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A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions

Abstract

We establish necessary and sufficient conditions on a weight pair (v,w) governing the boundedness of the Riesz potential operator I α defined on a homogeneous group G from L dec , r p (w,G) to L q (v,G), where L dec , r p (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 α 1 , α 2 defined on a product of two homogeneous groups G 1 × G 2 . 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 G 1 × G 2 , 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 δ t =exp(Alogt), t>0, where A is a diagonalized linear operator in G with positive eigenvalues. In the homogeneous group G the mappings expo δ t o exp 1 , t>0, are automorphisms in G, which will be again denoted by δ t . The number Q=trA 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:

  1. (i)

    r(x)=r( x 1 ) for every xG;

  2. (ii)

    r( δ t x)=tr(x) for every xG and t>0;

  3. (iii)

    r(x)=0 if and only if x=e;

  4. (iv)

    there exists c 0 >0 such that

    r(xy) c 0 ( r ( x ) + r ( y ) ) ,x,yG.

In the sequel we denote by B(a,t) an open ball with the center a and radius t>0, i.e.

B(a,t):= { y G ; r ( a y 1 ) < t } .

It can be observed that δ t B(e,1)=B(e,t).

Let us fix a Haar measure || in G such that |B(e,1)|=1. Then | δ t E|= t Q |E|. In particular, |B(x,t)|= t Q for xG, t>0.

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 [[1], 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:GC defined by the norm

f L p ( ρ , G ) = ( G | f ( x ) | p ρ ( x ) d x ) 1 p <.

If ρ1, then we use the notation L p (G).

Denote by DR(G) the class of all radially decreasing functions on G with values in R + , i.e. the fact that ϕDR(G) means that there is a decreasing ϕ ¯ : R + R + such that φ(x)= ϕ ¯ (r(x)). In the sequel we will use the symbol ϕ itself for ϕ ¯ ; ϕDR(G) will be written also by the symbol φr. Let G 1 and G 2 be homogeneous groups. We say that a function ψ: G 1 × G 2 R + is radially decreasing if it is such in each variable separately uniformly to another one. The fact that ψ is radially decreasing on G 1 × G 2 will be denoted as ψDR( G 1 × G 2 ).

Let

( I α f)(x)= G f(y) ( r ( x y 1 ) ) α Q dy,0<α<Q,

be the Riesz potential defined on G, where r is the homogeneous norm and dy is the normalized Haar measure on G. The operator I α 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 G 1 and G 2 be homogeneous groups with homogeneous norms r 1 and r 2 and homogeneous dimensions Q 1 and Q 2 , respectively. We define the potential operator on G 1 × G 2 as follows:

( I α , β f ) ( x , y ) = G 1 × G 2 f ( t , τ ) ( r 1 ( x t 1 ) ) α Q 1 ( r 2 ( y τ 1 ) ) β Q 2 d t d τ , ( x , y ) G 1 × G 2 , 0 < α < Q 1 , 0 < β < Q 2 .

Our aim is to derive two-weight criteria for I α on the cone of radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels I α , β 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 I α , β 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 R + n with respect to the cone of non-negative decreasing functions on R + n 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 R + n 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 B p 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 I α on R n were given by Rakotondratsimba [9]. In particular, the author showed that I α is bounded from L dec , r p (w, R n ) to L q (v, R n ) if the weighted Hardy operators (Hf)(x)= 1 | x | n α | y | < | x | f(y)dy and ( H f)(x)= | y | > | x | f ( y ) | y | n α dy are bounded from L p (w, R n ) to L q (v, R n ). 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 AB we mean that there are positive constants c 1 and c 2 (depending on appropriate parameters) such that c 1 AB c 2 A; AB means that there is a positive constant c such that AcB; integral over a product set E 1 × E 2 from g will be denoted by E 1 × E 2 g(x,y)dxdy or E 1 E 2 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 B ( e , t ) w(x)dx and B ( e i , t ) w i (x)dx respectively; for a weight w on G 1 × G 2 , we denote W(t,τ):= B ( e 1 , t ) × B ( e 2 , τ ) w(x,y)dxdy, 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 line of inequalities) will be denoted by c or C. The symbol p stands for the conjugate number of p: p =p/(p1), where 1<p<.

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 S={xG:r(x)=1}. There is a (unique) Radon measure σ on S such that for all u L 1 (G),

G u(x)dx= 0 S u( δ t y ¯ ) t Q 1 dσ( y ¯ )dt.

Let a be a positive number. The two-weight inequality for the Hardy-type transforms

( H a f ) (x)= B ( e , a r ( x ) ) f(y)dy,xG,( H ˜ a f(x)= G B ( e , a r ( x ) ) f(y)dy,xG,

reads as follows (see [10], Chapter 1 for more general case, in particular, for quasi-metric measure spaces):

Theorem A Let 1<pq< and let a be a positive number. Then

  1. (i)

    The operator H a is bounded from L p ( u 1 ,G) to L q ( u 2 ,G) if and only if

    sup t > 0 ( G B ( e , t ) u 2 ( x ) d x ) 1 / q ( B ( e , a t ) u 1 1 p ( x ) d x ) 1 / p <.
  2. (ii)

    The operator H ˜ a is bounded from L p ( u 1 ,G) to L q ( u 2 ,G) if and only if

    sup t > 0 ( B ( e , t ) u 2 ( x ) d x ) 1 / q ( G B ( e , a t ) u 1 1 p ( x ) d x ) 1 / p <.

We refer also to [11] 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 Barza, Johansson and Persson [12].

Proposition B Let w be a weight function on G and let 1<p<. If fDR(G) and g0, then

sup f r G f ( x ) g ( x ) d x ( G f ( x ) p w ( x ) d x ) 1 / p w L 1 ( G ) 1 / p g L 1 ( G ) + ( G H p ( r ( x ) ) W p ( r ( x ) ) w ( x ) d x ) 1 / p ,

where H(t)= B ( e , t ) g(x)dx, W(t)= B ( e , t ) w(x)dx.

The proof of Proposition B repeats the arguments (for R n ) used in the proof of Theorem 3.1 of [12] taking Proposition A and the following lemma into account.

Lemma A Let 1<p<. For a weight function w, the inequality

G w(x) ( G B ( e , r ( x ) ) f ( y ) d y ) p dxp G f p (x) W p ( r ( x ) ) w 1 p (x)dx,f0,

holds.

Proof The proof of this lemma is based on Theorem A (part (ii)) taking a=1, p=q, u 2 (x)=v(x), u 1 = w 1 p (x) W p (r(x)) there. Details are omitted. □

Corollary A Let the conditions of Proposition  B be satisfied and let G w(x)dx=. Then the following relation holds:

sup f r G f ( x ) g ( x ) d x ( G f p ( x ) w ( x ) d x ) 1 / p ( G H p ( r ( x ) ) W ( r ( x ) ) w ( x ) d x ) 1 / p .

Corollary A implies the following duality result, which follows in the standard way (see [3, 12] for details).

Proposition C Let 1<p,q< and let v, w be weight functions on G with G w(x)dx=. Then the integral operator T defined on functions on G is bounded from L dec , r p (w,G) to L q (v,G) if and only if

( G ( B ( e , r ( x ) ) ( T g ) ( y ) d y ) p W p ( r ( x ) ) w ( x ) d x ) 1 / p C ( G g q ( x ) v 1 q ( x ) d x ) 1 / q
(2.1)

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<pq< and let v and w be weights on G such that w L 1 ( G ) =. Then H is bounded from L dec , r p (w,G) to L v q (v,G) if and only if

  1. (i)
    sup t > 0 ( B ( e , t ) w ( x ) d x ) 1 / p ( B ( e , t ) v ( x ) r Q q ( x ) d x ) 1 / q <;
  2. (ii)
    sup t > 0 ( B ( e , t ) r Q p ( x ) W p ( r ( x ) ) w ( x ) d x ) 1 / p ( G B ( e , t ) v ( x ) d x ) 1 / q <.

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 (ρDC(G)) if there is a positive constant b>1 such that for all t>0 the following inequality holds:

B ( e , 2 t ) ρ(x)dxb B ( e , t ) ρ(x)dx.

Further, we say that wD C γ , p (G), where 1<p<, 0<γ<Q/p, if there is a positive constant b such that for all t>0

G B ( e , t ) r γ p (x) W p ( r ( x ) ) w(x)dxb G B ( e , 2 t ) r γ p (x) W p ( r ( x ) ) w(x)dx.

Remark 2.1 It can be checked that if a weight w satisfies the doubling condition et e in the strong sense, i.e., wDC(G) and B ( e , 2 t ) w(x)dxc B ( e , 2 t ) B ( e , t ) w(x)dx with a constant c independent of t, then wD C γ , p (G).

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 (ρDC(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:

B ( e 2 , 2 t ) ρ(x,y)dyc B ( e 2 , t ) ρ(x,y)dy.

Analogously is defined the class of weights DC(x).

3 Riesz potentials on G

The main result of this section reads as follows.

Theorem 3.1 Let 1<pq< and let v and w be weights such that either wD C α , p (G) or vDC(G); let w L 1 ( G ) =. Then the operator I α is bounded from L dec , r p (w,G) to L q (v,G) if and only if

  1. (i)
    sup t > 0 ( B ( e , t ) w ( x ) d x ) 1 / p ( B ( e , t ) r α q ( x ) v ( x ) d x ) 1 / q <;
    (3.1)
  2. (ii)
    sup t > 0 ( B ( e , t ) r p Q ( x ) W p ( r ( x ) ) w ( x ) d x ) 1 / p ( G B ( e , t ) r ( α Q ) q ( x ) v ( x ) d x ) 1 / q <;
    (3.2)
  3. (iii)
    sup t > 0 ( B ( e , t ) v ( x ) d x ) 1 / q ( G B ( e , t ) r α p ( x ) W p ( r ( x ) ) w ( x ) d x ) 1 / p <.
    (3.3)

To prove this result we need to prove some auxiliary statements.

Lemma 3.1 Let 0<α<Q and let c 0 be the constant from the triangle inequality of r. Then there is a positive constant c depending only on Q, α, and c 0 such that for all sB(e,r(x)/2),

I(x,y):= B ( e , r ( x ) ) B ( e , 2 c 0 r ( y ) ) r ( t y 1 ) α Q dtcr ( x y 1 ) α .
(3.4)

Proof We have

I ( x , y ) = 0 | { t G : r ( t y 1 ) α Q > λ } B ( e , r ( x ) ) B ( e , 2 c 0 r ( y ) ) | d λ = 0 r ( x y 1 ) α Q ( ) + r ( x y 1 ) α Q ( ) = : I ( 1 ) ( x , y ) + I ( 2 ) ( x , y ) .

Observe that, by the triangle inequality for r, we have r Q (x) c 0 Q 2 Q 1 ( r Q (x y 1 )+ r Q (y)). This implies that r Q (x) ( 2 c 0 ) Q r Q (y) c 0 Q 2 Q 1 r Q (x y 1 ). Hence,

I ( 1 ) ( x , y ) r ( x y 1 ) α Q | B ( e , r ( x ) ) B ( e , 2 c 0 r ( y ) ) | = r ( x y 1 ) α Q ( r Q ( x ) ( 2 c 0 ) Q r Q ( y ) ) c r ( x y 1 ) α .

Further, it is easy to see that

I ( 2 ) (x,y)cr ( x y 1 ) α .

Finally we have (3.4). □

Let us introduce the following potential operators:

( J α f ) ( x ) = B ( e , 2 c 0 r ( x ) ) f ( y ) r α Q ( x y 1 ) d y , ( S α f ) ( x ) = G B ( e , 2 c 0 r ( x ) ) f ( y ) r α Q ( x y 1 ) d y , x G , 0 < α < Q .

It is easy to see that

I α f= J α f+ S α f.
(3.5)

We need also to introduce the following weighted Hardy operator:

( H α f)(x)=r ( x ) α Q (Hf)(x).

Proposition 3.1 The following relation holds for all fDR(G):

J α f H α f.
(3.6)

Proof We have

( J α f ) ( x ) = B ( e , r ( x ) / 2 c 0 ) f ( y ) r α Q ( x y 1 ) d y + B ( e , 2 c 0 r ( x ) ) B ( e , r ( x ) / ( 2 c 0 ) ) f ( y ) r α Q ( x y 1 ) d y = : ( J α ( 1 ) f ) ( x ) + ( J α ( 2 ) f ) ( x ) .

If yB(e,r(x)/2 c 0 ), then r(x) c 0 (r(x y 1 )+r(y)) c 0 r(x y 1 )+r(x)/2. Hence r(x)2 c 0 r(x y 1 ). Consequently,

( J α ( 1 ) f ) (x)c( H α f)(x).

Applying now the fact that fDR(G) we see that

( J α ( 2 ) f ) ( x ) f ( r ( x ) / 2 c 0 ) B ( e , r ( x ) / 2 c 0 ) B ( e , 2 c 0 r ( x ) ) r α Q ( x y 1 ) d y c f ( r ( x ) / 2 c 0 ) r ( x ) α c ( H α f ) ( x ) .

 □

Lemma 3.2 Let 1<pq< and let v and w be weights on G such that w L 1 ( G ) =. Then the operator S α is bounded from L dec , r p (w,G) to L q (v,G) if

sup t > 0 ( G B ( e , t ) r α p ( x ) W p ( r ( x ) ) w ( x ) d x ) 1 / p ( B ( e , t / ( 2 c 0 ) ) v ( x ) d x ) 1 / q <.

Conversely, if S α is bounded from L dec , r p (w,G) to L q (v,G), then the condition

sup t > 0 ( G B ( e , t ) r α p ( x ) W p ( x ) w ( x ) d x ) 1 / q ( B ( e , t / ( 4 c 0 ) ) v ( x ) d x ) 1 / p <

is satisfied. Furthermore, if either wD C α , p (G) or vDC(G), then the operator S α is bounded from L dec , r p (w,G) to L q (v,G) if and only if

sup t > 0 ( G B ( e , t ) r α p ( x ) W p ( r ( x ) ) w ( x ) d x ) 1 / q ( B ( e , t ) v ( x ) d x ) 1 / p <.

Proof Applying Proposition C, S α is bounded from L dec , r p (w,G) to L q (v,G) if and only if

( G ( B ( e , r ( x ) ) ( S α f ) ( y ) d y ) p W p ( r ( x ) ) w ( x ) d x ) 1 / p c ( G g q ( x ) v 1 q ( x ) d x ) 1 / q ,

where

( S α f ) (x)= B ( e , r ( x ) / ( 2 c 0 ) ) f(y) r α Q ( x y 1 ) dy.

Now we show that

c 1 r α ( x ) B ( e , r ( x ) / ( 4 c 0 ) ) g ( s ) d s B ( e , r ( x ) ) ( S α g ) ( y ) d y c 2 r α ( x ) B ( e , r ( x ) / ( 2 c 0 ) ) g ( s ) d s , g 0 .
(3.7)

To prove the right-hand side estimate in (3.7) observe that by Tonelli’s theorem and Lemma 3.1 we have

B ( e , r ( x ) ) ( S α g ) ( y ) d y = B ( e , r ( x ) / ( 2 c 0 ) ) f ( s ) ( B ( e , r ( x ) ) B ( e , 2 c 0 r ( s ) ) r α Q ( s y 1 ) d y ) d s c 2 r ( x ) α B ( e , r ( x ) / ( 2 c 0 ) ) f ( s ) d s .

On the other hand,

B ( e , r ( x ) ) ( S α g ) ( y ) d y c r α Q ( x ) ( B ( e , r ( x ) ) B ( e , r ( x ) / 2 ) ( B ( e , r ( y ) / ( 2 c 0 ) ) f ( s ) d s ) d y ) c 1 r α ( x ) ( B ( e , r ( x ) / ( 4 c 0 ) ) f ( s ) d s ) .

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 find that J α is bounded from L dec , r p (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)

( G B ( e , t ) W ( r ( x ) ) w ( x ) d x ) 1 / p = ( B ( e , t ) w ( x ) d x ) 1 / p

we see that S α is bounded from L dec , r p (w,G) to L q (v,G) if and only if (i) is satisfied. □

4 Multiple potentials on G 1 × G 2

Let us now investigate the two-weight problem for the operator I α 1 , 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 [6]. The proof is omitted 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 relation

sup 0 f r G 1 × G 2 f ( x , y ) g ( x , y ) d x d y ( G 1 × G 2 f p ( x , y ) w ( x , y ) ) 1 / p i = 1 4 I k

holds for a non-negative measurable function g, where

I 1 : = w L 1 ( G 1 × G 2 ) 1 / p g L 1 ( G 1 × G 2 ) , I 2 : = w 2 L 1 ( G 1 ) 1 / p ( G 1 ( B ( e 1 , r 1 ( x ) ) g ( t , ) L 1 ( G 2 ) d t ) p W 1 p ( r 1 ( x ) ) w 1 ( x ) d x ) 1 / p , I 3 : = w 1 L 1 ( G 1 ) 1 / p ( G 2 ( B ( e 2 , r 2 ( y ) ) g ( , τ ) L 1 ( G 1 ) d τ ) p W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p , I 4 : = ( G 1 × G 2 ( G 1 × G 2 g ( t , τ ) d t d τ ) p W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p .

Applying Proposition D together with the duality arguments we can get the following statement (cf. [6]).

Proposition E Let 1<p< and let v and w be weights on G 1 × G 2 such that w(x,y)= w 1 (x) w 2 (y), w L 1 ( G 1 × G 2 ) =. Then an integral operator T defined for functions from DR( G 1 × G 2 ) is bounded from L dec , r p (w, G 1 × G 2 ) to L p (v, G 1 × G 2 ) if and only if for all non-negative measurable g on G 1 × G 2 ,

( G 1 × G 2 ( B ( e 1 , r 1 ( x ) ) × B ( e 2 , r 2 ( y ) ) ( T g ) ( t , τ ) d t d τ ) p W p ( x , y ) w ( x , y ) d x d y ) 1 / p C ( G 1 × G 2 g q ( x , y ) v 1 q ( x , y ) d x d y ) 1 / q .

The next statements deal with the double Hardy-type operators defined on G 1 × G 2 :

( H a , b f ) ( x , y ) = B ( e 1 , a r 1 ( x ) ) B ( e 2 , b r 2 ( x ) ) f ( t , τ ) d t d τ , ( x , y ) G 1 × G 2 , ( H ˜ a , b f ) ( x , y ) = G 1 B ( e 1 , a r 1 ( x ) ) G 2 B ( e 2 , b r 2 ( x ) ) f ( t , τ ) d t d τ , ( x , y ) G 1 × G 2 , ( H 1 a , b f ) ( x , y ) = B ( e 1 , a r 1 ( x ) ) G 2 B ( e 2 , b r 2 ( y ) ) f ( t , τ ) d t d τ , ( x , y ) G 1 × G 2 , ( H 2 a , b f ) ( x , y ) = G 1 B ( e 1 , a r 1 ( x ) ) B ( e 2 , b r 2 ( y ) ) f ( t , τ ) d t d τ , ( x , y ) G 1 × G 2 .

Proposition 4.1 Let 1<pq<. Suppose that v and w be weights on G 1 × G 2 such that either w(x,y)= w 1 (x) w 2 (y) or v(x,y)= v 1 (x) v 2 (y). Then:

  1. (i)

    The operator H a , b is bounded from L p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if

    A : = sup t > 0 , τ > 0 ( G 1 B ( e 1 , t ) G 2 B ( e 2 , τ ) v ( x , y ) d x d y ) 1 / q × ( B ( e 1 , a t ) B ( e 2 , b τ ) w 1 p ( x , y ) d x d y ) 1 / p < .
  2. (ii)

    The operator H ˜ a , b is bounded from L p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if

    sup t > 0 , τ > 0 ( B ( e 1 , t ) B ( e 2 , τ ) v ( x , y ) d x d y ) 1 / q ( G 1 B ( e 1 , a t ) G 2 B ( e 2 , b τ ) w 1 p ( x , y ) d x d y ) 1 / p <.
  3. (iii)

    The operator H 1 a , b is bounded from L p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if

    sup t > 0 , τ > 0 ( G 1 B ( e 1 , t ) B ( e 2 , τ ) v ( x , y ) d x d y ) 1 / q ( B ( e 1 , a t ) G 2 B ( e 2 , b τ ) w 1 p ( x , y ) d x d y ) 1 / p <.
  4. (iv)

    The operator H 2 a , b is bounded from L p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if

    sup t > 0 , τ > 0 ( B ( e 1 , t ) G 2 B ( e 2 , τ ) v ( x , y ) d x d y ) 1 / q ( G 1 B ( e 1 , a t ) B ( e 2 , b τ ) w 1 p ( x , y ) d x d y ) 1 / p <.

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 [13, 14] (see also Theorem 1.1.6 of [15]). 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:= G 2 w 2 1 p (y)dy=. Let { x k } k = + be a sequence of positive numbers for which the equality

2 k = B ( e 2 , b x k ) w 2 1 p (y)dy
(4.1)

holds for all kZ. 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

2 k = B ( e 2 , b x k + 1 ) B ( e 2 , b x k ) w 2 1 p (y)dy.

Let f0. We have

H a , b f L v q ( G 1 × G 2 ) q = G 1 × G 2 v ( x , y ) ( H a , b f ) q ( x , y ) d x d y k Z G 1 B ( e 2 , x k + 1 ) B ( e 2 , x k ) v ( x , y ) × ( B ( e 1 , a r 1 ( x ) ) × B ( e 2 , b r 2 ( x ) ) f ( t , τ ) d t d τ ) q d x d y k Z G 1 ( B ( e 2 , x k + 1 ) B ( e 2 , x k ) v ( x , y ) d y ) × ( B ( e 1 , a r 1 ( x ) ) ( B ( e 2 , b x k + 1 ) f ( t , τ ) d τ ) d t ) q d x = k Z G 1 V k ( x ) ( B ( e 1 , a r 1 ( x ) ) F k ( t ) d t ) q d x ,

where

V k (x):= B ( e 2 , x k + 1 ) B ( e 2 , x k ) v(x,y)dy; F k (t):= B ( e 2 , b x k + 1 ) f(t,τ)dτ.

It is obvious that

A q sup a > 0 j Z ( G 1 B ( e 1 , t ) v j ( y ) d y ) ( B ( e 1 , a t ) × B ( e 2 , b x j ) w 1 p ( x , y ) d x d y ) q / p .

Hence, by Theorem A

H a , b f L v q ( G 1 × G 2 ) q c A q j Z [ G 1 w 1 ( x ) ( B ( e 2 , b x j ) w 2 1 p ( y ) d y ) 1 p ( F k ( x ) ) p d x ] q / p c A q [ G 1 w 1 ( x ) j Z ( B ( e 2 , b x j ) w 2 1 p ( y ) d y ) 1 p × ( k = j B ( e 2 , b x k + 1 ) B ( e 2 , b x k ) f ( x , τ ) d τ ) p d x ] q / p .

On the other hand, (4.1) yields

k = n + ( B ( e 2 , b x k ) w 2 1 p ( y ) d y ) 1 p ( k = n B ( e 2 , b x k + 1 ) B ( e 2 , b x k ) w 2 1 p ( y ) d y ) p 1 = k = n + ( B ( e 2 , b x k ) w 2 1 p ( y ) d y ) 1 p ( B ( e 2 , b x n + 1 ) w 2 1 p ( y ) d y ) p 1 = ( k = n + 2 k ( 1 p ) ) 2 ( n + 1 ) ( p 1 ) c

for all nZ. Hence by the discrete Hardy inequality (see e.g. [16]) and Hölder’s inequality we have

H a , b f L v q ( G 1 × G 2 ) q c A q [ G 1 w 1 ( x ) j Z ( B ( e 2 , b x j + 1 ) B ( e 2 , b x j ) w 2 1 p ( y ) d y ) 1 p × ( B ( e 2 , b x j + 1 ) B ( e 2 , b x j ) f ( x , τ ) d τ ) p d x ] q / p c A q [ G 1 w 1 ( x ) j Z ( B ( e 2 , b x j + 1 ) B ( e 2 , b x j ) w 2 ( τ ) f p ( x , τ ) d τ ) d x ] q / p = c A q f L w p ( G 1 × G 2 ) q .

If S<, then without loss of generality we can assume that S=1. In this case we choose the sequence { x k } k = 0 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 [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 ( H a , b φ)(t,τ)= 0 a t 0 b τ φ(s,r)dsdr in the spirit of Theorem 1.1.6 in [15] and then pass to the case of G 1 × G 2 by Proposition A. □

The next statement will be useful for us.

Proposition 4.2 Let 1<pq<. Assume that v and w are weights on G 1 × G 2 . Suppose that w(x,y)= w 1 (x) w 2 (y) and that W i ()=, i=1,2. Then the operator H 1 , 1 is bounded from L dec , r p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if the following four conditions are satisfied:

  1. (i)
    sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) w ( x , y ) d x d y ) 1 / p × ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 1 Q 1 q ( x ) r 2 ( y ) Q 2 q v ( x , y ) d x d y ) 1 / q < ;
  2. (ii)
    sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 1 Q 1 p ( x ) r 2 ( y ) Q 2 p W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p × ( G 1 B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) v ( x , y ) d x d y ) 1 / q < ;
  3. (iii)
    sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) w 1 ( r 1 ( x ) ) d x ) 1 / p ( B ( e 2 , a 2 ) r 2 ( y ) Q 2 p W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p × ( B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) r 1 ( x ) Q 1 q v ( x , y ) d x d y ) 1 / q < ;
  4. (iv)
    sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) r 1 ( x ) Q 1 p W 1 p ( r 1 ( x ) ) w 1 ( x ) d t 1 ) 1 / p ( B ( e 2 , a 2 ) w 2 ( y ) d y ) 1 / p × ( G 1 B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 2 ( y ) Q 2 q v ( x , y ) d x d y ) 1 / q < .

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., w( x 1 , x 2 )= w 1 ( x 1 ) w 2 ( x 2 ), such that W i ()=, i=1,2, and v is any weight on G 1 × G 2 , then H 1 , 1 is bounded from L dec , r p (w, G 1 ) to L q (v, G 2 ) if and only if

( G 1 × G 2 ( B ( e 1 , r 1 ( x ) ) B ( e 2 , r 2 ( x ) ) [ G 1 B ( e 1 , r 1 ( t ) ) G 1 B ( e 2 , r 2 ( τ ) ) g ( s , ε ) d s d ε ] d t d τ ) p × W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p c ( G 1 × G 2 g q ( x , y ) v 1 q ( x , y ) d x d y ) 1 / q , g 0 .
(4.2)

Further, we have

B ( e 1 , r 1 ( x ) ) × B ( e 2 , r 2 ( x ) ) ( G 1 B ( e 1 , r 1 ( t ) ) G 2 B ( e 2 , r 2 ( t ) ) g ( s , ε ) d s d ε ) d t d τ = B ( e 1 , r 1 ( x ) ) B ( e 2 , r 2 ( x ) ) r 1 Q 1 ( t ) r 2 Q 2 ( τ ) g ( t , τ ) d t d τ + r 1 Q 1 ( x ) G 1 B ( e 1 , r 1 ( x ) ) B ( e 2 , r 2 ( y ) ) r 2 Q 2 ( τ ) g ( t , τ ) d t d τ + r 2 Q 2 ( y ) B ( e 1 , r 1 ( x ) ) G 2 B ( e 2 , r 2 ( y ) ) r 1 Q 1 ( t ) g ( t , τ ) d t d τ + r 1 Q 1 ( x ) r 2 Q 2 ( y ) G 1 B ( e 1 , r 1 ( x ) ) G 2 B ( e 2 , r 2 ( y ) ) g ( t , τ ) d t d τ = : I ( 1 ) ( x , y ) + I ( 2 ) ( x , y ) + I ( 3 ) ( x , y ) + I ( 4 ) ( x , y ) .

It is obvious that (4.2) holds if and only if

( G 1 × G 2 ( I ( j ) ) p ( x , y ) W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p c ( G 1 × G 2 g q ( x , y ) v 1 q ( x , y ) d x d y ) 1 / q
(4.3)

for j=1,2,3,4. By using Proposition 4.1 (part (i)) we find that

( G 1 × G 2 ( I ( 1 ) ) p ( x , y ) W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p c ( G 1 × G 2 g q ( x , y ) v 1 q ( x , y ) d x d y ) 1 / q

if and only if

( G 1 B ( e 1 , t ) G 2 B ( e 2 , τ ) W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p × ( B ( e 1 , t ) × B ( e 2 , τ ) ( v 1 q ( x , y ) r 1 Q 1 q ( x ) r 2 Q 2 q ( y ) ) 1 q d x d y ) 1 / q = c p ( B ( e 1 , t ) × B ( e 2 , τ ) w ( x , y ) d x d y ) 1 / p × ( B ( e 1 , t ) × B ( e 2 , τ ) v ( x , y ) r 1 Q 1 q ( x ) r 2 Q 2 q ( y ) d x d y ) 1 / q C .

In the latter equality we used the equality

( G i B ( e i , t ) W i p ( r i ( x ) ) w i ( x ) d x ) 1 / p = ( B ( e i , t ) w i ( x ) d x ) 1 / p ,i=1,2,

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 j=4 if and only if condition (ii) is satisfied, while Proposition 4.1 (parts (iii) and (iv)) and the following observation:

sup a 1 , a 2 > 0 ( G 1 B ( e 1 , a 1 ) w 1 ( x ) W 1 p ( r 1 ( x ) ) d x ) 1 / p ( B ( e 2 , a 2 ) r 2 p Q 2 ( y ) W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p × ( B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) r 1 Q 1 q ( x ) v ( x , y ) d x d y ) 1 / q = c p sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) w 1 ( x ) d x ) 1 / p ( B ( e 2 , a 2 ) r 2 Q 2 p ( y ) W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p × ( B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) r 1 Q q ( x ) v ( x , y ) d x d y ) 1 / q < ; sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) r 1 Q 1 p ( x ) W 1 p ( r 1 ( x ) ) w 1 ( x ) d x ) 1 / p ( G 2 B ( e 2 , a 2 ) w 2 ( y ) W 2 p ( r 2 ( y ) ) d y ) 1 / p × ( G 1 B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 2 Q 2 q ( y ) v ( x , y ) d x d y ) 1 / q = c p sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) r 1 Q 1 p ( x ) W 1 p ( r 1 ( x ) ) w 1 ( x ) d x ) 1 / p ( B ( e 2 , a 2 ) w 2 ( t 2 ) d t 2 ) 1 / p × ( G 1 B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 2 Q 2 q ( y ) v ( x , y ) d x d y ) 1 / q <

yield (4.3) for j=2,3. □

Let

( J α 1 , α 2 f ) ( x , y ) = B ( e 1 , 2 c 0 r 1 ( x ) ) B ( e 2 , 2 c 0 r 2 ( y ) ) f ( t , τ ) r 1 ( x t 1 ) α 1 Q 1 r 2 ( y τ 1 ) α 2 Q 2 d t d τ , ( J α 1 S α 2 f ) ( x , y ) = B ( e 1 , 2 c 0 r 1 ( x ) ) G 2 B ( e 2 , 2 c 0 r 2 ( y ) ) f ( t , τ ) r 1 ( x t 1 ) α 1 Q 1 r 2 ( y τ 1 ) α 2 Q 2 d t d τ , ( S α 1 J α 2 f ) ( x , y ) = G 1 B ( e 1 , 2 c 0 r 1 ( x ) ) B ( e 2 , 2 c 0 r 2 ( y ) ) f ( t , τ ) r 1 ( x t 1 ) α 1 Q 1 r 2 ( y τ 1 ) α 2 Q 2 d t d τ , ( S α 1 , α 2 f ) ( x , y ) = G 1 B ( e 1 , 2 c 0 r 1 ( x ) ) G 2 B ( e 2 , 2 c 0 r 2 ( y ) ) f ( t , τ ) r 1 ( x t 1 ) α 1 Q 1 r 2 ( y τ 1 ) α 2 Q 2 d t d τ ,

where c 0 is the constant from the triangle inequality for the homogeneous norms r 1 and  r 2 .

It is obvious that

I α 1 , α 2 f= 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.
(4.4)

Now we formulate the main result of this section.

Theorem 4.1 Let 1<pq<. 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 D C α i , p (G), i=1,2, or vDC(x)DC(y). Then the operator I α 1 , α 2 is bounded from L dec , r p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if the following conditions are satisfied:

  1. (i)
    A 1 : = sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) w ( x , y ) d x d y ) 1 / p × ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) ( r 1 α 1 ( x ) r 2 α 2 ( y ) ) q v ( x , y ) d x d y ) 1 / q < ;
  2. (ii)
    A 2 : = sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 1 Q 1 p ( x ) r 2 Q 2 p ( y ) W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p × ( G 1 B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) ( r 1 α 1 Q 1 ( x ) r 2 α 2 Q 2 ( y ) ) q v ( x , y ) d x d y ) 1 / q < ;
  3. (iii)
    A 3 : = sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) w 1 ( x ) d x ) 1 / p ( B ( e 2 , a 2 ) r 2 Q 2 p ( y ) W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p × ( B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) r 1 α 1 q ( x ) r 2 q ( α 2 Q 2 ) ( y ) v ( x , y ) d x d y ) 1 / q < ;
  4. (iv)
    A 4 : = sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) r 1 Q 1 p ( x ) W 1 p ( r 1 ( x ) ) w 1 ( x ) d x ) 1 / p ( B ( e 2 , a 2 ) w 2 ( y ) d y ) 1 / p × ( G 1 B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 1 q ( α 1 Q 1 ) ( x ) r 2 q α 2 ( y ) v ( x , y ) d x d y ) 1 / q < ;
  5. (v)
    A 5 : = sup a 1 , a 2 > 0 ( G 1 B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) r 1 α 1 p ( x ) r 2 α 2 p ( y ) W p ( r 1 ( x ) , r 2 ( y ) ) w ( x , y ) d x d y ) 1 / p × ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) v ( x , y ) d x d y ) 1 / q < ;
  6. (vi)
    A 6 : = sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) w 1 ( x ) d x ) 1 / p ( G 2 B ( e 2 , a 2 ) r 2 α 2 p ( y ) W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p × ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 1 α 1 q ( x ) v ( x , y ) d x d y ) 1 / q < ;
  7. (vii)
    A 7 : = sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) r 1 Q 1 p ( x ) W 1 p ( r 1 ( x ) ) w 1 ( x ) d x ) 1 / p × ( G 2 B ( e 2 , a 2 ) r 2 α 2 p ( y ) W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p × ( G 1 B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 1 ( α 1 Q 1 ) q ( x ) v ( x , y ) d x d y ) 1 / q < ;
  8. (viii)
    A 8 : = sup a 1 , a 2 > 0 ( G 2 B ( e 1 , a 1 ) r 1 α 1 p ( x ) W 1 p ( r 1 ( x ) ) w 1 ( x ) d x ) 1 / p ( B ( e 2 , a 2 ) w 2 ( y ) d y ) 1 / p × ( B ( e 1 , a 1 ) B ( e 2 , a 2 ) r 2 α 2 q ( x ) v ( x , y ) d x d y ) 1 / q < ;
  9. (ix)
    A 9 : = sup a 1 , a 2 > 0 ( B ( e 1 , a 1 ) r 2 Q 2 p ( y ) W 2 p ( r 2 ( y ) ) w 2 ( y ) d y ) 1 / p × ( G 1 B ( e 1 , a 1 ) r 1 α 1 p ( x ) W 1 p ( r 1 ( x ) ) w 1 ( x ) d x ) 1 / p × ( B ( e 1 , a 1 ) G 2 B ( e 2 , a 2 ) r 2 ( α 2 Q 2 ) q ( y ) v ( x , y ) d x d y ) 1 / q < .

Proof Let us assume that vDC(x)DC(y). The case when w i D C 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.

Since fDR( G 1 × G 2 ) by using the arguments of the proof of Proposition 3.1 it can be checked that

( J α 1 , α 2 f)(x,y) r 1 α 1 Q 1 (x) r 2 α 2 Q 2 (y) B ( e 1 , r 1 ( x ) ) × B ( e 2 , r 2 ( y ) ) f(t,τ)dtdτ

(see also [4] for a similar estimate in the case of the multiple one-sided potentials on R + 2 ). Hence, by Proposition 4.2 we find that J α 1 , α 2 is bounded from L dec , r p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) if and only if conditions (i)- (iv) hold.

Observe that the dual to S α 1 , α 2 is given by

( S α 1 , α 2 g ) (x,y)= B ( e 1 , r 1 ( x ) / ( 2 c 0 ) ) × B ( e 2 , r 2 ( y ) / ( 2 c 0 ) ) g(t,τ) r 1 α 1 Q 1 ( x t 1 ) r 2 α 2 Q 2 ( y τ 1 ) dtdτ.

Further, Tonelli’s theorem together with Lemma 3.1 for both variables implies that there are positive constants c 1 and c 2 such that for all (x,y) G 1 × G 2 for the dual (see also the proof of Lemma 3.2),

r 1 α 1 ( x ) r 2 α 2 ( y ) B ( e 1 , r 1 ( x ) / ( 4 c 0 ) ) × B ( e 2 , r 2 ( y ) / ( 4 c 0 ) ) g ( t , τ ) d t d τ c 1 B ( e 1 , r 1 ( x ) ) × B ( e 2 , r 2 ( y ) ) ( S α 1 , α 2 g ) ( t , τ ) d t d τ c 2 r 1 α 1 ( x ) r 2 α 2 ( y ) B ( e 1 , r 1 ( x ) / ( 2 c 0 ) ) × B ( e 2 , r 2 ( y ) / ( 2 c 0 ) ) g ( t , τ ) d t d τ .

Applying Propositions 4.1 and 4.2 with the condition that vDC( G 1 × G 2 ) we find that the operator S α 1 , α 2 is bounded from L dec , r p (w, G 1 × G 2 ) to L q (v, G 1 × G 2 ) 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

( J α 1 S α 2 f)(x,y)( H α 1 S α 2 f)(x,y),

where

( H α 1 S α 2 f)(x,y)= r 1 α 1 Q 1 (x) B ( e 1 , 2 c 0 r 1 ( x ) ) G 2 B ( e 2 , 2 c 0 r 2 ( y ) ) f(t,τ) r 2 ( y τ 1 ) α 2 Q 2 dtdτ.

Dual of H α 1 S α 2 is given by

( H α 1 S α 2 g ) (t,τ)= G 1 B ( e 1 , r ( t ) ) B ( e 2 , r ( τ ) / 2 c 0 ) r 1 α 1 Q 1 (s) r 2 α 2 Q 2 ( ε τ 1 ) f(s,ε)dsdε.

Further, we have

T ( x , y ) : = B ( e 1 , r 1 ( x ) ) × B ( e 2 , r 2 ( y ) ) ( H α 1 S α 2 g ) ( t , τ ) d t d τ = B ( e 1 , r 1 ( x ) ) × B ( e 2 , r 2 ( y ) ) ( B ( e 1 , r 1 ( x ) ) B ( e 1 , r ( t ) ) B ( e 2 , r ( τ ) / 2 c 0 ) r 1 α 1 Q 1 ( s ) × r 2 α 2 Q 2 ( τ ε 1 ) f ( s , ε ) d s d ε ) d t d τ + B ( e 1 , r 1 ( x ) ) × B ( e 2 , r 2 ( y ) ) ( G 1 B ( e 1 , r 1 ( x ) ) B ( e 2 , r ( τ ) / ( 2 c 0 ) ) r 1 α 1 Q 1 ( s ) × r 2 α 2 Q 2 ( τ ε 1 ) f ( s , ε ) d s d ε ) d t d τ = : T 1 ( x , y ) + T 2 ( x , y ) .

Tonelli’s theorem for G 1 , the inequality r 2 α 2 Q 2 (τ ε 1 )c r 2 α 2 Q 2 (y) for τB( e 2 ,r(y)), εB( e 2 ,r(τ)/(2 c 0 )), and the fact that the integral B ( e 1 , τ ) f(s,ε)ds is decreasing in τ uniformly to ε yield

T 1 ( x , y ) c r 2 α 2 Q 2 ( y ) × B ( e 1 , r 1 ( x ) ) B ( e 2 , r 2 ( y ) ) B ( e 2 , r 2 ( y ) / 2 ) ( B ( e 1 , r 1 ( x ) ) B ( e 1 , r ( t ) ) B ( e 2 , r 2