Potential Operators on Cones of Nonincreasing Functions
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 ∞.
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 . Two-weight Hardy inequality criteria on cones of nonincreasing functions were derived in the paper . 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 .
In the paper  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  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  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 .
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 ; ; ; .
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 .
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 .
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 () for , and by Kokilashvili  and Bradley  for (see also the monograph by Maz'ya  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  and the references cited therein.
The next result deals with the case (see ).
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 ). 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 ). 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 . 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 .
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:
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.
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  (see also the proof of Theorem 1.1.6 in ).
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 ), 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