WEIGHTED INEQUALITIES FOR THE LAPLACE TRANSFORM

WEIGHTED INEQUALITIES

FOR THE LAPLACE TRANSFORM

Yves Rakotondratsimba

(Received November 11, 1997)

Abstract. Necessary conditions and sucient conditions are given in order

that the Laplace Transform is bounded between two Lebesgue spaces with weights. Such a boundedness is characterized for a large class of weights. AMS1991 Mathematics Subject Classi
cation. 26D15, 26A33.

Key words and phrases. Weighted inequalities, Laplace transform, Hardy op-erators.

x

1 Introduction and Results

The Laplace Transform is de
ned as (L

f

)(

x

) =

Z

1

0

f

(

y

)exp;

xy

]

dy

0

< x <

1

:

Throughout this paper it is assumed that 1

< p
q <

1

p

0 =

_p

p

;1

q

0 =

_q

q

;1

r

=

_p

qp

;

q

and

u

(

:

)

v

(

:

)

v

1;p 0 (

:

)

are weight functions, i.e. nonnegative and locally integrable functions. Our purpose is to derive necessary conditions and sucient conditions on

u

(

:

) and

v

(

:

) for which L is bounded from the Lebesgue space

L

pv =

L

p_(]0

1

v

(

x

)

dx

) into

L

qu =

L

q(]0

1

u

(

x

)

dx

). That is for some constant

C >

0 (1.1)
Z 1 0 (L

f

)q(

x

)

u

(

x

)

dx

 1 q 

C

Z 1 0

f

p₍

_x

₎

_v

₍

_x

₎

_dx

 1 p _{for all}

_f

₍

_:

₎ 0

:

For convenience this boundedness is also denoted by L:

L

pv!

L

qu.

This problem has been investigated by many authors. Indeed a sucient condition ensuring (1

:

1) was given by K. Andersen and H. Heinig An-Hg],

Hg] (see also Hz2], An]). And a sucient condition close to be necessary

was found by S. Bloom Bm].

Our present contribution is
rst to provide variant sucient conditions for

L :

L

pv !

L

qu improving the result of S. Bloom Bm] whenever

q >

4. The

technic used in Bm] is to reduce the problem (1

:

1) to weighted estimates involving Hardy operator and Hardy antidierentiation operator. As in An-Hg] and An-Hg], our approach is based on the weighted inequality

(1.2)
Z 1 0 (

Tf

)q(

x

)

u

(

x

)

dx

 1 q 

C

Z 1 0

f

p(

x

)

v

(

x

)

dx

 1 p _{for all}

_f

₍

_:

₎ 0 where (

Tf

)(

x

) = Z x 0

f

(

y

)exp;

xy

;1]

dy

0

< x <

1

:

Although a sucient condition for (1

:

2) is available in An-Hg], Hg] and Hz2], here we give a new one which is not too far to be necessary for

T

:

L

_pv!

L

qu.

The second contribution in this work is to provide another approach for L :

L

pv !

L

qu which can be extended to treat boundednesses problems for many

integral operators in higher dimension.

Our method for dealing with (1

:

2) is
rst to break the operator

T

into small pieces and next to do summations just by using Holder inequalities and the fast increase of the exponential function.

Our
rst result reads as

Theorem 1.

Suppose that L:

L

pv!

L

qu. Then for some constant

A >

0

(1.3)
Z R ;1 0

u

(

x

)

dx

 1 q
Z R 0

v

1;p 0 (

y

)

dy

 1 p0 

A

for all

R >

0 and (1.4)
Z 1 R;1

u

(

x

)exp;

qRx

]

dx

 1 q
Z R 0

v

1;p 0 (

y

)

dy

 1 p0 

A

for all

R >

0

:

Conversely let

p



q

. Then L :

L

pv !

L

qu whenever for some

A >

0 and

0

< "

1 (1.5)
Z 1 R;1

u

(

x

)exp;4 ;1 (1;

"

)

qRx

]

dx

 1 q 
Z R 2 ;1R

v

1;p 0 (

y

)

dy

 1 p0 

A

for all

R >

0

and the condition (1

:

3) is satised.

A similar result was previously obtained by S. Bloom Bm] with the condi-tion (1

:

5) replaced by (1.6)
Z 1 R;1

u

(

x

)exp;

Rx

]

dx

 1 q
Z R 0

v

1;p 0 (

y

)

dy

 1 p0 

A

for all

R >

0

:

Therefore our Theorem improves Bloom's one whenever

q >

4.

In order to get a characterization result for certain weights, it would be useful to see connections between conditions (1

:

3) and (1

:

5).

Lemma 2.

Condition (1

:

3) implies (1

:

5) under one of the following assump-tions i)

u

(

:

) is a decreasing function ii)

v

1;p 0 (

:

)2

D

 iii)

u

(

:

)2

D



iv)

v

(

:

) is a increasing function.

Here

w

(

:

)2

D

means that for some

C >

0 Z 2R 0

w

(

y

)

dy



C

Z R 0

w

(

y

)

dy

for all

R >

0

:

Theorem 1 and Lemma 2 can be combined to get

Corollary 3.

For

p



q

, the boundedness L :

L

pv !

L

qu is equivalent to

condition (1

:

3) whenever one of the following assumptions is satised: i)

u

(

:

) is a decreasing function

ii)

v

1;p 0

(

:

)2

D



iii)

v

(

:

) is an increasing function iv)

u

(

:

)2

D

.

Our second main result concerning the case

q < p

is

Theorem 4.

Let

q < p

. Suppose that L:

L

pv!

L

qu. Then

(1.7) Z 1 0 
Z x ;1 0

u

(

y

)

dy

 1 q
Z x 0

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1 and (1.8) Z 1 0 
Z 1 x;1

u

(

y

)exp;

qxy

]

dy

 1 q
Z x 0

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1

:

ConverselyL:

L

pv!

L

qu whenever for some 0

< "

1

(1.9) Z 1 0 
Z 1 x;1

u

(

y

)exp;4 ;1(1 ;

"

)

qxy

]

dy

 1 q 
Z x 2 ;1x

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1

:

and the condition (1

:

7) is satised.

Following S. Blom Bm] for

q < p

, the boundedness L :

L

pv !

L

qu holds

whenever (1.10) Z 1 0 
Z 1 x;1

u

(

y

)exp;

xy

]

dy

 1 q
Z x 0

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1

:

and the condition (1

:

7) is satis
ed. So this author's result is improved in Theorem 4 whenever

q >

4.

As we have announced above, Theorems 1 and 4 are based on a sucient condition for the boundedness

T

:

L

pv!

L

qu, which is described by

Theorem 5.

For

p



q

, the boundedness

T

:

L

pv !

L

qu holds if for some

constants

A >

0 and 0

< "

1 (1.11)
Z R 2 ;1R

u

(

x

)

dx

 1 q
Z R 0

v

1;p 0 (

y

)exp;4 ;1(1 ;

"

)

p

0

Ry

;1]

dy

 1 p0 

A

for all

R >

0.

And for

q < p

, then

T

:

L

pv!

L

qu whenever

(1.12) Z 1 0 
Z x 2 ;1x

u

(

z

)

dz

 1 p 
Z 2x 0

v

1;p 0 (

y

)exp;4 ;1(1 ;

"

)

p

0

xy

;1]

dy

 1 p0 r

u

(

x

)

dx <

1

:

Actually the above boundedness holds under the condition (1.13) 1 X k=;1 
Z 2 ;k 2 ;(k+1)

u

(

x

)

dx

 1 q 
Z 2 ;k 0

v

1;p 0 (

y

)exp;4 ;1(1 ;

"

)

p

02;k

y

;1]

dy

 1 p0 r

<

1 which is implied by (1

:

12).

A necessary condition for

T

:

L

pv !

L

qu to be true is that

(1.14)
Z R 2 ;1R

u

(

x

)

dx

 1 q
Z 2 ;1R 0

v

1;p 0 (

y

)exp;

p

0

Ry

;1 ]

dy

 1 p0 

A

for all

R >

0.

Condition (1

:

11) seems new and quite dierent from those given in An-Hg], An] and Hz1]. Moreover it appears not too far from the necessary condition (1

:

14).

While this paper was submitted, the author has been extended the present approach to treat boundednesses problems for the two-dimensional Laplace transform.

x

2 Proofs of Results

First we prove Theorem 1, Lemma 2 and Theorem 4. And next we give the proof of Theorem 5 on which Theorems 1 and 4 are based.

Proof of Theorem 1

The Necessary Part.

Suppose that L :

L

pv !

L

qu. To get condition (1

:

4), let

R >

0 and

f

(

:

) a

nonnegative function whose support is included in ]0

R

. Since for 0

< x <

1

(L

f

)(

x

) = Z R 0

f

(

y

)exp;

yx

]

dy


exp;

Rx

]  Z R 0

f

(

y

)

dy

inequality (1

:

1) yields
Z R 0

f

(

y

)

dy

q Z 1 R;1

u

(

x

)exp;

qRx

]

dx



C

q
Z R 0

f

p₍

_x

₎

_v

₍

_x

₎

_dx

qp

:

Taking

f

(

:

) =

v

1;p 0

(

:

) on its support ]0

R

 and using the identity (1;

p

0)

p

+1 =

(1;

p

0) then condition (1

:

4) appears immediately.

To get condition (1

:

3) the key is to observe that (L

f

)(

x

)exp(;1)](

Hf

)(

x

;1) = exp( ;1)] Z x ;1 0

f

(

y

)

dy

0

< x <

1

:

So inequality (1

:

1) yields

H

:

L

pv !

L

qw with

w

(

x

) =

u

(

x

;1)

x

;2

:

As it is well-known in Oc-Kr], this last boundedness implies that

(2.1)
Z 1 R

w

(

x

)

dx

 1 q
Z R 0

v

1;p 0 (

y

)

dy

 1 p0 

A

for all

R >

0

and for some
xed constant

A >

0. By the de
nition of

w

(

:

), (2

:

1) is nothing else than condition (1

:

3).

The SucientPart.

As used in An-Hg] and Hg], the main point is to observe that (L

f

)(

x

)

(

Hf

)(

x

;1) + (

T



f

)(

x

;1) 

0

< x <

1

where the dual

T

 of the operator

T

is de
ned by

(

T



f

)(

y

) = Z 1 y

f

(

x

)exp;

xy

;1]

dx

0

< y <

1

and Z 1 0 (

T



f

)(

y

)

g

(

y

)

dy

= Z 1 0

f

(

x

)(

Tg

)(

x

)

dx

. With the above observation to derive (1

:

1) it remains to get

(2.2.)

H

:

L

_pv !

L

qw and

T

:

L

pv !

L

qw

As it is known in Oc-Kr], for

p



q

, the boundedness

H

:

L

pv!

L

qw holds

whenever the test condition (2

:

1) (which is the same as (1

:

3)) holds. By duality arguments,

T

:

L

pv !

L

qw is equivalent to

T

:

L

q0 w1;q 0 !

L

p0 v1;p 0.

And by Theorem 5, this last boundedness holds whenever for some

A >

0 and 0

< "

1
Z R 0

w

(

y

)exp;4 ;1(1 ;

"

)

qRy

;1]

dy

 1 q
Z R 2 ;1R

v

1;p 0 (

x

)

dx

 1 p0 

A

for all

R >

0. Using the de
nition of

w

(

:

), clearly the validity of this last inequality is ensured by condition (1

:

5).

Proof of Lemma 2

Condition (1

:

3) implies (1

:

5) whenever

u

(

:

) is a decreasing function since for

R >

0 Z 1 R;1

u

(

x

)exp;4 ;1(1 ;

"

)

qRx

]

dx



u

(

R

;1) Z 1 R;1 exp;4 ;1(1 ;

"

)

qRx

]

dx

=

u

(

R

;1 )

R

;1 Z 1 1 exp;4 ;1 (1;

"

)

qz

]

dz

=

c

1

u

(

R

;1 )

R

;1 

c

1 Z R ;1 0

u

(

y

)

dy:

The implication (1

:

3) =)(1

:

5) for

v

1;p

0

(

:

)2

D

can be seen as follows
Z 1 R;1

u

(

x

)exp;4 ;1(1 ;

"

)

qRx

]

dx


Z R 2 ;1R

v

1;p 0 (

y

)

dy

qp 0 = 1 X k=1
Z 2kR ;1 2k ;1R;1

u

(

x

)exp;4 ;1(1 ;

"

)

qRx

]

dx


Z R 2 ;1R

v

1;p 0 (

y

)

dy

qp 0  1 X k=1 h exp;8 ;1(1 ;

"

)

q

2k] i
Z 2kR ;1 2k ;1R;1

u

(

x

)

dx


Z R 0

v

1;p 0 (

y

)

dy

qp 0  1 X k=1 2k
h exp;8 ;1(1 ;

"

)

q

2k] i
Z 2kR ;1 0

u

(

x

)

dx


Z 2 ;kR 0

v

1;p 0 (

y

)

dy

qp 0 

A

q 1 X k=1 2k
h exp;8 ;1(1 ;

"

)

q

2k] i =

c

2

A

q

_:

Here we have used the fact that

w

(

:

)2

D

implies that for some

 >

0 Z 2mR 0

w

(

y

)

dy

2m
Z R 0

w

(

y

)

dy

for all

R >

0 and all integers

m

1

:

The implication (1

:

3) =) (1

:

5), for

u

(

:

) 2

D

, can be easily seen as above.

This implication is also true provided

v

(

:

) is an increasing function, since in this case

v

1;p

0

(

:

) is a decreasing function and consequently

v

1;p 0

(

:

)2

D

.

Proof of Theorem 4

The Necessary part

As in the proof of Theorem 1, to get the condition (1

:

7), the key is to observe that the boundedness L :

L

pv !

L

qu implies the weighted Hardy inequality

H

:

L

pv!

L

qw with

w

(

:

) de
ned as in (2

:

2). As it is well-known in Oc-Kr], for

q < p

, this last boundedness implies that

(2.3) Z 1 0 
Z 1 x

w

(

y

)

dy

 1 q
Z x 0

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1

:

Using the de
nition of

w

(

:

), then (2

:

3) is nothing else than condition (1

:

7). Condition (1

:

8) can be derived immediatly from (L

u

)(

z

)(

T



u

)(

z

;1) and the inequality Z 1 0 
L

u

)(

qx

)  1 q
Z x 0

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1

:

The fact that this last is a necessary condition forL:

L

pv!

L

qu was proved in

Bm].

The Sucientpart.

As we have explained in the proof of Theorem 1, it remains to get the boundednesses of

H

and

T

as in (2

:

2).

For

q < p

, again by well-known results as written in Oc-Kr], then

H

:

L

_pv!

L

qw whenever the test condition (2

:

3) (which is the same as (1

:

7)) holds.

Theorem 5 applied to

q < p

and

T

:

L

q0

w1;q 0

!

L

p0

v1;p

0, lead to state that

the boundedness

T

:

L

pv !

L

qw holds whenever Z 1 0 
Z x 2 ;1x

v

1;p 0 (

y

)

dy

1 q0 
Z 2x 0

w

(

z

)exp;4 ;1(1 ;

"

)

qxz

;1]

dz

 1 qr

v

1;p 0 (

x

)

dx <

1

:

Clearly, after using the de
nition of

w

(

:

), the validity of this inequality is equivalent to Z 1 0 
Z 1 (2x) ;1

u

(

y

)exp;4 ;1 (1;

"

)

qxy

]

dy

 1 q 
Z x 2 ;1x

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1

:

This inequality will be a consequence of condition (1

:

9) and Z 1 0 
Z x;1 (2x) ;1

u

(

y

)exp;4 ;1(1 ;

"

)

qxy

]

dy

 1 q 
Z x 2 ;1x

v

1;p 0 (

z

)

dz

 1 q0 r

v

1;p 0 (

x

)

dx <

1

:

This last inequality is true because of condition (1

:

7) and exp;4 ;1(1 ;

"

)

qxy

]

<

exp;8 ;1(1 ;

"

)

q

] whenever (2

x

) ;1

< y < x

;1.

Proof of Theorem 5

The Sucientpart

The main point to get

T

:

L

_pv!

L

qu is to cut the operator as

(2.4) Z 1 0 (

Tf

)q(

x

)

u

(

x

)

dx

 1 X k=;1  1 X j=0



j
Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

 1 p_ k q

for all functions

f

(

:

)0. Here



j = exp;

"

2 j_{] with 0}

_{< "}

1 and k=k(

p
q
v
u

) =
Z 2 ;k 2 ;(1+k)

u

(

y

)

dy

1 q
Z 2 ;k 0

v

1;p 0 (

x

)exp;4 ;1(1 ;

"

)

p

02;k

x

;1]

dx

 1 p0

:

It can be noted that 1 X

l=0



l =

c

0

<

1. We will postpone below the proof of

(2

:

4).

Now consider the case

p



q

. By condition (1

:

11) (with

R

= 2

;k) then

k 

A:

Using this last fact and the cutting out (2

:

4), the boundedness

T

:

L

pv !

L

qu

appears as follows Z 1 0 (

Tf

)q(

x

)

u

(

x

)

dx

 1 X k=;1  1 X j=0



j
Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

 1 p_ k q 

A

q 1 X k=;1  1 X j=0



j
Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

 1 pq



A

q 1 X k=;1  1 X j=0



j Z 2 ;(j+k) 2 ;(2+j+k)

f

p₍

_x

₎

_v

₍

_x

₎

_dx

qp h 1 X l=0



l iqp 0 

c

1

A

q 1 X j=0



j 1 X k=;1 Z 2 ;(j+k) 2 ;(2+j+k)

f

p₍

_x

₎

_v

₍

_x

₎

_dx

qp since

q

_p

1 

c

2

A

qh 1 X j=0



j iqp
Z 1 0

f

p₍

_x

₎

_v

₍

_x

₎

_dx

qp =

c

3

A

q
Z 1 0

f

p₍

_x

₎

_v

₍

_x

₎

_dx

qp

:

For the case

q < p

, observe that by condition (1

:

13)

1

X

k=;1

rk

< A

r

:

We will dier below the proof of the fact that (1

:

13) is implied by condition (1

:

12).

Using this observation, the boundedness

T

:

L

pv!

L

qu appears as follows Z 1 0 (

Tf

)q(

x

)

u

(

x

)

dx

 1 X k=;1  1 X j=0



j
Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

 1 p_kq  1 X k=;1  1 X j=0



j Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

qp q_kh 1 X l=0



l iqp 0 

c

1 1 X k=;1  1 X j=0



j Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

qp q_k 

c

1  1 X k=;1 1 X j=0



j Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

qp 1 X m=;1 rm  1;qp 

c

1

A

q 1 X j=0



j 1 X k=;1 Z 2 ;(j+k) 2 ;(2+j+k)

f

p(

x

)

v

(

x

)

dx

qp 

c

2

A

qh 1 X j=0



j iqp
Z 1 0

f

p(

x

)

v

(

x

)

dx

qp 

c

3

A

q
Z 1 0

f

p(

x

)

v

(

x

)

dx

qp

:

It is now time to prove (2

:

4). This inequality is true since

Z 1 0 (

Tf

)q(

x

)

u

(

x

)

dx

= Z 1 0 h 1 X j=0 Z 2 ;jx 2 ;(j+1)x

f

(

y

)exp;

xy

;1]

dy

iq

u

(

x

)

dx

= 1 X k=;1 Z 2 ;k 2 ;(k+1) h 1 X j=0 Z 2 ;jx 2 ;(j+1)x

f

(

y

)exp;

xy

;1]

dy

iq

u

(

x

)

dx

 1 X k=;1  1 X j=0
Z 2 ;(j+k) 2 ;(j+2+k)

f

(

y

)

dy

 exp;2j] q
Z 2 ;k 2 ;(k+1)

u

(

x

)

dx

  1 X k=;1  1 X j=0
Z 2 ;(j+k) 2 ;(j+2+k)

f

p₍

_y

₎

_v

₍

_y

₎

_dy

 1 p 
Z 2 ;(j+k) 2 ;(j+2+k)

v

1;p 0 (

z

)

dz

 1 p0
Z 2 ;k 2 ;(k+1)

u

(

x

)

dx

 1 q _exp ;2j] q  1 X k=;1  1 X j=0 exp;

"

2j]
Z 2 ;(j+k) 2 ;(j+2+k)

f

p₍

_y

₎

_v

₍

_y

₎

_dy

 1 p 
Z 2 ;(j+k) 2 ;(j+2+k)

v

1;p 0 (

z

)exp;4 ;1 (1;

"

)

p

0 2;k

z

;1 ]

dz

 1 p0 
Z 2 ;k 2 ;(k+1)

u

(

x

)

dx

 1 qq  1 X k=;1  1 X j=0 exp;

"

2j]
Z 2 ;(j+k) 2 ;(j+2+k)

f

p₍

_y

₎

_v

₍

_y

₎

_dy

 1 p 
Z 2 ;k 0

v

1;p 0 (

z

)exp;4 ;1(1 ;

"

)

p

02;k

z

;1]

dz

 1 p0
Z 2 ;k 2 ;(k+1)

u

(

x

)

dx

 1 qq = 1 X k=;1  1 X j=0



j
Z 2 ;(j+k) 2 ;(j+2+k)

f

p₍

_y

₎

_v

₍

_y

₎

_dy

 1 p_ k q

:

Now we can show how does the condition (1

:

12) imply (1

:

13). Here the main point is the elementary equality

Z b a

u

(

z

)

dz

rq =

r

_q

Z b a
Z x a

u

(

z

)

dz

rp

u

(

x

)

dx

0

< a < b <

1

:

For shortness the term 4;1(1 ;

"

)

p

0 is merely denoted by

C

. The implication

(1

:

12) =)(1

:

13) can be seen as follows 1 X k=;1 
Z 2 ;k 0

v

1;p 0 (

y

)exp;

C

2 ;k

y

;1]

dy

 1 p0
Z 2 ;k 2 ;(k+1)

u

(

x

)

dx

 1 qr 1 X k=;1
Z 2 ;k 0

v

1;p 0 (

y

)exp;

C

2 ;k

y

;1]

dy

rp 0  Z 2 ;k 2 ;(k+1)
Z x 2 ;(k+1)

u

(

z

)

dz

rp

u

(

x

)

dx

= 1 X k=;1 Z 2 ;k 2 ;(k+1) 
Z x 2 ;(k+1)

u

(

z

)

dz

 1 p 
Z 2 ;k 0

v

1;p 0 (

y

)exp;

C

2 ;k

y

;1]

dy

1 p0 r

u

(

x

)

dx

 1 X k=;1 Z 2 ;k 2 ;(k+1) 
Z x 2 ;1x

u

(

z

)

dz

 1 p 
Z 2x 0

v

1;p 0 (

y

)exp;

Cxy

;1]

dy

 1 p0 r

u

(

x

)

dx

= Z 1 0 
Z x 2 ;1x

u

(

z

)

dz

 1 p
Z 2x 0

v

1;p 0 (

y

)exp;

Cxy

;1 ]

dy

 1 p0 r

u

(

x

)

dx



A

r _{by using (1}

_:

₁₂₎

_:

The Necessary part

To get the condition (1

:

14) from the boundedness

T

:

L

pv !

L

qu, by a duality

argument, it can be assumed that

T

 :

L

q 0 u1;q 0 !

L

p0 v1;p 0 or equivalently for some constant

C >

0
Z 1 0 h Z 1 y

f

(

x

)

u

(

x

)exp;

xy

;1]

dx

ip 0

v

1;p 0 (

y

)

dy

 1 p0 

C

Z 1 0

f

q0 (

z

)

u

(

z

)

dz

 1 q0 for all

f

(

:

)0

:

Take

R >

0 and

f

(

:

) a nonnegative function whose support is included ]2;1

R
R

. Since for 0

< y <

2;1

R

Z 1 y

f

(

x

)

u

(

x

)exp;

xy

;1]

dx

= Z R 2 ;1R

f

(

x

)

u

(

x

)exp;

xy

;1]

dx


Z R 2 ;1R

f

(

x

)

u

(

x

)

dx

 exp;

Ry

;1]

the above inequality yields

Z R 2 ;1R

f

(

x

)

u

(

x

)

dx

p 0 Z 2 ;1R 0

v

1;p 0 (

y

)exp;

p

0

Ry

;1]

dy



C

p 0
Z R 2 ;1R

f

q0 (

z

)

u

(

z

)

dz

p 0 q0

:

Taking

f

(

:

) = 1 on its support ]2;1

R
R

 then condition (1

:

14) appears

immedi-ately.

Acknowledgement

The author would like to thank the referee for his suggestions on improve-ments of some results in the preliminary version of this work.

References

An] K. Andersen, Weighted inequalities for convolutions, Proc. Amer. Math. Soc.

123(1995), 1129-1136.

An-Hg] K. Andersen, H. Heinig, Weighted norm inequalities for certain integral oper-ators, SIAM J. Math. Anal.14(1983), 1983.

Bm] S. Bloom, Hardy integral estimates for the Laplace Transform, Proc. Amer. Math. Soc.116(1992), 417-426.

Hg] H. Heinig, Weighted inequalities for certain integral operators II, Proc. Amer. Math. Soc.95(1985), 387-395.

Hz1] E. Hernandez, Factorization and extrapolation of pairs of weights, Studia Math.

95(1989), 179-193.

Hz2] E. Hernandez, Weighted inequalities through factorization, Publicacions Mat.

35(1991), 141-153.

Oc-Kr] B. Opic, A. Kufner, Hardy-type inequalities, Harlow: Longman Sci & Techn. (1990).

Yves Rakotondratsimba

Institut polytechnique St Louis, EPMI

13 bd de l'Hautil 95 092 Cergy Pontoise France E-mail: y.rakoto@ipsl.tethys-software.fr