Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 474681, 26 pagesdoi:10.1155/2012/474681

Research Article

Potential Operators on Cones of Nonincreasing Functions

Alexander Meskhi1,2 and Ghulam Murtaza³

¹A. Razmadze Mathematical Institute, Ivane Javakhishvili Tbilisi State University, 2 University Street, 0143 Tbilisi, Georgia²Faculty of Informatics and Control Systems, Georgian Technical University, 77 Kostava Street, Tbilisi, Georgia³Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town, Lahore, Pakistan

Received 16 May 2011; Accepted 15 June 2011

Academic Editor: V. M. Kokilashvili

Copyright © 2012 Alexander Meskhi and Ghulam Murtaza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators and on the cone of nonincreasing functions are derived. In the case of , we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.

1. Introduction

Our aim is to derive necessary and sufficient conditions on weight pairs governing the boundedness of the following potential operators: from to , where .

Historically, necessary and sufficient condition on a weight function , for which the boundedness of the one-dimensional Hardy transform from to holds, was established in [1]. Two-weight Hardy inequality criteria on cones of nonincreasing functions were derived in the paper [2]. The multidimensional analogues of these results were studied in [3–5]. Some characterizations of the two-weight inequality for the single integral operators involving Hardy-type transforms for monotone functions were given in [6–8]. The same problem for the Riesz potentials for nonnegative nonincreasing radial functions was studied in [9].

In the paper [10] necessary and sufficient conditions governing the boundedness of the multiple Riemann-Liouville transform from to were derived, provided that is a product of one-dimensional weights. Earlier, the problem of the boundedness of the two-dimensional Hardy transform from to was studied in [4] under the condition that and have the following form: .

It should be emphasized that the two-weight problem for the Hardy-type transforms and fractional integrals with single kernels has been already solved. For the weight theory and history of these operators in classical Lebesgue spaces, we refer to the monographs [11–15] and references cited therein.

The monograph [13] is dedicated to the two-weight problem for multiple integral operators in classical Lebesgue spaces (see also the papers [16–18] for criteria guaranteeing trace inequalities for potential operators with product kernels).

Unfortunately, in the case of double potential operator, we assume that the right-hand weight is of product type and the left-hand one satisfies the doubling condition with respect to one of the variables. Even under these restrictions the two-weight criteria are written in terms of several conditions on weights. We hope to remove these restrictions on weights in our future investigations.

Some of the results of this paper were announced without proofs in [19].

Finally we mention that constants (often different constants in the same series of inequalities) will generally be denoted by or ; by the symbol , where and are linear positive operators defined on appropriate classes of functions, we mean that there are positive constants and independent of and such that ; denotes the interval and means the number for ; ; ; .

2. Preliminaries

We say that a function is nonincreasing if is nonincreasing in each variable separately.

Let be the class of all nonnegative nonincreasing functions on . Suppose that is measurable a.e. positive function (weight) on . We denote by , , the class of all nonnegative functions on for which By the symbol we mean the class .

The next statement regarding two-weight criteria for the Hardy operator on the cone of nonincreasing functions was proved in [2].

Theorem A. Let and be weight functions on , and let . (i)Suppose that . Then the inequality holds if and only if the following two conditions are satisfied: (ii)Let . Then is bounded from to if and only if the following two conditions are satisfied: where .

The following statement was proved in [2] for . For we refer to [4].

Proposition A. Let . Suppose that is a positive integral operator defined on functions , which are nonincreasing in each variable separately. Suppose that is its formal adjoint. Let be a product weight such that , . Let be a general weight on . Then the operator is bounded from to if and only if the inequality holds for all .

Let be the Riemann-Liouville transform with single kernel

If , then is the Hardy transform. The boundedness for was characterized by Muckenhoupt ([20]) for , and by Kokilashvili [21] and Bradley [22] for (see also the monograph by Maz'ya [23] for these and relevant results).

In the case when , the Riemann-Liouville transform has singularity. For the results regarding the two-weight problem, in this case we refer, for example, to the monograph [11] and the references cited therein.

The next result deals with the case (see [24]).

Theorem B. Let . Then the operator is bounded from to if and only if for and for , where is defined as follows: .

Theorem C (see [10]). Let , and let , . Assume that and are weights on . Suppose also that for some one-dimensional weights and and that , . Then the following conditions are equivalent: (a) is bounded from to ;(b)the following four conditions hold simultaneously:(i)(ii)(iii)(iv)

In particular, Theorem C yields the trace inequality criteria on the cone of nonincreasing functions.

Corollary A (see [10]). Let , and let , . Then the following conditions are equivalent: (a)the boundedness of from to holds for ;(b)(c)(d)(e)

3. Potentials on

In this section we discuss the two-weight problem for the operator . We begin with the following lemma.

Lemma 3.1. The following relation holds for nonnegative and nonincreasing function : where is the Hardy operator defined above.

Proof. We follow the proof of Proposition  3.1 of [10]. We have Observe that if , then . Hence, Further, since is nonincreasing, we have that Finally we have the upper estimate for .The lower estimate is obvious because for .

In the next statement we assume that is the operator given by

Lemma 3.2. Let , and let . Suppose that . Then the operator is bounded from to if and only if

Proof. Taking Proposition A into account (for ), an integral operator is bounded from to if and only if where is a formal adjoint to .We have Taking and , we derive the desired result.

Now we formulate the main results of this section.

Theorem 3.3. Let , and let . Suppose that . Then is bounded from to if and only if

Theorem 3.4. Let , and let . . Then is bounded from to if and only if where .

Proof of Theorems 3.3 and 3.4. By using the representation the obvious equality Theorems A and B and Lemmas 3.1 and 3.2, we have the desired results.

Corollary 3.5. Let , and let . Then the operator is bounded from to if and only if

Proof. Necessity follows immediately taking the test function in the two-weight inequality and observing that for .Sufficiency. By Theorem 3.3, it is enough to show that where , (see Theorem 3.3 for the definition of ).The estimates , , are obvious. We show that for . We have Further, by the condition , we have that

Definition 3.6. Let be a locally integrable a.e. positive function on . We say that satisfies the doubling condition if there is a positive constant such that for all the following inequality holds:

Remark 3.7. It is easy to check that if , then satisfies the reverse doubling condition: there is a positive constant such that Indeed by (3.22) we have Then Analogously, Finally, we have (3.23).

Corollary 3.8. Let , and let . Suppose that . Suppose also that . Then is bounded from to if and only if condition (3.11) is satisfied.

Proof. Observe that by Remark 3.7, for , the inequality holds for all , where is defined in (3.23).Let . Then there is such that . By applying (3.27) and the doubling condition for , we find that So, we have seen that (3.11)⇒(3.10). Let us check now that (3.13)⇒(3.12).Indeed, for , we choose so that . Then, by using the condition and Remark 3.7, Hence, (3.13)⇒(3.12) follows. Implication (3.11)⇒(3.13) follows in the same way as in the case of implication (3.11)⇒(3.10). The details are omitted.

4. Potentials with Multiple Kernels

In this section we discuss two-weight criteria for the potentials with product kernels .

To derive the main results, we introduce the following multiple potential operators: where , , and , .

Definition 4.1. One says that a locally integrable a.e. positive function on satisfies the doubling condition with respect to the second variable () if there is a positive constant such that for all and almost every the following inequality holds:

Analogously is defined the class of weights .

Remark 4.2. If , then satisfies the reverse doubling condition with respect to the second variable; that is, there is a positive constant such that

Analogously, . This follows in the same way as the single variable case (see Remark 3.7).

Theorem C implies the next statement.

Corollary B. Let the conditions of Theorem C be satisfied. (i)If , then for the boundedness of from to , it is necessary and sufficient that conditions (2.10) and (2.12) are satisfied.(ii)If , then is bounded from to if and only If conditions (2.10) and (2.11) are satisfied.(iii)If , then is bounded from to if and only if the condition (2.10) is satisfied.

Proof of Corollary B. The proof of this statement follows by using the arguments of the proof of Corollary 3.8 (see Section 2) but with respect to each variable separately (also see Remark 4.2). The details are omitted.

The following result concerns with the two-weight criteria for the two-dimensional operator with (see [25], [13, Section  1.6]).

Theorem D. Let , and let . (i)Suppose that . Then the operator is bounded from to if and only if Moreover, .(ii)Let . Then the operator is bounded from to if and only if Moreover, .

Let us introduce the following multiple integral operators:

Now we prove some auxiliary statements.

Proposition 4.3. Let , and let . Suppose that either or for some one-dimensional weights , , , and . (i)The operator is bounded from to if and only if Moreover, .(ii)The operator is bounded from to if and only if Moreover, .(iii)The operator is bounded from to if and only if Moreover, .(iv)The operator is bounded from to if and only if Moreover, .

Proof. Let . The proof of the case is followed by duality arguments. We prove, for example, part (i). Proofs of other parts are similar and, therefore, are omitted. We follow the proof of Theorem  3.4 of [25] (see also the proof of Theorem  1.1.6 in [13]).Sufficiency. First suppose that . Let be a sequence of positive numbers for which the equality holds for all . It is clear that is increasing and . Moreover, it is easy to verify that Let . We have that where It is obvious that Hence, by using the two-weight criteria for the one-dimensional Riemann-Liouville operator without singularity (see [24]), we find that where .On the other hand, (4.11) yields for all . Hence by Hardy’s inequality in discrete case (see, for example, [25, 26]) and Hölder’s inequality we have that If , then without loss of generality we can assume that . In this case we choose the sequence for which (4.11) holds for all . Arguing as in the case of , we finally obtain the desired result.Necessity follows by choosing the appropriate test functions. The details are omitted.To prove, for example, (iii), we choose the sequence so that (notice that is decreasing) and argue as in the proof of (i).

Proposition 4.4. Let , and let . Suppose that either or for some one-dimensional weights: , , , and . (i)The operator is bounded from to if and only if Moreover, .(ii)The operator is bounded from to if and only if Moreover, .(iii)The operator is bounded from to if and only if Moreover, .(iv)The operator is bounded from to if and only if Moreover, .

Proof of this proposition is similar to Proposition 4.3 by changing the order of variables.

Theorem 4.5. Let , and let . Suppose that the weight function on is of product type, that is, . Suppose also that . (i)If , then is bounded from to if and only if (ii)If , then is bounded from to if and only if (iii)If , then is bounded from to if and only if

Proof. By using Proposition A we see that the operator is bounded from to if and only if the inequality holds for all . Further, it is easy to see that Hence is bounded from to if and only if is bounded from to .By using Theorem D, (i) and (ii) follow immediately.To prove (iii) we show that if , then (4.26) implies (4.23) and (4.24). Let . Then for some . By using the doubling condition with respect to the first variable uniformly to the second one and Remark 4.2, we see that Hence, . In a similar manner we can show that .For necessity, let us see, for example, that (4.23) implies (4.26). For , by using the doubling condition for with respect to the first variable and Remark 4.2, we have Hence, taking the supremum with respect to and , we find that .

The following statements give analogous statement for the mixed-type operator and .

Theorem 4.6. Let , and let . Suppose that the weight function on is of product type, that is, . Suppose also that . (i)The operator is bounded from to if and only if (ii)The operator is bounded from to if and only if

Proof. We prove part (i). The proof of part (ii) is similar by changing the order of variables.First we show that the two-sided pointwise relation , , holds. Indeed, by using the fact that is nonincreasing in the first variable, we find that The inequality is obvious because for .Further, it is easy to check that Hence, since the boundedness of from to is equivalent to the inequality (see also [4]) we can conclude that Proposition 4.4 yields the desired result.

Proposition 4.7. Let the conditions of Theorem 4.6 be satisfied. Then (i)if , then is bounded from to if and only if (4.33) and (4.34) hold;(ii)if , then is bounded from to if and only if (4.32) and (4.34) are satisfied;(iii)if , then is bounded from to if and only if (4.34) holds.

Proof. (i) Taking into account the arguments used in Theorem 4.5, we can prove that (4.34) implies (4.32) and (4.33) implies (4.31).(ii) It can be checked that (4.32) implies (4.31) and (4.34) implies (4.33). To show that, for example, (4.32) implies (4.31), we take . Then for some integer . By using the doubling condition for with respect to the second variable, we have By a similar manner it follows that (4.34) implies (4.33). The proof of (iii) is similar, and we omit it.

The proof of the next statement is similar to that of Proposition 4.7.

Proposition 4.8. Let the conditions of Theorem 4.6 be satisfied. Then (i)if , then is bounded from to if and only if (4.36) and (4.38) hold;(ii)if , then is bounded from to if and only if (4.37) and (4.38) are satisfied;(iii)if , then is bounded from to if and only if (4.38) holds.

Now we are ready to discuss the operators on the cone of nonincreasing functions.

Theorem 4.9. Let , and let . Suppose that the weight belongs to the class . Let for some one-dimensional weight functions and and . Then the operator is bounded from to if and only if conditions (2.10), (2.11), (4.23), (4.24), (4.32), (4.34), (4.37), and (4.38) are satisfied.

Theorem 4.10. Let , and let . Suppose that the weight belongs to the class . Let for some one-dimensional weight functions and and . Then the operator is bounded from to if and only if conditions (2.10), (2.12), (4.25), (4.33), (4.34), (4.36), and (4.38) are satisfied.

Theorem 4.11. Let , and let . Suppose that the weight . Let for some one-dimensional weight functions and and . Then the operator is bounded from to if and only if conditions (2.10), (4.26), (4.34), and (4.38) are satisfied.

Proofs of these statements follow immediately from the pointwise estimate Corollary B, Theorem 4.5, and Propositions 4.7 and 4.8.

The next statement shows that the two-weight inequality for can be characterized by one condition when .

Corollary 4.12. Let , and let . Suppose that . Then the operator is bounded from to if and only if

Proof. Necessity can be derived by substituting the test function in the two-weight inequality for .Sufficiency follows by using Theorems 4.9 and 4.10 and the arguments of the proof of Corollary 3.5 with respect to each variable. Details are omitted.

Acknowledgments

The first author was partially supported by the Shota Rustaveli National Science Foundation Grant (Project no. GNSF/ST09_23_3-100). A part of this work was carried out at the Abdus Salam School of Mathematical Sciences, GC University, Lahore. The authors are thankful to the Higher Education Commission, Pakistan, for the financial support. The first author expresses his gratitude to Professor V. M. Kokilashvili for drawing his attention to the two-weight problem for potentials with product kernels.

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