A Characterization of the Two-weight Inequality for Riesz Potentials on Cones of Radially Decreasing Functions

We establish necessary and sufficient conditions on a weight pair $(v,w)$ governing the boundedness of the Riesz potential operator $I_{\alpha}$ defined on a homogeneous group $G$ from $L^p_{dec,r}(w, G)$ to $L^q(v, G)$, where $L^p_{dec,r}(w, G)$ 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 $I_{\alpha_1, \alpha_2}$ defined on a product of two homogeneous groups $G_1\times G_2$. In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces.


Introduction
A homogeneous group is a simply connected nilpotent Lie group G on a Lie algebra g with the one-parameter group of transformations δ t = exp(A log t), t > 0, where A is a diagonalized linear operator in G with positive eigenvalues. In the homogeneous group G the mappings exp o δ t o exp −1 , t > 0, are automorphisms in G, which will be again denoted by δ t . The number Q = tr A 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 r : G → [ 0, ∞) which is continuous on G, smooth on G\{e} and satisfies the conditions: (i) r(x) = r(x −1 ) for every x ∈ G; (ii) r(δ t x) = tr(x) for every x ∈ G and t > 0; (iii) r(x) = 0 if and only if x = e ; (iv) There exists c o > 0 such that r(xy) ≤ c o (r(x) + r(y)), x, y ∈ G.
In the sequel we denote by B(a, t) an open ball with the center a and radius t > 0, i.e.
It can be observed that δ t B(e, 1) = B(e, t).
Examples of homogeneous groups are: the Euclidean n-dimensional space R n , the Heisenberg group, upper triangular groups, etc. For the definition and basic properties of the homogeneous group we refer to [9], p. 12.
An everywhere positive function ρ on G will be called a weight. Denote by L p (ρ, G) (1 < p < ∞) the weighted Lebesgue space, which is the space of all measurable functions f : G → C defined by the norm 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 R n + with respect to the cone of non-negative decreasing functions on R n + were studied in [16], [15]. It should be emphasized that the two-weight problem for multiple Hardy operator for the cone of decreasing functions on R n + was investigated by S. Barza, H. P. Heinig and L. -E. Persson [4] under the restriction that both weights are of product type.
Historically the one-weight inequality for the classical Hardy operator was characterize by M. A. Arino and B. Muckenhoupt [3] under the so called B p condition. The same problem for multiple Hardy transform was studied by N. Arcozzi, S. Barza, J. L. Garcia-Domingo and J. Soria [2]. This problem in the the two-weight setting was solved by E. Sawyer [18]. Some sufficient conditions guaranteeing the two-weight inequality for the Riesz potential I α on R n was given by Y. Rakotondratsimba [17]. In particular, the author showed that 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 A ≈ B we mean that there are positive constants c 1 and c 2 (depending on appropriate parameters) such that c 1 A ≤ B ≤ c 2 A; A ≪ B means that there is a positive constant c such that A ≤ cB; integral over a product set E 1 × E 2 from g will be denoted by g(x, y)dxdy; for a weight functions w and w i on G, by the symbols W(t) and W i (t) will be denoted the integrals where e 1 and e 1 are neutral elements in G 1 and G 2 respectively. Finally we mention that constants (often different constants in one and the same lines of inequalities) will be denoted by c or C. The symbol p ′ stands for the conjugate number of p:

Preliminaries
We begin this section with the statements regarding polar coordinates in G (see e.g., [9], P. 14).
Proposition A. Let G be a homogeneous group and let S = {x ∈ G : r(x) = 1}. There is a (unique) Radon measure σ on S such that for all u ∈ L 1 (G), Let a be a positive number. The two-weight inequality for the Hardy-type transforms reeds as follows (see [8], Ch.1 for more general case, in particular for quasi-metric measure spaces): We refer also to [7] for the Hardy inequality written for balls with center at the origin.
In the sequel we denote H 1 by H.
The following statement for Euclidean spaces was derived by S. Barza, M. Johansson and L. -E. Persson [5].
Proposition B. Let w be a weight function on G and let 1 < p < ∞. If f ∈ DR(G) and g ≥ 0, then The proof of Proposition B repeats the arguments (for R n ) used in the proof of Theorem 3.1 of [5] taking Proposition A and the following lemma into account.

holds.
Proof of this lemma is based on Theorem A (part (ii)) taking a = 1, p = q, u 2 there. Details are omitted.
Corollary A implies the following duality result which follows by the standard way (see [18], [5] for details).
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 DR(G). In particular the following statement is true: Theorem B. Let 1 < p ≤ q < ∞ and let v and w be weights on G such that w L 1 (G) = ∞. Then H is bounded from Proof of this statement follows by the standard way applying Proposition C (see e.g. [18], [5]). .

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 ( ρ ∈ DC(G) ) if there is a positive constant b > 1 such that for all t > 0 the following inequality holds: Definition 2.2. We say that a locally integrable a.e. positive function ρ on G 1 × G 2 satisfies the doubling condition with respect to the second variable ( ρ ∈ DC (s) (y) ) uniformly to the first one if there is a positive constant c such that for all t > 0 and almost every x ∈ G 1 the following inequality holds: Analogously is defined the class of weights DC (s) (x).

Riesz Potentials on G
The main result of this section reeds as follows: Theorem 3.1. Let 1 < p ≤ q < ∞ and let v and w be weights such that either w ∈ DC α,p (G) or v ∈ DC(G); let To prove this result we need to prove some auxiliary statements.
Proof. We have Observe that, by the triangle inequality for r, we have . 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 f ∈ DR(G) If y ∈ B(e, r(x)/2c 0 ), then r(x) ≤ c 0 (r(xy −1 ) + r(y)) ≤ c 0 r(xy −1 ) + r(x)/2. Hence r(x) ≤ 2c 0 (r(xy −1 ). Consequently, . Applying now the fact that f ∈ DR(G) we see that G\B(e,t) 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 that 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 J α f and S α f. By applying Proposition 3.1 and Theorem B we have that J α is bounded from L p dec,r (w, G) to L q (v, G) 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)

Multiple Potentials on G 1 × G 2
Let us now investigate the two-weight problem for the operator I α,α 2 on the cone DR(G 1 × G 2 ). In the sequel without loss of generality we denote the triangle inequality constants for G 1 and G 2 by one and the same symbol c 0 .
The following statement can be derived just in the same way as Theorem 3.1 was obtained in [4]. The proof is omitted because to avoid repeating those arguments.
Proposition D. Let 1 < p < ∞ and let w(x, y) = w 1 (x)w 2 (y) be a product weight on G 1 × G 2 . Then the following relation sup 0≤ f ↓r holds for a non-negative measurable function g, where , Applying Proposition D together with the duality arguments we can get the following statement (cf. [4]).

The next statements deals with the double Hardy-type operators defined on
Proposition 4.1. Let 1 < p ≤ q < ∞. Suppose that v and w be weights on G 1 ×G 2 such that either w(x, y) = w 1 (x)w 2 (y) Proof. Let w(x, y) = w 1 (x)w 2 (y). Then the proposition follows in the same way as the appropriate statements regarding the Hardy operators defined on R 2 + in [14], [12] (see also Theorem 1.1.6 of [13]). If v is a product weight, i.e. v(x, y) = v 1 (x)v 2 (y), then the result follows from the duality arguments. We give the proof, for example, for H a,b in the case when w(x, y) = w 1 (x)w 2 (y).
First suppose that S := k=−∞ be a sequence of positive numbers for which the equality holds for all k ∈ Z. This equality follows because of the continuity in t of the integral over the ball B(e 2 , bt). It is clear that {x k } is increasing and R + = ∪ k∈Z [x k , x k+1 ). Moreover, it is easy to verify that Let f ≥ 0. We have that

Hence, by Theorem A
On the other hand, (4.1) yields that for all n ∈ Z. Hence by the discrete Hardy inequality (see e.g. [6]) and Hölder's inequality we have If S < ∞, then without loss of generality we can assume that S = 1. In this case we choose the sequence {x k } 0 k=−∞ for which (4.1) holds for all k ∈ Z − . Arguing as in the case S = ∞ and using slight modification of the discrete Hardy inequality (see also [13], Chapter 1 for similar arguments), we finally obtain the desired result. Finally we notice that the part (i) can be also proved if we first establish the boundedness of the operator (H a,b ϕ)(t, τ) = at 0 bτ 0 ϕ(s, r)dsdr in the spirit of Theorem 1.1.6 in [13] and then pass to the case of G 1 × G 2 by Proposition A.
The next statement will be useful for us.
Theorem 4.1. Let 1 < p ≤ q < ∞. Assume that v and w are weights on G 1 × G 2 such that w(x, y) = w 1 (x)w 2 (y). Suppose that either w i ∈ DC α i ,p , i = 1, 2, or v ∈ DC(x)∩DC(y). Then the operator I α 1 ,α 2 is bounded from L p dec,r (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if the following conditions are satisfied: (i) (v) v(x, y)dxdy (vi) Proof. Let us assume that v ∈ DC(x) ∩ DC(y). The case when w i ∈ DC α i ,p (G i ), i = 1, 2 follows analogously. By using representation (4.4) we have to investigate the boundedness of the operators J α 1 ,α 2 f , J α 1 S α 2 f , S α 1 J α 2 f , S α 1 ,α 2 f separately.
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 J α 1 S α 2 is bounded from L p dec,r (w, G 1 ) to L q (v, G 2 ) if and only if the conditions (vi) and (vii) are satisfied.
By the similar manner (changing the roles of the first and second variables) we can get that S α 1 J α 2 is bounded from L p dec,r (w, G 1 ) to L q (v, G 2 ) 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 I α 1 ,α 2 . Namely the following statement holds: Theorem 4.2. Let 1 < p ≤ q < ∞ and let 0 < α i < Q i /p, i = 1, 2. Then I α 1 ,α 2 is bounded from L p dec,r (G 1 × G 2 ) to L q (v, v(x, y)dxdy Proof. Sufficiency is a consequence of the inequality max{A 1 , · · · , A 9 } ≤ cB, while necessity follows immediately by taking the test function f a 1 ,a 2 (x, y) = χ B(e 1 ,a 1 ) (x)χ B(e 2 ,a 2 ) (y), a 1 , a 2 > 0.