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A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions
Journal of Inequalities and Applications volume 2014, Article number: 383 (2014)
Abstract
We establish necessary and sufficient conditions on a weight pair governing the boundedness of the Riesz potential operator defined on a homogeneous group G from to , where is the Lebesgue space defined for non-negative radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels defined on a product of two homogeneous groups . In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces. To get the main results we use Sawyer-type duality theorems (which are also discussed in this paper) and two-weight Hardy-type inequalities on G and , respectively.
MSC:42B20, 42B25.
1 Introduction
A homogeneous group is a simply connected nilpotent Lie group G on a Lie algebra g with the one-parameter group of transformations , , where A is a diagonalized linear operator in G with positive eigenvalues. In the homogeneous group G the mappings , , are automorphisms in G, which will be again denoted by . The number is the homogeneous dimension of G. The symbol e will stand for the neutral element in G.
It is possible to equip G with a homogeneous norm which is continuous on G, smooth on , and satisfies the conditions:
-
(i)
for every ;
-
(ii)
for every and ;
-
(iii)
if and only if ;
-
(iv)
there exists such that
In the sequel we denote by an open ball with the center a and radius , i.e.
It can be observed that .
Let us fix a Haar measure in G such that . Then . In particular, for , .
Examples of homogeneous groups are: the Euclidean n-dimensional space , the Heisenberg group, upper triangular groups, etc. For the definition and basic properties of the homogeneous group we refer to [[1], p.12].
An everywhere positive function ρ on G will be called a weight. Denote by () the weighted Lebesgue space, which is the space of all measurable functions defined by the norm
If , then we use the notation .
Denote by the class of all radially decreasing functions on G with values in , i.e. the fact that means that there is a decreasing such that . In the sequel we will use the symbol ϕ itself for ; will be written also by the symbol . Let and be homogeneous groups. We say that a function is radially decreasing if it is such in each variable separately uniformly to another one. The fact that ψ is radially decreasing on will be denoted as .
Let
be the Riesz potential defined on G, where r is the homogeneous norm and dy is the normalized Haar measure on G. The operator plays a fundamental role in harmonic analysis, e.g., in the theory of Sobolev embeddings, in the theory of sublaplacians on nilpotent groups etc. Weighted estimates for multiple Riesz potentials can be applied, for example, to establish Sobolev and Poincaré inequalities on product spaces (see, e.g., [2]).
Let and be homogeneous groups with homogeneous norms and and homogeneous dimensions and , respectively. We define the potential operator on as follows:
Our aim is to derive two-weight criteria for on the cone of radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels defined on a product of two homogeneous groups, where only the right-hand side weight is of product type. As far as we know the derived results for are new, even in the case of Euclidean spaces. The proofs of the main results are based on Sawyer (see [3]) type duality theorem which is also true for homogeneous groups (see Propositions C and E below) and Hardy-type two-weight inequalities in homogeneous groups. Analogous results for multiple potential operators defined on with respect to the cone of non-negative decreasing functions on were studied in [4, 5]. It should be emphasized that the two-weight problem for a multiple Hardy operator for the cone of decreasing functions on was investigated by Barza, Heinig and Persson [6] under the restriction that both weights are of product type.
Historically, the one-weight inequality for the classical Hardy operator on decreasing functions was characterize by Arino and Muckenhoupt [7] under the so called condition. The same problem for multiple Hardy transform was studied by Arcozzi, Barza, Garcia-Domingo and Soria [8]. This problem in the two-weight setting was solved by Sawyer [3]. Some sufficient conditions guaranteeing the two-weight inequality for the Riesz potential on were given by Rakotondratsimba [9]. In particular, the author showed that is bounded from to if the weighted Hardy operators and are bounded from to . In fact, the author studied the problem on the cone of monotone decreasing functions.
Now we give some comments regarding the notation: in the sequel under the symbol we mean that there are positive constants and (depending on appropriate parameters) such that ; means that there is a positive constant c such that ; integral over a product set from g will be denoted by or ; for a weight functions w and on G, by the symbols and will be denoted the integrals and respectively; for a weight w on , we denote , where and are neutral elements in and , respectively. Finally, we mention that constants (often different constants in one and the same line of inequalities) will be denoted by c or C. The symbol stands for the conjugate number of p: , where .
2 Preliminaries
We begin this section with the statements regarding polar coordinates in G (see e.g., [[1], p.14]).
Proposition A Let G be a homogeneous group and let . There is a (unique) Radon measure σ on S such that for all ,
Let a be a positive number. The two-weight inequality for the Hardy-type transforms
reads as follows (see [10], Chapter 1 for more general case, in particular, for quasi-metric measure spaces):
Theorem A Let and let a be a positive number. Then
-
(i)
The operator is bounded from to if and only if
-
(ii)
The operator is bounded from to if and only if
We refer also to [11] for the Hardy inequality written for balls with center at the origin.
In the sequel we denote by H.
The following statement for Euclidean spaces was derived by Barza, Johansson and Persson [12].
Proposition B Let w be a weight function on G and let . If and , then
where , .
The proof of Proposition B repeats the arguments (for ) used in the proof of Theorem 3.1 of [12] taking Proposition A and the following lemma into account.
Lemma A Let . For a weight function w, the inequality
holds.
Proof The proof of this lemma is based on Theorem A (part (ii)) taking , , , there. Details are omitted. □
Corollary A Let the conditions of Proposition B be satisfied and let . Then the following relation holds:
Corollary A implies the following duality result, which follows in the standard way (see [3, 12] for details).
Proposition C Let and let v, w be weight functions on G with . Then the integral operator T defined on functions on G is bounded from to if and only if
holds for every positive measurable g on G.
The next statement yields the criteria for the two-weight boundedness of the operator H on the cone . In particular the following statement is true.
Theorem B Let and let v and w be weights on G such that . Then H is bounded from to if and only if
-
(i)
-
(ii)
Proof The proof of this statement follows by the standard way applying Proposition C (see e.g. [3, 12]). □
Definition 2.1 Let ρ be a locally integrable a.e. positive function on G. We say that ρ satisfies the doubling condition at e () if there is a positive constant such that for all the following inequality holds:
Further, we say that , where , , if there is a positive constant b such that for all
Remark 2.1 It can be checked that if a weight w satisfies the doubling condition et e in the strong sense, i.e., and with a constant c independent of t, then .
Definition 2.2 We say that a locally integrable a.e. positive function ρ on satisfies the doubling condition with respect to the second variable () uniformly to the first one if there is a positive constant c such that for all and almost every the following inequality holds:
Analogously is defined the class of weights .
3 Riesz potentials on G
The main result of this section reads as follows.
Theorem 3.1 Let and let v and w be weights such that either or ; let . Then the operator is bounded from to if and only if
-
(i)
(3.1)
-
(ii)
(3.2)
-
(iii)
(3.3)
To prove this result we need to prove some auxiliary statements.
Lemma 3.1 Let and let be the constant from the triangle inequality of r. Then there is a positive constant c depending only on Q, α, and such that for all ,
Proof We have
Observe that, by the triangle inequality for r, we have . This implies that . Hence,
Further, it is easy to see that
Finally we have (3.4). □
Let us introduce the following potential operators:
It is easy to see that
We need also to introduce the following weighted Hardy operator:
Proposition 3.1 The following relation holds for all :
Proof We have
If , then . Hence . Consequently,
Applying now the fact that we see that
□
Lemma 3.2 Let and let v and w be weights on G such that . Then the operator is bounded from to if
Conversely, if is bounded from to , then the condition
is satisfied. Furthermore, if either or , then the operator is bounded from to if and only if
Proof Applying Proposition C, is bounded from to if and only if
where
Now we show that
To prove the right-hand side estimate in (3.7) observe that by Tonelli’s theorem and Lemma 3.1 we have
On the other hand,
Thus, Theorem A completes the proof. □
Proof of Theorem 3.1 By (3.5) it is enough to estimate the terms with and . By applying Proposition 3.1 and Theorem B we find that is bounded from to if and only if the conditions (ii) and (iii) are satisfied. Now by Lemma 3.2 and the equality (which is a consequence of Proposition A)
we see that is bounded from to if and only if (i) is satisfied. □
4 Multiple potentials on
Let us now investigate the two-weight problem for the operator on the cone . In the sequel without loss of generality we denote the triangle inequality constants for and by one and the same symbol .
The following statement can be derived just in the same way as Theorem 3.1 was obtained in [6]. The proof is omitted to avoid repeating those arguments.
Proposition D Let and let be a product weight on . Then the relation
holds for a non-negative measurable function g, where
Applying Proposition D together with the duality arguments we can get the following statement (cf. [6]).
Proposition E Let and let v and w be weights on such that , . Then an integral operator T defined for functions from is bounded from to if and only if for all non-negative measurable g on ,
The next statements deal with the double Hardy-type operators defined on :
Proposition 4.1 Let . Suppose that v and w be weights on such that either or . Then:
-
(i)
The operator is bounded from to if and only if
-
(ii)
The operator is bounded from to if and only if
-
(iii)
The operator is bounded from to if and only if
-
(iv)
The operator is bounded from to if and only if
Proof Let . Then the proposition follows in the same way as the appropriate statements regarding the Hardy operators defined on in [13, 14] (see also Theorem 1.1.6 of [15]). If v is a product weight, i.e. , then the result follows from the duality arguments. We give the proof, for example, for in the case when .
First suppose that . Let be a sequence of positive numbers for which the equality
holds for all . This equality follows because of the continuity in t of the integral over the ball . It is clear that is increasing and . Moreover, it is easy to verify that
Let . We have
where
It is obvious that
Hence, by Theorem A
On the other hand, (4.1) yields
for all . Hence by the discrete Hardy inequality (see e.g. [16]) and Hölder’s inequality we have
If , then without loss of generality we can assume that . In this case we choose the sequence for which (4.1) holds for all . Arguing as in the case and using slight modification of the discrete Hardy inequality (see also [15], Chapter 1 for similar arguments), we finally obtain the desired result.
Finally we notice that the part (i) can also be proved if we first establish the boundedness of the operator in the spirit of Theorem 1.1.6 in [15] and then pass to the case of by Proposition A. □
The next statement will be useful for us.
Proposition 4.2 Let . Assume that v and w are weights on . Suppose that and that , . Then the operator is bounded from to if and only if the following four conditions are satisfied:
-
(i)
-
(ii)
-
(iii)
-
(iv)
Proof We follow the proof of Theorem 5.3 in [6]. First of all observe that by Proposition E, if w is a product weight, i.e., , such that , , and v is any weight on , then is bounded from to if and only if
Further, we have
It is obvious that (4.2) holds if and only if
for . By using Proposition 4.1 (part (i)) we find that
if and only if
In the latter equality we used the equality
which is a direct consequence of integration by parts and Proposition A. Taking now Proposition 4.1 (part (ii)) into account we find that (4.3) holds for if and only if condition (ii) is satisfied, while Proposition 4.1 (parts (iii) and (iv)) and the following observation:
yield (4.3) for . □
Let
where is the constant from the triangle inequality for the homogeneous norms and .
It is obvious that
Now we formulate the main result of this section.
Theorem 4.1 Let . Assume that v and w are weights on such that . Suppose that either , , or . Then the operator is bounded from to if and only if the following conditions are satisfied:
-
(i)
-
(ii)
-
(iii)
-
(iv)
-
(v)
-
(vi)
-
(vii)
-
(viii)
-
(ix)
Proof Let us assume that . The case when , , follows analogously. By using representation (4.4) we have to investigate the boundedness of the operators , , , separately.
Since by using the arguments of the proof of Proposition 3.1 it can be checked that
(see also [4] for a similar estimate in the case of the multiple one-sided potentials on ). Hence, by Proposition 4.2 we find that is bounded from to if and only if conditions (i)- (iv) hold.
Observe that the dual to is given by
Further, Tonelli’s theorem together with Lemma 3.1 for both variables implies that there are positive constants and such that for all for the dual (see also the proof of Lemma 3.2),
Applying Propositions 4.1 and 4.2 with the condition that we find that the operator is bounded from to if and only if condition (v) is satisfied.
Further, observe that due to the fact that f is radially decreasing with respect to the first variable we have
where
Dual of is given by
Further, we have
Tonelli’s theorem for , the inequality for , , and the fact that the integral is decreasing in τ uniformly to ε yield
Here we used the notation
Taking into account that the function is decreasing in λ uniformly to s, the inequality for , , and Tonelli’s theorem for we find that
To get the upper estimate, observe that Tonelli’s theorem for and Lemma 3.1 for yield
Similarly,
Summarizing these estimates we see that there are positive constants and depending only on , , , and such that
Taking Propositions 4.1 and E into account together with the doubling condition for v with respect to the second variable we see that the operator is bounded from to if and only if the conditions (vi) and (vii) are satisfied.
In a similar manner (changing the roles of the first and second variables) we can see that is bounded from to if and only if the conditions (viii) and (ix) are satisfied. □
Theorem 4.1 and Remark 2.1 imply criteria for the trace inequality for . Namely the following statement holds.
Theorem 4.2 Let and let , . Then is bounded from to if and only if the following condition holds:
Proof Sufficiency is a consequence of the inequality , while necessity follows immediately by taking the test function , . □
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Acknowledgements
The first author is grateful to Professor V Kokilashvili for drawing his attention to the two-weight problem for multiple Riesz potentials. The first author was partially supported by the Shota Rustaveli National Science Foundation Grant (Contract Numbers: D/13-23 and 31/47). The authors are grateful to the referees for their remarks.
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GM and MS established Propositions B, C, D, E and checked the proofs of the statements throughout the paper. All authors read and approved the final manuscript.
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Meskhi, A., Murtaza, G. & Sarwar, M. A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions. J Inequal Appl 2014, 383 (2014). https://doi.org/10.1186/1029-242X-2014-383
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DOI: https://doi.org/10.1186/1029-242X-2014-383