus, the desired inequality is proved. Further, let us introduce the following function:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ25_HTML.gif)
It is clear that because
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ26_HTML.gif)
for some positive constant . Then, by using this inequality, the definition of the function
, the condition
, and the obvious estimate
, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ27_HTML.gif)
Consequently, . Hence,
. Analogously taking into account the fact that
and arguing as above, we find that
. Thus, summarizing these estimates we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ28_HTML.gif)
Lemma 2.14 for spaces defined with respect to the Lebesgue measure was derived in [24] (see also [22] for
,
, and
).
3. Hardy-Type Transforms
In this section, we derive two-weight estimates for the operators:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ29_HTML.gif)
Let be a positive constant, and let
be a measurable function defined on
. Let us introduce the notation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ30_HTML.gif)
Remark 3.1.
If we deal with a quasimetric measure space with , then we will assume that
. Obviously,
and
in this case.
Theorem 3.2.
Let be a quasimetric measure space. Assume that
and
are measurable functions on
satisfying the condition
. In the case when
, suppose that
const,
const, outside some ball
. If the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ31_HTML.gif)
holds, then is bounded from
to
.
Proof.
Here we use the arguments of the proofs of Theorem 1.1.4 in [31, (see page 7)] and of Theorem 2.1 in [21]. First, we notice that for all
. Let
and let
. First, assume that
. We denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ32_HTML.gif)
Suppose that , then
for some
. Let us denote
, and
. Then
is a nondecreasing sequence. It is easy to check that
for
, and
. If
, then
if and only if
. If
, then we take
. Since
for every
, we have that
. It is obvious that
. Further, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ33_HTML.gif)
Let us denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ34_HTML.gif)
Notice that . Consequently, by this estimate and Hölder's inequality with respect to the exponent
we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ35_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ36_HTML.gif)
Observe now that . Hence, this fact and the condition
imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ37_HTML.gif)
It follows now that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ38_HTML.gif)
Since , it is obvious that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ39_HTML.gif)
Finally, . Thus,
is bounded if
.
Let us now suppose that . We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ40_HTML.gif)
By using the already proved result for and the fact that
, we find that
because
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ41_HTML.gif)
Further, observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ42_HTML.gif)
It is easy to see that (see also [31, Theorems 1.1.3 or 1.1.4]) the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ43_HTML.gif)
guarantees the boundedness of the operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ44_HTML.gif)
from to
. Thus,
is bounded. It remains to prove that
is bounded. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ45_HTML.gif)
Observe, now, that the condition guarantees that the integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ46_HTML.gif)
is finite. Moreover, . Indeed, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ47_HTML.gif)
Further,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ48_HTML.gif)
For , we have that
. Since
and condition (2.1) holds, there exists a point
such that
. Consequently,
and
, where
. Consequently, the condition
yields
. Finally, we have that
. Hence,
is bounded from
to
.
The proof of the following statement is similar to that of Theorem 3.2; therefore, we omit it (see also the proofs of Theorem 1.1.3 in [31] and Theorems 2.6 and 2.7 in [21] for similar arguments).
Theorem 3.3.
Let be a quasimetric measure space. Assume that
and
are measurable functions on
satisfying the condition
. If
, then, one assumes that
const,
const outside some ball
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ49_HTML.gif)
then is bounded from
to
.
Remark 3.4.
If const, then the condition
in Theorem 3.2 (resp.,
in Theorem 3.3) is also necessary for the boundedness of
(resp.,
) from
to
. See [31, pages 4-5] for the details.
4. Potentials
In this section, we discuss two-weight estimates for the potential operators and
on quasimetric measure spaces, where
. If
, then we denote
and
by
and
, respectively.
The boundedness of Riesz potential operators in spaces, where
is a domain in
was established in [5, 6, 36, 37].
For the following statement we refer to [11].
Theorem A..
Let be an
. Suppose that
and
. Assume that if
, then
outside some ball. Let
be a constant satisfying the condition
. One sets
. Then,
is bounded from
to
.
Theorem B (see [9]).
Let be a nonhomogeneous space with
and let
be a constant defined by
, where the constants
and
are taken from the definition of the quasimetric
. Suppose that
and that
is upper Ahlfors 1-regular. One defines
, where
. Then
is bounded from
to
.
For the statements and their proofs of this section, we keep the notation of the previous sections and, in addition, introduce the new notation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ50_HTML.gif)
where and
are constants defined in Definition 2.11 and the triangle inequality for
, respectively. We begin this section with the following general-type statement.
Theorem 4.1.
Let be an SHT without atoms. Suppose that
and
is a constant satisfying the condition
. Let
. One sets
. Further, if
, then one assumes that
const outside some ball
. Then the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ51_HTML.gif)
holds if the following three conditions are satisfied:
(a) is bounded from
to
;
(b) is bounded from
to
;
(c)there is a positive constant such that one of the following inequalities hold: (1)
for
a.e.
; (2)
for
a.e.
.
Proof.
For simplicity, suppose that . The proof for the case
is similar to that of the previous case. Recall that the sets
and
are defined in Section 2. Let
and let
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ52_HTML.gif)
where .
Observe that if and
, then
. Consequently, the triangle inequality for
yields
, where
. Hence, by using Remark 2.4, we find that
. Applying condition (a) now, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ53_HTML.gif)
Further, observe that if and
, then
. By condition (b), we find that
.
Now we estimate . Suppose that
. Theorem A and Lemma 2.14 yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ54_HTML.gif)
The estimate of for the case when
is similar to that of the previous one. Details are omitted.
Theorems 4.1, 3.2, and 3.3 imply the following statement.
Theorem 4.2.
Let be an SHT. Suppose that
and
is a constant satisfying the condition
. Let
. One sets
. If
, then, one supposes that
const outside some ball
. Then inequality (4.2) holds if the following three conditions are satisfied:
-
(i)
(4.6)
-
(ii)
(4.7)
-
(iii)
condition (c) of Theorem 4.1 holds.
Remark 4.3.
If const on
, then the conditions
,
, are necessary for (4.2). Necessity of the condition
follows by taking the test function
in (4.2) and observing that
for those
and
which satisfy the conditions
and
(see also [31, Theorem 6.6.1, page 418] for the similar arguments) while necessity of the condition
can be derived by choosing the test function
and taking into account the estimate
for
and
.
The next statement follows in the same manner as the previous one. In this case, Theorem B is used instead of Theorem A. The proof is omitted.
Theorem 4.4.
Let be a nonhomogeneous space with
. Let
be a constant defined by
. Suppose that
and that
is upper Ahlfors 1-regular. We define
, where
. Then the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ57_HTML.gif)
holds if
-
(i)
(4.9)
-
(ii)
(4.10)
and (iii) condition (c) of Theorem 4.1 is satisfied.
Remark 4.5.
It is easy to check that if and
are constants, then conditions (i) and (ii) in Theorem 4.4 are also necessary for (4.8). This follows easily by choosing appropriate test functions in (4.8) (see also Remark 4.3).
Theorem 4.6.
Let be an SHT without atoms. Let
and let
be a constant with the condition
. One sets
. Assume that
has a minimum at
and that
. Suppose also that if
, then
is constant outside some ball
. Let
and
be positive increasing functions on
. Then the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ60_HTML.gif)
holds if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ61_HTML.gif)
for ;
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ62_HTML.gif)
for .
Proof.
We prove the theorem for . The proof for the case when
is similar. Observe that by Lemma 2.10 the condition
implies
. We will show that the condition
implies the inequality
for all
, where
and
are constants defined in Definition 2.11 and the triangle inequality for
, respectively. Indeed, let us assume that
, where
is a small positive constant. Then, taking into account the monotonicity of
and
and the facts that
(for small
) and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ63_HTML.gif)
Hence, . Further, if
, where
is a large number, then since
and
are constants, for
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ64_HTML.gif)
In the last inequality we used the fact that satisfies the reverse doubling condition.
Now we show that the condition implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ65_HTML.gif)
Due to monotonicity of functions and
, the condition
, Proposition 2.6, Lemmas 2.9 and 2.10, and the assumption that
has a minimum at
, we find that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ66_HTML.gif)
Now, Theorem 4.2 completes the proof.
Theorem 4.7.
Let be an SHT with
. Suppose that
,
and
are measurable functions on
satisfying the conditions:
and
. Assume that
and there is a point
such that
. Suppose also that
is a positive increasing function on
. Then the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ67_HTML.gif)
holds if the following two conditions are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ68_HTML.gif)
Proof.
For simplicity, assume that . First observe that by Lemma 2.10 we have
and
. Suppose that
and
. We will show that
.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ69_HTML.gif)
First, observe that by virtue of the doubling condition for , Remark 2.4, and simple calculation we find that
. Taking into account this estimate and Theorem 3.2 we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ70_HTML.gif)
Further, it is easy to see that if , then the triangle inequality for
and the doubling condition for
yield that
. Hence, due to Proposition 2.7, we see that
for such
and
. Therefore, Theorem 3.3 implies that
.
It remains to estimate . Let us denote:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ71_HTML.gif)
Then we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ72_HTML.gif)
Using Hölder's inequality for the classical Lebesgue spaces we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ73_HTML.gif)
Denote the first inner integral by and the second one by
.
By using the fact that , where
, we see that
, while by applying Lemma 2.9, for
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ74_HTML.gif)
Summarizing these estimates for and
we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ75_HTML.gif)
By applying monotonicity of , the reverse doubling property for
with the constants
and
(see Remark 2.12), and the condition
we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ76_HTML.gif)
Due to the facts that and
is increasing, for
, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ77_HTML.gif)
Analogously, the estimate for follows. In this case, we use the condition
and the fact that
when
. The details are omitted. The theorem is proved.
Taking into account the proof of Theorem 4.6, we can easily derive the following statement, proof of which is omitted.
Theorem 4.8.
Let be an SHT with
. Suppose that
,
and
are measurable functions on
satisfying the conditions
and
. Assume that
. Suppose also that there is a point
such that
and
has a minimum at
. Let
and
be a positive increasing function on
satisfying the condition
(see Theorem 4.6). Then inequality (4.11) is fulfilled.
Theorem 4.9.
Let be an SHT with
and let
be upper Ahlfors 1-regular. Suppose that
and that
. Let
have a minimum at
. Assume that
is constant satisfying the condition
. We set
. If
and
are positive increasing functions on
satisfying the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ78_HTML.gif)
then the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ79_HTML.gif)
holds.
Proof.
The proof is similar to that of Theorem 4.6, we only discuss some details. First, observe that due to Remark 2.5 we have that , where
. It is easy to check that the condition
implies that
for all t, where the constant
is defined in Definition 2.11 and
is from the triangle inequality for
. Further, Lemmas 2.9 and 2.10, the fact that
has a minimum at
, and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ80_HTML.gif)
where the constant does not depend on
and
, yield that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F329571/MediaObjects/13660_2010_Article_2124_Equ81_HTML.gif)
Theorem 4.4 completes the proof.
Example 4.10.
Let and
, where
and
are constants satisfying the condition
,
. Then
satisfies the conditions of Theorem 4.6.
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Acknowledgments
The first and second authors were partially supported by the Georgian National Science Foundation Grant (project numbers: GNSF/ST09/23/3-100 and GNSF/ST07/3-169). A part of this work was fulfilled in Abdus Salam School of Mathematical sciences, GC University, Lahore. The second and third authors are grateful to the Higher Educational Commission of Pakistan for financial support. The authors express their gratitude to the referees for their very useful remarks and suggestions.
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Kokilashvili, V., Meskhi, A. & Sarwar, M. Potential Operators in Variable Exponent Lebesgue Spaces: Two-Weight Estimates. J Inequal Appl 2010, 329571 (2010). https://doi.org/10.1155/2010/329571
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DOI: https://doi.org/10.1155/2010/329571