Potential Operators on Cones of Nonincreasing Functions

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators Iαf x ∫∞ 0 f t /|x − t|1−α dt and Iα1 ,α2f x, y ∫∞ 0 ∫∞ 0 f t, τ / |x − t|1−α1 |y − τ |1−α2 dtdτ on the cone of nonincreasing functions are derived. In the case of Iα1 ,α2 , we assume that the right-hand side weight is of product type. The same problem for other mixedtype double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and∞.


Introduction
Our aim is to derive necessary and sufficient conditions on weight pairs governing the boundedness of the following potential operators: 1.1 from L p dec to L q , where 1 < p, q < ∞.
Historically, necessary and sufficient condition on a weight function u, for which the boundedness of the one-dimensional Hardy transform from L p dec u, R to L p u, R 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 L p dec w, R 2 to L p v, R 2 were derived, provided that w is a product of one-dimensional weights.Earlier, the problem of the boundedness of the two-dimensional Hardy transform H 2 R 1,1 from L p dec w, R 2 to L p v, R 2 was studied in 4 under the condition that w and v have the following form: w x, y w 1 x w 2 y , v x, y v 1 x v 2 y .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 c or C; by the symbol Tf ≈ Kf, where T and K are linear positive operators defined on appropriate classes of functions, we mean that there are positive constants c 1 and c 2 independent of f and x such that Tf x ≤ c 1 Kf x ≤ c 2 Tf x ; R denotes the interval 0, ∞ and p means the number p/ p − 1 for 1 < p < ∞; W x : x 0 w t dt; W j x j : x j 0 w j t dt; W t 1 , . . ., t n : Π n i 1 W i t i .
Journal of Function Spaces and Applications 3

Preliminaries
We say that a function f : R n → R is nonincreasing if f is nonincreasing in each variable separately.
Let D be the class of all nonnegative nonincreasing functions on R n .Suppose that u is measurable a.e.positive function weight on R n .We denote by L p u, R n , 0 < p < ∞, the class of all nonnegative functions on R n for which

2.1
By the symbol L p dec u, R n we mean the class L p u, R n ∩ D. The next statement regarding two-weight criteria for the Hardy operator H on the cone of nonincreasing functions was proved in 2 .
Theorem A. Let v and w be weight functions on R , and let W ∞ ∞.
i Suppose that 1 < p ≤ q < ∞.Then the inequality holds if and only if the following two conditions are satisfied: R if and only if the following two conditions are satisfied:
The following statement was proved in 2 for n 1.For n ≥ 1 we refer to 4 .
Proposition A. Let 1 < p, q < ∞.Suppose that T is a positive integral operator defined on functions f : R n → R , which are nonincreasing in each variable separately.Suppose that T * is its formal adjoint.Let w x 1 , . . ., x n w holds for all g ≥ 0.
Let R α be the Riemann-Liouville transform with single kernel If α 1, then R α is the Hardy transform.The L p w, R → L q v, R boundedness for R 1 was characterized by Muckenhoupt 20 for p q, and by Kokilashvili 21 and Bradley 22 for p < q see also the monograph by Maz'ya 23 for these and relevant results .
In the case when 0 < α < 1, 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 α > 1 see 24 .
Theorem C see 10 .Let 1 < p ≤ q < ∞, and let 0 < α i < 1, i 1, 2. Assume that v and w are weights on R 2 .Suppose also that w x 1 , x 2 w 1 x 1 w 2 x 2 for some one-dimensional weights w 1 and w 2 and that W i ∞ ∞, i 1, 2. Then the following conditions are equivalent: R 2 ; b the following four conditions hold simultaneously:

2.12
In particular, Theorem C yields the trace inequality criteria on the cone of nonincreasing functions.
Corollary A see 10 .Let 1 < p ≤ q < ∞, and let 0 < α i < 1, i 1, 2. Then the following conditions are equivalent: Journal of Function Spaces and Applications c 2.16

Potentials on R
In this section we discuss the two-weight problem for the operator I α .We begin with the following lemma.where H is the Hardy operator defined above.
Proof.We follow the proof of Proposition 3.1 of 10 .We have Further, since f is nonincreasing, we have that Finally we have the upper estimate for R α .The lower estimate is obvious because x − t α−1 ≥ x α−1 for t ≤ x.
In the next statement we assume that W α is the operator given by , g ≥ 0. 3.6 Proof.Taking Proposition A into account for n 1 , an integral operator where T * is a formal adjoint to T .We have

3.9
Taking T W α and T * R α , we derive the desired result.

Journal of Function Spaces and Applications
Now we formulate the main results of this section.

3.16
Theorems A and B and Lemmas 3.1 and 3.2, we have the desired results.

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Corollary 3.5.Let 1 < p ≤ q < ∞, and let 0 < α < 1/p.Then the operator I α is bounded from 3.17 Proof.Necessity follows immediately taking the test function f a x χ 0,a x in the twoweight inequality and observing that where The estimates A i ≤ cB, i 1, 4, are obvious.We show that A i ≤ cB for i 2, 3. We have

3.21
Definition 3.6.Let ρ be a locally integrable a.e.positive function on R .We say that ρ satisfies the doubling condition ρ ∈ DC R if there is a positive constant b > 1 such that for all t > 0 the following inequality holds: 3.26 Finally, we have 3.23 .
Corollary 3.8.Let 1 < p ≤ q < ∞, and let 0 < α < 1. Suppose that W ∞ ∞.Suppose also that v ∈ DC R .Then I α is bounded from L p dec w, R to L q v, R if and only if condition 3.11 is satisfied.
Proof.Observe that by Remark 3.7, for m 0 ∈ Z, the inequality holds for all k > m 0 , where b 1 is defined in 3.23 .
Let a > 0. Then there is m 0 ∈ Z such that a ∈ 2 m 0 , 2 m 0 1 .By applying 3.27 and the doubling condition for v, we find that a 0 w t dt

3.28
So, we have seen that 3.11 ⇒ 3.10 .Let us check now that 3.13 ⇒ 3.12 .Indeed, for a > 0, we choose m 0 so that a ∈ 2 m 0 , 2 m 0 1 .Then, by using the condition v ∈ DC R and Remark 3.7,

3.29
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.

Potentials with Multiple Kernels
In this section we discuss two-weight criteria for the potentials with product kernels I α 1 ,α 2 .
To derive the main results, we introduce the following multiple potential operators: where Definition 4.1.One says that a locally integrable a.e.positive function ρ on R 2 satisfies the doubling condition with respect to the second variable ρ ∈ DC y if there is a positive constant c such that for all t > 0 and almost every x > 0 the following inequality holds: Analogously, ρ ∈ DC x ⇒ ρ ∈ RDC x .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.
, it is necessary and sufficient that conditions 2.10 and 2.12 are satisfied.iii The following result concerns with the two-weight criteria for the two-dimensional operator R α 1 ,α 2 with α 1 , α 2 > 1 see 25 , 13, Section 1.6 .
R 2 if and only if
ii Let w 1−p ∈ DC x .Then the operator R α 1 ,α 2 is bounded from L p w, R 2 to L q v, R if and only if
Let us introduce the following multiple integral operators:

4.6
Now we prove some auxiliary statements.
i The operator RH α 1 ,α 2 is bounded from L p w, R 2 to L q v, R if and only if
ii The operator WH α 1 ,α 2 is bounded from L p w, R 2 to L q vR if and only if iii The operator RH α 1 ,α 2 is bounded from L p w, R 2 to L q v, R if and only if
iv The operator WH α 1 ,α 2 is bounded from L p w, R 2 to L q v, R if and only if
Proof.Let w x 1 , x 2 w 1 x 1 w 2 x 2 .The proof of the case v x 1 , x 2 v 1 x 1 v 2 x 2 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 .

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Sufficiency.First suppose that S : x 2 dx 2 .

4.12
Let f ≥ 0. We have that

4.14
It is obvious that

4.15
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 n ∈ Z. Hence by Hardy's inequality in discrete case see, for example, 25, 26 and H ölder's inequality we have that

4.18
If S < ∞, then without loss of generality we can assume that S 1.In this case we choose the sequence {a k } 0 k −∞ for which 4.11 holds for all k ∈ Z − .Arguing as in the case of S ∞, 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 {x k } so that x dx 2 k notice that x k is decreasing and argue as in the proof of i .
i The operator HR α 1 ,α 2 is bounded from L p w, R 2 to L q v, R 2 if and only if
ii The operator HW α 1 ,α 2 is bounded from L p w, R 2 to L q v, R if and only if
iii The operator H R α 1 ,α 2 is bounded from L p w, R 2 to L q v, R if and only if

4.22
Moreover, Proof of this proposition is similar to Proposition 4.3 by changing the order of variables.

4.26
Proof.By using Proposition A we see that the operator holds for all g ≥ 0. Further, it is easy to see that

4.28
Hence By using Theorem D, i and ii follow immediately.
To prove iii we show that if v ∈ DC x ∩ DC y , then 4.26 implies 4.23 and 4.24 .Let a, b > 0. Then a ∈ 2 m 0 , 2 m 0 1 for some m 0 ∈ Z.By using the doubling condition with respect to the first variable uniformly to the second one and Remark 4.2, we see that

4.29
Hence, A 1 ≤ C 1 .In a similar manner we can show that A 2 ≤ C 1 .
For necessity, let us see, for example, that 4.23 implies 4.26 .For a ∈ 2 m 0 , 2 m 0 1 , by using the doubling condition for v with respect to the first variable and Remark 4.2, we have

4.30
Hence, taking the supremum with respect to a and b, we find that C 1 ≤ cA 1 .

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ii The operator WR α 1 ,α 2 is bounded from

4.38
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 RW α 1 ,α 2 f ≈ HW α 1 ,α 2 f, f ↓, holds.Indeed, by using the fact that f is nonincreasing in the first variable, we find that

4.39
The inequality Further, it is easy to check that

4.41
Hence, since the boundedness of HW α 1 ,α 2 from L p dec w, R 2 to L q v, R 2 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  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 a, b > 0. Then b ∈ 2 m 0 , 2 m 0 1 for some integer m 0 .By using the doubling condition for v with respect to the second variable, we have

4.43
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. Proofs of these statements follow immediately from the pointwise estimate 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.

Lemma 3 . 1 .
The following relation holds for nonnegative and nonincreasing function f: R α f x ≈ x α Hf x , 3.1

22 Remark 3 . 7 .
2t 0 ρ x dx ≤ b min t 0 ρ x dx, 2t t ρ x dx .3.It is easy to check that if ρ ∈ DC R , then ρ satisfies the reverse doubling condition: there is a positive constant b 1 > 1 such that 2t 0 ρ x dx ≥ b 1 max

Remark 4 . 2 .
the class of weights DC x .If ρ ∈ DC y , then ρ satisfies the reverse doubling condition with respect to the second variable; that is, there is a positive constant c 1 such that 2t 0 ρ x, y dy ≥ c 1 max Theorem 4.5, and Propositions 4.7 and 4.8.The next statement shows that the two-weight inequality for I α 1 ,α 2 can be characterized by one condition when w ≈ 1.Corollary 4.12.Let 1 < p ≤ q < ∞, and let0 < α 1 , α 2 < 1/p.Suppose that v ∈ DC x ∪ DC y .Then the operator I α 1 ,α 2 is bounded from L p dec 1, R 2 to L q v, R 2 if and only if D : sup a,b>0 a α 1 − 1/p b α 2 − 1/pcan be derived by substituting the test function f a,b x χ 0,a × 0,b x in the two-weight inequality for I α 1 ,α 2 .
and only If conditions 2.10 and 2.11 are satisfied.
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.
and only if 4.33 and 4.34 hold; ii if v ∈ DC y , then RW α 1 ,α 2 is bounded from L 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 .
R 2 to L q v, R 2 if and only if 4.36 and 4.38 hold;ii if v ∈ DC y , then WR α 1 ,α 2 is bounded from L p dec w, R 2 to L q v, R2 if and only if 4.37 and 4.38 are satisfied;iii if v ∈ DC x ∩ DC y , then WR α 1 ,α 2 is bounded from L p dec w, R 2 to L q v, R 2 if and only if 4.38 holds.Now we are ready to discuss the operators I α 1 ,α 2 on the cone of nonincreasing functions.Let 1 < p ≤ q < ∞, and let 0 < α 1 , α 2 < 1. Suppose that the weight v belongs to the class DC y .Let w x 1 , x 2 w 1 x 1 w 2 x 2 for some one-dimensional weight functions w 1 and w 2 andW 1 ∞ W 2 ∞ ∞.Then the operator I α 1 ,α 2 is bounded from L p dec w, R 2 to L q v, R 2 ifand 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 1 < p ≤ q < ∞, and let 0 < α 1 , α 2 < 1. Suppose that the weight v belongs to the class DC x .Let w x 1 , x 2 w 1 x 1 w 2 x 2 for some one-dimensional weight functions w 1 and w 2 and W 1 ∞ W 2 ∞ ∞.Then the operator I α 1 ,α 2 is bounded from L p dec w, R 2 to L q v, R 2 if and only if conditions 2.10 , 2.12 , 4.25 , 4.33 , 4.34 , 4.36 , and 4.38 are satisfied.Let 1 < p ≤ q < ∞, and let 0 < α 1 , α 2 < 1. Suppose that the weight v ∈ DC x ∩ DC y .Let w x 1 , x 2 w 1 x 1 w 2 x 2 for some one-dimensional weight functions w 1 and w 2 and W 1 ∞ W 2 ∞ ∞.Then the operator I α 1 ,α 2 is bounded from L p dec w, R 2 to L q v, R 2 if and only if conditions 2.10 , 4.26 , 4.34 , and 4.38 are satisfied.
7.Proposition 4.8.Let the conditions of Theorem 4.6 be satisfied.Theni if v ∈ DC x , then WR α 1 ,α 2 is bounded from L p dec w,