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Weighted kernel operators in variable exponent amalgam spaces
Journal of Inequalities and Applications volume 2013, Article number: 173 (2013)
Abstract
The paper is devoted to weighted inequalities for positive kernel operators in variable exponent amalgam spaces. In particular, a characterization of a weight v governing the boundedness/compactness of the weighted kernel operators and , defined on and ℝ, respectively, under the log-Hölder continuity condition on exponents of spaces is established. These operators involve, for example, weighted variable parameter fractional integrals. The results are new even for constant exponent amalgam spaces.
MSC:46E30, 47B34.
1 Introduction
In the paper, we derive necessary and sufficient conditions on a weight function v governing the boundedness/compactness of the positive kernel operators
in variable exponent amalgam spaces (VEAS) under the log-Hölder continuity condition on exponents of spaces. It should be emphasized that the results are new even for constant exponent amalgam spaces.
Historically, the boundedness problem for the two-weighted Hardy transform from to was studied in the papers [1, 2] in different terms on weights (see also [3] for related topics). In [1], the authors explored also the compactness problem for . The boundedness for fractional integral operators in (weighted) variable exponent Lebesgue spaces defined on Euclidean spaces was investigated by many authors (see, e.g., the papers [4–14], etc.). The compactness (resp. non-compactness) of fractional and singular integrals in weighted spaces was studied in [15]. We refer also to the monograph [16] for related topics.
The space is a special case of the Musielak-Orlicz space (see [17, 18]). The first systematic study of modular spaces is due to Nakano [19].
Variable exponent Lebesgue and Sobolev spaces arise, e.g., in the study of mathematical problems related to applications to mechanics of the continuum medium (see [16, 20] and references cited therein).
The manuscript consists of four sections. In Section 2, we recall some well-known facts about variable exponent Lebesgue spaces . In Section 3, we recall the definition, history and some essential properties of amalgam spaces with a constant exponent, and also known results about the boundedness of some integral operators in these spaces; boundedness criteria for the operators and in VEAS are also established. The compactness of positive kernel operators in VEAS is studied in Section 4.
Throughout the paper, constants (often different constants in the same series of inequalities) will mainly be denoted by c or C; by the symbol , we denote the function , ; the relation means that there are positive constants and such that .
2 Preliminaries
We begin this section by the definition and essential properties of variable exponent Lebesgue spaces.
Let E be a measurable set in ℝ with positive measure. We denote
for a measurable function p on E. Suppose that . Denote by ρ a weight function on E (i.e., ρ is an almost everywhere positive measurable function). We say that a measurable function f on E belongs to (or to ) if
It is a Banach space with respect to the norm (see, e.g., [21–24])
If , then we use the symbol (resp. ) instead of (resp. ). It is clear that .
In the sequel, we will denote by ℤ and the set of all integers and the set of non-positive integers, respectively.
To prove the main results, we need some known statements.
Let E be a measurable subset of ℝ. Suppose that . Then
-
(i)
-
(ii)
Hölder’s inequality
holds, where , .
Let and let E be a bounded subset of ℝ. Then the following inequality
holds.
Definition 2.1 We say that p satisfies the weak Lipschitz (log-Hölder continuity) condition on (), if there is a positive constant A such that for all x and y in E with the inequality
holds.
Lemma A ([25])
Let I be an interval in ℝ. Then if and only if there exists a positive constant c such that
for all intervals with . Moreover, the constant c does not depend on I.
For the next statement we refer to [2] in the case of finite interval, and [26] for infinite interval.
Proposition C Let p and q be measurable functions on () satisfying the condition , . Let . Suppose also that if , then , outside some large interval . Then there is a positive constant c depending only on p and q such that for all , and all sequences of intervals , where are disjoint intervals satisfying the condition , the inequality
holds. Moreover, the value of is defined as follows: if and if .
Let v and w be a.e. positive measurable function on , , and let
Further, we denote
Let us recall the two-weight criterion for the Hardy operator in classical Lebesgue spaces:
Let r and s be constants such that . Suppose that . Let v and w be non-negative measurable functions on . Then the Hardy inequality
holds if and only if
Moreover, if c is the best constant in the Hardy inequality, then there are positive constants and depending only on r and s such that .
For the Hardy inequalities, we also refer the books [29, 30].
The following statement was proved in [2] for finite interval and in [12] for the case of infinite interval, but we give the proof because of the upper and lower bound of the norm of .
Theorem B Let and let p and q be measurable functions on satisfying the conditions: , . We assume that , outside some large interval if . Then is bounded from to if and only if
Moreover, there are positive constants and independent of the interval I such that
where the constant is defined in Proposition C.
Proof Sufficiency. Let . Suppose that and that for some integer . We construct a sequence so that
It is easy to check that . Let g be a function satisfying the condition . Applying Hölder’s inequality for variable exponent Lebesgue spaces and Proposition C we have that
where is the constant defined in Proposition C. Taking now the supremum with respect to g, we have sufficiency for .
Let now . Then
By applying already used arguments, we have that , where . Further, due to Hölder’s inequality and Theorem A, we find that
To get the lower bound for is trivial by choosing the appropriate test function , in the boundedness of from to . □
Corollary A Let p and q be defined on and satisfy the conditions of Theorem B. Then for all ,
where , is defined in Theorem B and the constant c depends only on p and q.
Proof By the hypothesis, p and q are constant outside some large interval . Let for some integer . Then by Theorem B for , we have
where the positive constant c depends only on p and q. If , then p and q are constants on the intervals . In this case taking the proof of Theorem B into account, we find that
□
Theorem C ([5])
Let and be measurable functions on an interval . Suppose that and . If
where k is a non-negative kernel, then the operator
is compact from to .
Lemma B (see, e.g. [31])
Let and . Suppose that and are sequences of positive real numbers. The following statements are equivalent:
-
(i)
There exists such that the inequality
holds for all sequences of real numbers.
-
(ii)
.
Lemma C (see e.g. [32])
Let p, q be constants such that . Suppose that , , . Then there exists a constant such that
holds for all non-negative sequence , if and only if
-
(i)
in case ,
-
(ii)
in case ,
where .
Definition 2.2 Let , . We say that a kernel belongs to () if there exists a constant such that for all x, y, t with the inequality
holds.
Definition 2.3 Let r be a measurable function on , with values in . We say a kernel k belongs to if there exists a positive constant such that for a.e. , the inequality
is fulfilled.
Example 2.1 (Lemma 3 of [26])
Let , where . Let α be a measurable function on I satisfying the condition . Suppose that r is a function on I with values in satisfying the condition . Suppose that outside some interval when . Then when .
The next examples of kernels can be checked easily:
Example 2.2 Let , where . Suppose that α is a measurable function on I satisfying the condition . Let r be a function on I with the values in satisfying the condition where . Suppose that outside some interval when . Then when and .
Example 2.3 Let , . Let r be a function on I with the values in satisfying the condition and let r be increasing on I. Suppose that outside some interval when . Further, let and . Then .
For other examples of kernel k satisfying the condition , where r is constant, we refer to [33] (see also [34], p.163).
3 Boundedness on VEAS
This section is devoted to the boundedness of weighted kernel operators in VEAS.
3.1 Amalgam spaces
Let I be ℝ or and be a cover of I consisting of disjoint half-open intervals , each of the form , whose union is I. Let
we define the general amalgams with variable exponent
If , then is denoted by .
Let and . Then we have the usual irregular amalgam (see [35]); if and , then is the amalgam space introduced by Wiener (see [36, 37]) in connection with the development of the theory of generalized harmonic analysis.
We call irregular weighted amalgam spaces with variable exponent. If , then will be denoted by .
Let and . We denote weighted dyadic amalgam with variable exponent by . Some properties regarding general amalgams with variable exponent can be derived in the same way as for usual irregular amalgams , where p is constant. Irregular amalgams were introduced in [38] and studied in [35].
Theorem D Let p be a measurable function on I with and q be constant with . The irregular amalgams with variable exponent is a Banach space whose dual space is . Further, Hölder’s inequality holds in the following form:
Proof Since is a Banach space and (see [22]), from general arguments (see [35, 39–41]) we have the desired result. □
The next statement for more general case, i.e., when amalgams are defined with respect to Banach spaces, can be found in [35].
Theorem E Let p be measurable function on I and , then
Other structural properties of amalgams are investigated, e.g., in [41] and [35].
The next statement is a generalization of Theorem 4 in [35] for variable exponent amalgams with weights.
Proposition D Let p, q be measurable functions on I such that and . Then the space is continuously embedded in if
Conversely, if , then condition (3.1) is also necessary for the continuous embedding of into .
Proof It is known (see [42]) that the continuous embedding () holds if and only if
Moreover, the estimate
holds, where the positive constant c depends only on p and q; Id is the identity operator.
Let condition (3.1) hold. Then
Hence, .
Conversely, let the continuous embedding hold and let . By taking functions supported in we can derive the estimate
By applying the left-hand side inequality of (3.1′) and Proposition A, we conclude that condition (3.1) is satisfied. □
3.2 General operators in VEAS
We begin this subsection by the following definition.
Definition 3.1 ([31])
Let T be an operator defined on a set of real measurable functions f on ℝ. Define a sequence of local operators
Let us assume that there is a discrete operator satisfying the following conditions:
-
(i)
There exists a positive constant c such that for all non-negative functions f, and arbitrary the inequality
holds.
-
(ii)
There is such that for all sequences of non-negative real numbers and , the inequality
holds for all and all non-negative f, where , . It is also assumed that T satisfies the conditions
We will say that an operator T satisfying all the above mentioned conditions is admissible on ℝ.
For example, Hardy operators, Hardy-Littlewood maximal operators, fractional integral operators, fractional maximal operators are admissible on ℝ (see [31]). Carton-Leburn, Heinig and Hoffmann [32] established two weighted criteria for the Hardy transform in amalgam spaces defined on ℝ (see also [43, 44] for related topics). In [32], the authors derived some sufficient conditions for the two-weight boundedness of the kernel operator where k is non-decreasing in the second variable and non-increasing in the first one. In the paper [45], the two-weight problem for generalized Hardy-type kernel operators including the fractional integrals of order greater than one (without singularity) was solved.
General type results for the admissible operators read as follows.
Theorem F ([31])
Let , and let w and v be weight functions on ℝ. Suppose that T is an admissible operator on ℝ. Then the inequality
holds for all measurable f if and only if
-
(i)
is bounded from to , where , .(ii)
-
(a)
for .
-
(b)
, where for .
Our aim is to establish weighted characterization of the boundedness of kernel operators involving fractional integrals of variable parameter of order less than one in variable exponent amalgam spaces. For the continuous part of amalgam spaces, we take variable exponent Lebesgue spaces defined on I.
It should be emphasized that the following fact holds: by the change of variable it is possible to get appropriate boundedness or compactness results from dyadic amalgams to amalgams defined on ℝ.
Analyzing the proof of Theorem 1 of [31], we can formulate the next statement and give the proof for completeness.
Proposition 3.1 Let , be measurable functions on ℝ satisfying , . Suppose that q and are constants satisfying . Assume that w and v are weight functions on ℝ and that T is an admissible operator on ℝ. Then the inequality
holds if
-
(i)
is bounded from to where , . (ii)
-
(a)
for .
-
(b)
with for .
Conversely, let (3.2) hold. Then
-
(1)
conditions (ii) are satisfied;
-
(2)
condition (i) is satisfied for or for p and being constants outside some large interval , .
Proof
Let (i) and (ii) hold. We have
Let . By the hypothesis and Hölder’s inequality for variable exponents and , we have that
Let us estimate . Suppose that . Since the operators are uniformly bounded, we find that
If , then by using Hölder’s inequality we derive
Conversely, suppose that (3.2) holds. Let and let f be a non-negative function supported in . Then
On the other hand,
Now due to inequality (3.2), we conclude that (a) of (ii) holds. Let us now show that if , then (b) of (ii) is satisfied.
Since , we have that for each n, there exists a non-negative measurable function , with the support in and with , such that . Thus, it is sufficient to prove that .
Let be a sequence of non-negative real numbers and . For each , and then for all .
Consequently,
Hence, inequality (3.2) yields that
Finally, by Lemma B, we see that (b) of (ii) holds.
Now let us prove that (i) holds when . If is a sequence of non-negative real numbers and
then , and and by the properties of T, we have
Applying the two-weight inequality, we find that
Hence, (i) holds.
Suppose now that w is a general weight and there is a positive integer such that p, are constants outside . Taking
it is easy to see that . Moreover, by virtue of Proposition A and the fact that
we have for ,
where the positive constant c depends on . Since
using again Proposition A, we find that
□
Definition 3.2 Let T be an operator defined on a set of real measurable functions f on . We say that an operator T is admissible on if the conditions of Definition 3.1 are satisfied replacing n by , .
The next statement can be obtained in the similar manner as Proposition 3.1 was proved; therefore, we omit the proof.
Proposition 3.2 Let , be measurable functions on satisfying , . Suppose that q and are constants satisfying . Suppose also that w and v are weight functions on and that T is an admissible operator on .
Then the inequality
holds if
-
(i)
is bounded from to where , . (ii)
-
(a)
for .
-
(b)
with for .
Conversely, if (3.3) holds, then
-
(1)
conditions (ii) are satisfied;
-
(2)
condition (i) is also satisfied but for or for p and satisfying the condition , outside some large interval , .
Proposition 3.2 gives criteria for the boundedness of in dyadic amalgams on but by the next statement we prove the two-weight inequality under slightly different conditions.
Proposition 3.3 Let and let . Let . Suppose that and that outside some large interval . Then the inequality
with a positive constant independent of f holds if
-
(i)
in the case ,
-
(a)
-
(b)
-
(ii)
in the case ,
-
(a)
, where
-
(b)
where .
Proof Let . Suppose that . We represent:
We have
Let . Then by the discrete Hardy inequality (see Lemma C) and Hölder’s inequality with respect to the exponents and we derive
Further, by Corollary A and Theorem E, we have that
Let . Using representation (3.4), we derive
We estimate and .
By the two-weight inequality for the discrete Hardy transform (see Lemma C), we have
Now we estimate . Using Corollary A for intervals and Hölder’s inequality, we find that
□
3.3 Kernel operators on amalgams and
The conditions of general-type statements (see Propositions 3.1 and 3.2) are not easily verifiable for general kernel operators as well as for some concrete fractional integral operators such as the Riemann-Liouville fractional integral transform with variable parameter. That is why we investigate mapping properties of general kernel operators independently from general-type statements.
Let
One of our aims is to characterize a class of weights v governing the boundedness of from to .
We will use the notation:
Theorem 3.1 Let , and let . Suppose that and q are constants such that . Let and outside some large interval , . Let . Then is bounded from to if and only if , where .
Proof Sufficiency. Using the representation:
we have that
Further, taking Proposition 3.3 and the condition into account, we find that
Now observe that by the condition , Proposition A and Lemma A we obtain
where
Let us now observe that by Proposition A and Lemma A, , where
Necessity. Let be the sequence defined above. Considering the test function in the boundedness of from to and taking the condition into account we have that
It is easy to see that
-
(i)
(3.9)
for ;
-
(ii)
(3.10)
for .
Denoting and taking (3.9), Proposition A and Lemma A into account we have for ,
Similarly if , then by (3.10),
Hence, .
Let now f be a function supported in . Then due to the boundedness of from to and the condition we have that
where the positive constant c does not depend on n. Using Theorem B with respect to the intervals and the weight pair , where and , it follows that . □
Remark 3.1 We have noticed in the proof of Theorem 3.1 that , where is defined in the same proof.
Now we formulate the boundedness criteria for the kernel operator
on amalgams defined on ℝ.
Let be a kernel on and v, p, be defined on ℝ. For the next statement we define , , and as follows:
Theorem 3.2 Let and let . Let and q are constants such that . Assume that and outside some large interval . Let . Then is bounded from to if and only if
Proof The proof follows from Theorem 3.1 by the change of variable . □
Let
where and .
By virtue of Theorem 3.2 and Example 2.1 we can easily deduce the next statement.
Corollary 3.1 Let p, , q and be constants. Suppose that α is a measurable function on ℝ and that , , . Then the operator is bounded from to if and only if
Moreover, there are positive constants and depending on p, , q, and α such that .
4 Compactness of kernel operators on VEAS
In this section, we derive compactness necessary and sufficient conditions for kernel operators on VEAS. Since for the amalgam norm we have the property when a.e. (), therefore, the following statement holds (see [46], Chap. XI).
Proposition 4.1 Let p, be measurable functions on I such that . Let q, be constants satisfying the condition . Then the set of all functions of the form
is dense in the mixed norm space , where , ( are measurable disjoint sets of I) and .
The next statement gives sufficient condition for the kernel operator to be compact on amalgams defined on .
Proposition 4.2 Let and be measurable functions on an interval . Suppose that , . Let q, be constants such that . If
where k is a non-negative kernel, then the operator
is compact from to .
Proof
By Proposition 4.1 the set of functions
is dense in . By Hölder’s inequality for amalgam spaces (see Theorem D), we have
Hence,
This means that .
Now we prove the compactness of K. For each , let
Note that
where
This means that is a finite rank operator, i.e., it is compact. Further, let . Using the above-mentioned arguments, we have that there is such that for ,
Thus K can be represented as a limit of finite rank operators. Hence, K is compact. □
Theorem 4.1 Let and let . Let and q be constants such that . Assume that . Suppose that and outside some large interval . Then is compact from to if and only if
where, and are defined by (3.5) and (3.6), respectively, and
Proof Sufficiency. Let , be integers such that . Then we represent as follows:
It is clear that
where if and if . Then
Denoting , , , we represent as
Now we estimate , and separately
Further,
Considering separately the cases and , by using Proposition A and Lemma A we find that
Consequently, since , we have
Since we find that
So, by Proposition 4.2, we conclude that is a compact operator. Further, write as follows:
where if and if . Then we have
Denoting and considering both cases when and separately, we derive as previously that
and since we have
Hence, by Proposition 4.2, is compact.
Let us denote
Following the proofs of Theorems 3.1, 3.2 and applying Proposition A and Lemma A, we have that
as because . Further, applying Theorem 3.1, we find that
as .
Hence,
as , , . Hence is compact, since it is the limit of compact operators.
Necessity. First we show that . Let , where is defined in the proof of Theorem 3.1. Then weakly in as . Indeed, let . Then
as .
Observe now that
Hence, because is compact and if .
Further, (4.2) implies that
as .
To show that we represent as follows:
where and is defined by (4.1). Observe now that
as because as . The latter convergence follows from the convergence of the series.
Further,
as because as (see (4.2)). Hence, .
Further, it is easy to see that for and ,
Hence,
as or .
The conditions and follow from the fact that every compact operator is bounded. □
Now we formulate the compactness criteria for the kernel operator defined on ℝ.
Theorem 4.2 Let and let . Let and q be constants such that . Assume that and outside some large interval . Let . Then is compact from to if and only if
where
, and and are defined in Section 3.
Proof The proof follows from Theorem 4.1 by the change of variable . □
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Acknowledgements
The first and second authors were supported by the Shota Rustaveli National Science Foundation grant (Contract No. D/13-23). The part of this work is carried out at Abdus Salam School of Mathematical Sciences, GC University, Lahore. The second and third authors are thankful to the Higher Education Commission, Pakistan for the financial support. The authors are grateful to the anonymous referees for their remarks and suggestions.
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Kokilashvili, V., Meskhi, A. & Zaighum, M.A. Weighted kernel operators in variable exponent amalgam spaces. J Inequal Appl 2013, 173 (2013). https://doi.org/10.1186/1029-242X-2013-173
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DOI: https://doi.org/10.1186/1029-242X-2013-173