A new proof of the characterization of the weighted Hardy inequality

A **new** **pro of**

**inequality**

A.L. Bernardis, F.J. Martín-Reyes, and P. Ortega Salvador

Abstract. Maz’ja and Sinnamon proved a **characterization** **of** **the** boundedness

**of** **the** **Hardy** operator from L p (v) into L q (w) in **the** case 0 < q < p,

1 < p < ∞. We present here a **new** simple **pro of**

result.

Maz’ja [2] and Sinnamon [3] characterized **the** weights such that

(∫ ∞

(∫ x q 1/q (∫ ∞

) 1/p

(1)

f)

w(x) dx)

≤ C f p (x)v(x) dx

0

0

for all measurable f ≥ 0 with a constant independent **of** f, where 0 < q < p and

1 < p < ∞. The **characterization** reduces to **the** condition that **the** function

(∫ ∞

) 1/p (∫ x

) 1/p

′

Ψ(x) = w v 1−p′

x

belong to L r (w), where 1/r = 1/q − 1/p. The **pro of**

direct. However a direct and easy **pro of** can be found in [4]. The

sufficiency in [2] uses **the** Hölder **inequality** for three exponents and **the** **characterization**

**of** **Hardy**’s **inequality** for **the** case p = q. The **pro of**

is simpler but uses **the** **Hardy**’s **inequality** in **the** case p = q.

The aim **of** this paper is to present a **new** **pro of**

(1) that we believe is simple, elementary and standard because it is more similar

to **the** **pro of**s

**the** **characterization** in **the** case p = q. The key point is to show that Ψ ∈ L r (w)

implies that **the** function

Φ(x) =

sup Ψ(a) =

0

2 A.L. BERNARDIS, F.J. MARTÍN-REYES, AND P. ORTEGA SALVADOR

ideas in **the** **pro of** could be

**the** study **of** **the** **Hardy**’s inequalities with q < p. In fact, **the** ideas **of** this paper

are used in [1] to study **the** generalized **Hardy**-Steklov operator in **weighted** spaces.

Next we establish **the** **the**orem (due to Maz’ja and Sinnamon) and we give a

complete simple **pro of**

(a)⇒(b), is taken from [4] while **the** **pro of**s

contribution. We include **the** **pro of**

**the**orem.

Theorem 1. Let w and v be nonnegative measurable functions. Let q, p, p ′

and r be such that 0 < q < p < ∞, 1 < p < ∞, p + p ′ = pp ′ , 1/r = 1/q − 1/p. The

following statements are equivalent:

(a) There exists a positive constant C such that

(∫ ∞

(∫ x q 1/q (∫ ∞

) 1/p

f)

w(x) dx)

≤ C f p (x)v(x) dx

0

for all measurable f ≥ 0.

(b) The function

(∫ ∞

Ψ(x) =

belongs to L r (w).

(c) The function

0

Φ(x) =

belongs to L r (w).

x

) 1/p (∫ x

) 1/p

′

w v 1−p′

0

(∫ ∞

) 1/p (∫ a

) 1/p

′

sup w v 1−p′

0 λ}) + w({x : Ψ(x) ≤ λ < Φ 1 (x)}),

we only have to show that if E = {x : Ψ(x) ≤ λ < Φ 1 (x)} **the**n

w(E) ≤ w({x : Ψ(x) > λ}).

To prove this last **inequality** we only have to establish that

(2)

∫ ∞

x

w ≤ w({y : Ψ(y) > λ})

0

WEIGHTED HARDY INEQUALITY 3

for all x ∈ E. Let us fix a point x ∈ E. Then **the**re exists a, 0 < a < x, such that

(∫ ∞

) 1/p (∫ x

) 1/p

′ (∫ x

) 1/p (∫ a

) 1/p

′

w v 1−p′ ≤ λ < w v 1−p′ .

x

These inequalities imply that

0

∫ ∞

x

w <

If (a, x) ⊂ {y : Ψ(y) > λ} **the**n (2) follows immediately. Assume that **the** set

F = {y ∈ (a, x) : Ψ(y) ≤ λ} is non-empty and let y ∈ F . Then

(∫ ∞

) 1/p (∫ y

) 1/p

′ (∫ x

) 1/p (∫ a

) 1/p

′

w v 1−p′ ≤ λ < w v 1−p′ .

0

a

0

y

Since a < y it follows that

Thus

∫ ∞

y

w ≤

∫ x

a

∫ ∞

x

w =

w ≤

∫ x

a

∫ y

a

∫ y

a

a

w.

w +

for all y ∈ F . If β is **the** infimum **of** F **the** last **inequality** implies that

∫ ∞

x

w ≤

This **inequality** implies that (2) holds since (a, β) ⊂ {y : Ψ(y) > λ}.

(c)⇒(a). It will suffice to prove it for integrable functions f. Let us choose a

decreasing sequence x k such that x 0 = +∞ and ∫ x k

f = 2 ∫ −k ∞

f. By **the** Hölder

0 0

**inequality** with exponents p and p ′ , we obtain

∫ ∞

(∫ x

) q

f w(x) dx ≤ ∑ ∫ xk

(∫ xk

) q

w(x) dx f

0 0

k

x k+1 0

≤ 4 ∑ ∫ (

xk

∫ ) q

xk+1

q w(x) dx f

k

x k+1 x k+2

≤ 4 ∑ ( ∫ ) q/p xk+1

∫ (

xk

∫ ) q/p

′

xk+1

q f p v w(x) dx v 1−p′ .

k

x k+2

x k+1 x k+2

Applying **the** Holder **inequality** with exponents p/q and r/q and **the** definition **of**

Φ, we have that **the** last term is dominated by

( ∑

∫ ) ⎛ q/p xk+1

4 q f p v ⎝ ∑ ∫ (

xk

∫ ) r/p (

xk

∫ ) r/p

′⎞q/r

xk+1

w(x) dx w

v 1−p′ ⎠

k

x k+2

k

x k+1 x k+1 x k+2

∫ β

a

w.

w.

∫ x

y

w.

0

≤ 4 q (∫ ∞

0

) ( q/p ∑

f p v

k

∫ xk

x k+1

Φ r w

) q/r

≤ 4 q (∫ ∞

0

) q/p (∫ ∞ q/r

f p v Φ w) r .

0

4 A.L. BERNARDIS, F.J. MARTÍN-REYES, AND P. ORTEGA SALVADOR

(a)⇒(b). Let w 0 and v 0 be nonnegative integrable functions such that w 0 ≤ w

and v 0 ≤ v 1−p′ . Let

(∫ ∞

) r (∫

pq t

) r

p ′ −1 q

f(t) = w 0 v 0 v 0 (t).

Then

∫ x

0

(∫ ∞

) r

f(t) dt ≥ w 0

t

x

0

pq ∫ x

0

(∫ t

0

v 0

) r

p ′ q −1 v 0 (t) dt

(∫

= p′ q ∞

) r (∫

pq x

) r

p ′ q

w 0 v 0 .

r x

0

Then by (a)

∫ ∞

( p ′ ) q (∫

q ∞

) r/p (∫ x

) r/p

′ ∫ ∞

(∫ x q

w 0 v 0 w 0 (x) dx ≤ f)

w(x) dx

0 r x

0

0 0

(∫ ∞

) q (

≤ C q f p p ∫ ∞

(∫ ∞

) r/q (∫ t

) r/q

′

) q p

v = C

q

w 0 v 0 v p 0 (t)v(t) dt

0

( ∫ ∞

(∫ ∞

) r/q (∫ t

) r/q

′

≤ C q w 0 v 0 v 0 (t) dt

0

t

0

0

t

0

) q p

( ) (

p

= C q ′ q/p ∫ ∞

(∫ ∞

) r/p (∫ x

) r/p

′

w 0 v 0 w 0 (x) dx

q

0

x

where **the** last equality is integration by parts. Since w 0 and v 0 are integrable

functions, we get that

∫ ∞

(∫ ∞

) r/p (∫ x

) r/p

′

w 0 v 0 w 0 (x) dx ≤ C r (r/p ′ q) r (p ′ /q) r/p .

0

x

0

Approximating w and v 1−p′ by increasing sequences **of** integrable functions we

obtain (b).

□

0

References

[1] A.L. Bernardis, F.J. Martín-Reyes and P. Ortega Salvador, Weighted inequalities for **Hardy**-

Steklov operators, to appear in Can. J. Math.

[2] W.G. Maz’ya, Sobolev spaces., Springer-Verlag. xix, 486 p.,1985.

[3] G.J. Sinnamon, Weighted **Hardy** and Opial-type inequalities, J. Math. Anal. Appl. 160, No.2,

1991, 434–445.

[4] G.J. Sinnamon and W.D. Stepanov, The **weighted** **Hardy** **inequality**: New **pro of**s and

p = 1, J. Lond. Math. Soc., 54, No.1, 1996, 89–101.

IMAL-CONICET, Güemes 3450, (3000) Santa Fe, Argentina

E-mail address: bernard@ceride.gov.ar

Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga,

Spain

E-mail address: martin reyes@uma.es

Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga,

Spain

E-mail address: ortega@anamat.cie.uma.es

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