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165

Kragujev**ac** J. Math. 29 (2006) 165–173.

**ON** **WEIGHTED** **NORM** **INTEGRAL** **INEQUALITY** OF

G. H. HARDY’S TYPE

K. Rauf and J. O. Omolehin

Department of Mathematics, Unive**rs**ity of Ilorin, Ilorin, Nigeria

(e-mails: balk r@yahoo.com, omolehin joseph@yahoo.com)

(Received May 25, 2005)

Abstr**ac**t. In this paper, we give a necessary and sufficient condition on Hardy’s integral

inequality:

∫

∫

[T f] p wdµ ≤ C f p vdµ ∀f ≥ 0 (1)

X

where w, v are non-negative measurable functions on X, a non-negative function f defined

on (0, ∞), K(x, y) is a non-negative and measurable on X ×X, (T f)(x) = ∫ ∞

0

K(x, y)f(y)dy

and C is a constant depending on K, p but independent of f. This work is a continuation

of our recent result in [9].

X

1. INTRODUCTI**ON**

In the early twenties, G. H. Hardy proved the following result:

Theorem 1.1. Let 1 < p < ∞ and let f be a non-negative measurable function

defined on (0, ∞). Then,

∫ ∞ (∫ x

x −p

0

0

) ( ) p p

p

∫ ∞

f(t)dt dx ≤ f(x) p dx (2)

p − 1 0

166

holds.

For a proof (see [3]).

It has wide applications in differential and integral equations, Lapl**ac**e transforms

and Fourier series (See [1, 2, 4, 5, 8, 10] and [11]) for related works. Recently, B.

Mucheuhoupt [7] raised the question that given the weight w, under what condition

will there exist a weight function v, finite µ−almost everywhere on X such that the

inequality (1) holds. In their attempt to simplify this problem, R. Kerman and E.

Sawyer [6] provided partial solution to his question and two new open problems were

generated. In [9], the proof of the following theorem, among othe**rs**, on weighted norm

inequalities for positive sublinear operator was established:

Theorem 1.2. Let 1 < p < ∞ and suppose w is a weight on X. Define the

sublinear operator T by T (f + g)(x) = ∫ X K(x, y)(f + g)(y)dµ(y) then, there exists a

weight function v, finite µ−almost everywhere on X such that ∫ X [T (f + g)]p wdµ ≤

C(K, p) ∫ X (f p + g p )vdµ holds, for all f, g > 0, if and only if there is a positive

function Φ and θ on X with ∫ X (T Φ)p wdµ < ∞ and ∫ X (T θ)p wdµ < ∞ or equivalently

Φ 1−p T ∗ ((T Φ) p−1 w) < ∞ and Φ 1−p T ∗ ((T θ) p−1 w) < ∞ respectively, C(K, p) =

max [C 1 (K, p), C 2 (K, p)] is a constant independent of f and g.

The proof is ex**ac**tly as in [9]. Thus, we omit it.

This theorem is known to have provided partial solution to the open problem

number one in [6]. The aim of the present paper is to extent our recent result in [9] to

the case when T is a special integral operator and to consider the case of interchanging

non-negative weight functions w for which there are non trivial v ’s. These provided

partial solutions to the second open problem in [6].

Throughout this work, we let (X, ζ, µ) be a σ−finite measure sp**ac**e, K(x, y) be a

non-negative and measurable on X × X. Also, set (T f)(x) = ∫ ∞

0 K(x, y)f(y)dy and

(T ∗ f)(x) = ∫ ∞

0 K(y, x)f(y)dy for non negative functions f.

Furthermore, let p ′

denotes the conjugate index of p, p ≠ 0 and is defined by

1

+ 1 = 1 with p p ′ p′ = ∞ if p = 1.

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2. STATEMENT OF RESULTS

In this section, we state and prove our main results by induction and show that

for all n ∈ ℵ, the nth partial sum of the left hand side of the inequality (3) is less

than or equal to the nth partial sum of the right hand side.

Theorem 2.1. Let 1 < p < ∞ and suppose v is a weight on X. Then there is a

weight w, finite µ−almost everywhere on X, such that the weighted norm inequality:

∑

∫

∫

n∈ℵ

[T f] p wdµ ≤ ∑ f p vdµ (3)

X n n∈ℵ

X n

holds, for all f > 0, if and only if there is a positive function Φ on X with

∫

X

[T ∗ Φ] p v 1−p′ dµ < ∞ (4)

or equivalently,

Φ 1−p T ∗ [ (T Φ) p−1 w ] < ∞ (5)

µ−almost everywhere.

in (5).

We noted that the weighted norm inequality (3) holds with v equals to the weight

Proof. Suppose that ∫ X [T ∗ Φ] p v 1−p′ dµ < ∞, where 1 p + 1 p ′

= 1. Since X is a

σ−finite measure sp**ac**e. X = ∪ ∞ n=1X n with X n ∩ X m = ∅, n ≠ m and µ(X n ) < ∞.

We shall determine a necessary and sufficient condition on v such that the inequality

(3) holds. But,

and

∫

X

Thus we need to show that

∫

[T f] p wdµ = f]

∪ ∞ n=1 n[T p wdµ = ∑ ∫

[T f] p wdµ

X n∈ℵ

X n

∫

∑

∫

n∈ℵ

X

f p vdµ = ∑ n∈ℵ

∫

X n

[T f] p wdµ ≤ ∑ n∈ℵ

X n

f p vdµ

∫

X n

f p vdµ

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Let n = 1, we have

∫

[T f] p wdµ =

X 1

=

≤

∫X 1

[∫

∫X 1

[∫

K(x, y)f(y)dµ(y)] p

wdµ

K(x, y) 1 −

p f(y)Φ 1 p ′ K(x, y) 1 p ′ Φ 1 p

p

dµ(y)] ′ wdµ

[ ∫

∫

]

( K(x, y)f(y)

∫X p Φ − p p ′ dµ(y)) 1 p ( K(x, y)Φdµ(y)) 1 p

p ′ wdµ

1

By Holder’s inequality

∫ [

= (T f p Φ 1−p )(T Φ) p−1 w ] dµ

X 1

By a property of an inner product

∫

= f p vdµ

X 1

Hence, valid for n = 1.

Next, assume n = k > 1, we have

k∑

∫

[T f] p wdµ = [T f] p wdµ

n=1

∪

∫

k n=1 Xn ∫

∫

= [T f] p wdµ + [T f] p wdµ + . . . + [T f] p wdµ

X 1 X 2 X

∫

∫

∫

k

≤ f p vdµ + f p vdµ + . . . + f p vdµ

X 1 X 2 X k

By Holder’s inequality and using the above argument for n = 1

∫

= f p vdµ

Then, for n = k + 1, we have

k+1 ∑

∫

∪ k n=1 X n

[T f] p wdµ = ∑ k ∫

n=1 X n

[T f] p wdµ + ∫ X k+1

[T f] p wdµ

n=1

X n

k∑

∫

∫

≤ f p vdµ + [T f] p wdµ

n=1

X n X k+1

by assumption when n = k

≤

k∑

∫

∫

f p vdµ +

X n

f p vdµ

X k+1

n=1

since µ(X k+1 ) < ∞ and by the proof of k = 1. On the other hand, let us assume that

(3) holds for some v < ∞ µ−almost everywhere on X. By using the σ−finiteness of

µ, we can find a positive function Φ such that ∫ X Φp vdµ < ∞ and (4) holds.

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Finally, suppose (4) holds and let v denotes the weight in (5) then

k+1 ∑

n=1

∫

X n

Φ p vdµ =

=

=

=

=

∫

∫

∫

Φ p vdµ + Φ p vdµ + . . . + Φ p vdµ

X 1 X 2 X

∫

k+1

Φ [ p (Φ 1−p T ∗ (T Φ) p−1 w) ] ∫

dµ + Φ [ p (Φ 1−p T ∗ (T Φ) p−1 w) ] dµ

X 1 X

∫

2

+ . . . + Φ [ p (Φ 1−p T ∗ (T Φ) p−1 w) ] dµ

X

∫

k+1

[ [ ΦT

∗

(T Φ) p−1 w ]] ∫ [

dµ +

[ ΦT

∗

(T Φ) p−1 w ]] dµ

X 1 X

∫

2

[

+ . . . +

[ ΦT

∗

(T Φ) p−1 w ]] dµ

X

∫

k+1

[

(T Φ)(T Φ) p−1 w ] ∫ [

dµ + (T Φ)(T Φ) p−1 w ] dµ

X 1 X

∫

2

[

+ . . . + (T Φ)(T Φ) p−1 w ] dµ

X

∫

k+1

∫

∫

[(T Φ) p w] dµ + [(T Φ) p w] dµ + . . . + [(T Φ) p w] dµ

X 1 X 2 X k+1

∫

⇒

[(T f) p w] dµ =

X 1 +X 2 +...+X k+1

by (5).

k+1 ∑

n=1

∫

X n

[(T Φ) p w] dµ < ∞

We conclude that the result in (3) is valid for all n = 1, 2, . . . and the proof is complete.

✷

Remark. If we set n = 1 with X 1 = X in (3), then our result yields Theorem 1

obtained by Kerman and Sawyer [6].

let

Theorem 2.2. For X ∈ R and 1 < p < ∞. Let f(t) ≥ 0 and g(t) ≥ 0 and also

h(t) =

{

f(t) if t > 0

g(t) if t < 0

and zero otherwise, also suppose T is a completely arbitrary integral operator, then

∫

R

∫

[T h](t) p dµ ≤

R

∫

f p v 1 dµ +

If and only if there exists a positive function Φ on t with

X

R

g p v 2 dµ (6)

∫

v 1 = [T Φ] p dµ < ∞ (7)

170

and

or equivalently,

and

∫

v 2 =

X

[T ∗ Φ] p dµ < ∞ (8)

v 1 = Φ 1−p T ∗ (T Φ) p−1 < ∞ (9)

v 2 = Φ 1−p T (T ∗ Φ) p−1 < ∞ (10)

Proof. Suppose T f = ∫ ∞

0 f(t)dt and T ∗ is the dual of T .

Let

Then,

Therefore,

I =

=

∫

R

∫

R

= ∫ x

I = T h(t)

h(t)dt

[f(t) + g(t)] dt

0 f(t)dt + ∫ ∞

x g(t)dt

∫

∫ [∫ x ∫ ∞ p

[T h](t) p dµ = f(t)dt + g(t)dt]

dµ

R

R 0

x

∫ [∫ x

∫ ∞ ]

≤ f(t) p dt + g(t) p dt dµ

R 0

x

By Minkowski’s inequality

∫

∫

= (T f) p dµ + (T ∗ g) p dµ

R

R

∫ [

≤ (T f p Φ 1−p )(T Φ) p−1] ∫ [

dµ + (T ∗ g p Φ 1−p )(T ∗ Φ) p−1] dµ

∫R

[ R

= f p Φ 1−p T ∗ (T Φ) p−1] ∫ [

dµ + g p Φ 1−p T ∗∗ (T ∗ Φ) p−1] dµ

By definition of inner product

∫

∫

= f p v 1 dµ +

Conve**rs**ely, we can assume that (6) holds for some v < ∞ µ−almost everywhere.

R

R

R

g p v 2 dµ

By using the σ−finiteness of µ, we can obtain a positive function Φ such

that ∫ X Φp 1(v 1 )dµ < ∞ and ∫ X Φp 2(v 2 )dµ < ∞, then (7) and (8) holds.

R

171

Finally, suppose (7) and (8) holds and let v denotes the weight in (9) and (10)

then we can use the method of theorem 2.1 to show that

∫

∫

∫

Φ p 1(v 1 )dµ + Φ p 2(v 2 )dµ = [T h](t) p dµ

R

R

There is a similar result for the dual operator. This completes the proof of the

theorem.

✷

R

3. C**ON**SEQUENCE OF OUR MAIN RESULT

The next thorem treats the case of a convolution operator with radially decreasing

kernel on R + .

Theorem 3.1. Let 1 < p < ∞ and suppose that Φ, w(x) ≥ 0 are locally integrable

with respect to Lebesque measure on R + and that Φ(x) = Φ(|x|) is non-increasing as

a function of |x|. Define the convolution operator T by

T h(x) = (Φ ∗ h(x) =

∫ ∞

0

Φ(x − s)ϕ(h(s))dµ(s) (11)

Where ϕ is a scalar function and h(x) is as defined in Theorem 2.2. Then, there

exists a weight function v(x) finite µ−almost everywhere on X and C ≥ 0 such that:

∫

∫

(T h) p wdµ(s) ≤ C(K, p) h p vdµ(s) (12)

X

X

if and only if for all s ∈ R +

∫

X

Φ(x − s) p w(x)d(x) < ∞ (13)

Proof. Equation (11) can be transformed into linear form:

T h(x) =

and a nonlinear element

∫ x

0

Φ(x − s)U(s)dµ(s) (14)

U(t) = ϕ(h(s)) t ∈ R + (15)

The proof follows readily from the proof of theorem 3.1 in [9] and Theorem 3.2, if we

set K(x, y) ≡ Φ(x − s) and also theorem 2.2 of the current paper.

✷

172

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