p <1, q >1 it is known that inequality (1.1) doesn’t hold (see [4], p.46)

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http://www.math-analysis.org

THE HARDY INEQUALITY WITH ONE NEGATIVE PARAMETER

A. KUFNER¹, K. KULIEV^2∗ AND G. KULIEVA³ Dedicated to Professor Josip E. Peˇcari´c

Submitted by C. Park

Abstract. In this paper, necessary and sufficient conditions for the validity of the Hardy inequality for the caseq <0, p >0 and for the caseq >0, p <0 are derived.

1. Introduction and preliminaries The classical Hardy inequality

Z b a

Z x a

f(t)v(t)dt q

u(x)dx

1 q

≤C Z b

a

f(x)^pdx

1 p

(1.1) for all f ≥0, whereu, v are weight functions, is almost completely described for p, q such that

p≥1, q >0 (see [3], [4], [5]), while forp, q such that

0< p <1, q >1

it is known that inequality (1.1) doesn’t hold (see [4], p.46).

Date: Received: 19 February 2008; Accepted: 3 June 2008.

∗ Corresponding author.

2000Mathematics Subject Classification. Primary 26D10, 26D15; Secondary 47B38.

Key words and phrases. Inequalities, Hardy-type inequalities, weights, negative powers.

76

The so calledreverse Hardy inequality Z b

a

Z x a

f(t)v(t)dt q

u(x)dx ¹q

≥C Z b

a

f(x)^pdx ¹p

(1.2) was studied in [1] for

0< p, q <1 and p, q <0;

the second case was described in [6] and the case for

−∞< q ≤p < 0 was described in [2].

Here, we want to consider parameters p, q which satisfy either p <0, q >0

or

p > 0, q <0.

It will be shown that in the first case, the reverse inequality (1.2) hold (see Theorem 2.1) while in the second case, the reverse inequality (1.2) holds for 0 < p < 1, q < 0 (see Theorem 2.2) and the Hardy inequality (1.1) holds for p ≥ 1, q < 0 (see Theorem 2.4). The results can be extended to the ”adjoint”

inequalities Z b

a

Z b x

f(t)v(t)dt ^q

u(x)dx

!¹_q

≤C Z b

a

f(x)^pdx

1 p

(1.3) and

Z b a

Z b x

f(t)v(t)dt ^q

u(x)dx

!¹_q

≥C Z b

a

f(x)^pdx

1 p

(1.4) (see Remark 2.6).

The negative powers p, q force us to work with functions having values from the interval [0, +∞]. Therefore, we define the following arithmetics:







0 + (+∞) = a+ (+∞) = a·(+∞) = ^a₀ = +∞, a∈(0, +∞];

0·(+∞) = _+∞^a = 0, a∈[0, +∞);

0^−α = (+∞)^α = +∞, 0^α = (+∞)^−α = 0, α∈(0, +∞).

2. Main results Let us denote

A(t) :=

Z t a

v^p0(x)dx

_p¹0 Z b t

u(x)dx ¹q

, p⁰ = p p−1. Then we can formulate the following theorems:

Theorem 2.1. Let p < 0 and q > 0. Then inequality (1.2) holds if and only if there exists τ ∈(a, b) such that

A(τ)>0. (2.1)

Moreover, i) if

0< A^∗ := sup

(a,b)

A(t)<∞,

and C is the best possible constant of inequality (1.2) then A^∗ ≤C;

ii) if

A^∗ =∞,

then the best constant of inequality (1.2) does not exist, more precisely, the left hand side of (1.2) is infinite for all positive functions for which Rb

af^p(t)dt¹_p

>0.

Proof. Let τ ∈(a, b) be arbitrary. Then J :=

Z b a

Z x a

f(t)v(t)dt q

u(x)dx≥ Z b

τ

Z x a

f(t)v(t)dt q

u(x)dx

≥ Z b

τ

Z τ a

f(t)v(t)dt q

u(x)dx= Z b

τ

u(x)dx Z τ

a

f(t)v(t)dt q

.

Applying the reverse H¨older inequality with powersp andp⁰ = _p−1^p to the second integral of the last expression, we get that

J ≥ Z b

τ

u(x)dx Z τ

a

v(t)^p0dt

_p^q0 Z τ a

f(t)^pdt ^q_p

≥ Z b

τ

u(x)dx Z τ

a

v(t)^p0dt

_p^q0 Z b a

f(t)^pdt

q p

. Thus, we obtain that

Z b a

Z x a

f(t)v(t)dt q

u(x)dx≥A(τ)^q Z b

a

f(t)^pdt

q p

. (2.2)

It is easy to see that the condition (2.1) is equivalent with the validity of inequality (1.2), i.e. (2.2). If we suppose that condition (2.1) is satisfied, i.e. if there exists τ ∈ (a, b) such that A(τ) > 0, then from (2.2) we have inequality (1.2) with C ≥ A(τ). Conversely, let us suppose that inequality (1.2) holds, which means that for positive functionsf such that

Z b a

f(t)^pdt ¹p

>0,

the expression on the left hand side of inequality (1.2) is greater than zero, i.e.

Z b a

Z x a

f(t)v(t)dt q

u(x)dx ¹q

>0. (2.3)

If we define

a^∗ = sup{t∈[a, b), Z t

a

v^p0(s)ds = 0}; b^∗ = inf{t∈(a, b], Z b

t

u(s)ds= 0}, then

A(t) = 0 for all t∈(a, b) if and only if b^∗ ≤a^∗. This together with (2.3) implies that A(t) is positive for some t ∈(a, b).

From (2.2) we have that

A(τ)≤C = inf

f >0

Rb a

Rx

a f(t)v(t)dtq

u(x)dx ¹_q

Rb

a f(t)^pdt_p¹

.

The right hand side of the last estimate is independent on τ,so we get that A^∗ = sup

(a,b)

A(τ)≤C.

This ends the proof of i).

If A^∗ = ∞ then inequality (1.2) holds, since its left hand side is infinite for functionsf such that

Z b a

f^p(t)dt

1 p

>0,

which follows from (2.2).

Theorem 2.2. Let 0 < p < 1 and q < 0. Then inequality (1.2) holds for all functions f >0 if and only if the following condition is satisfied:

A_∗ := inf

(a,b)

A(t)>0.

Moreover, if C is the best possible constant in (1.2), then

1 + p⁰ q

_p¹0 1 + q

p^{0 1}_q

A∗ ≤C≤A∗.

Proof. (Sufficiency) Let α ∈(0,−_p¹0) be a parameter and denote V(t) :=

Z t a

v^p0(τ)dτ.

For

J :=

Z b a

Z x a

f(t)v(t)dt q

u(x)dx

= Z b

a

Z x a

f(t)v(t)dt

pZ x a

f(t)v(t)dt q−p

u(x)dx

= Z b

a

Z x a

f(t)V^−α(t)V^α(t)v(t)dt

pZ x a

f(t)v(t)dt q−p

u(x)dx

applying the reverse H¨older inequality with powerspand p⁰ = _p−1^p to the integral in the first brackets, we get,

J ≥ Z b

a

Z x a

f^p(t)V^−αp(t)dt

Z x a

V^αp0(t)v^p0(t)dt _p^p0

× Z x

a

f(t)v(t)dt q−p

u(x)dx

= 1

(1 +αp⁰)^pp0 Z b

a

Z x a

f^p(t)V^−αp(t)dt

V^(1+αp0)

p p0

(x)

× Z x

a

f(t)v(t)dt q−p

u(x)dx.

Now we again apply the reverse H¨older inequality with powers ^q_p and _q−p^q which yields

J ≥ 1

(1 +αp⁰)^pp0 Z b

a

Z x a

f^p(t)V^−αp(t)dt ^q_p

V^(1+αp0)pq0(x)u(x)dx

!^p_q

× Z b

a

Z x a

f(t)v(t)dt q

u(x)dx ¹⁻

p q

= J^1−pq (1 +αp⁰)

p p0

Z b a

Z x a

f^p(t)V^−αp(t)dt ^q_p

V^(1+αp0)pq0(x)u(x)dx

!^p_q .

The reverse Minkowski integral inequality with powerr = ^q_p yields

J ≥ J^1−pq (1 +αp⁰)

p p0

Z b a

f^p(t)V^−αp(t) Z b

t

V^(1+αp0)pq0(x)u(x)dx

p q

dt

≥ J^1−pq A^p∗(α) (1 +αp⁰)^pp0

Z b a

f^p(t)dt, where

A^∗(α) := inf

(a,b)A(t, α) = inf

(a,b)V^−α(t) Z b

t

V^(1+αp0)pq0(x)u(x)dx ¹_q

.

Therefore, we obtain that

J^1q ≥ A^∗(α) (1 +αp⁰)^p10

Z b a

f^p(t)dt ¹p

. (2.4)

Now we show that

A^∗(α)≥C₁A∗,

where C₁ depends only on α. Integration by parts leads to the estimate J1(t, α) :=

Z b t

V^(1+αp0)

q p0

(x)u(x)dx

= Z b

t

V^(1+αp0)pq0(x)d

− Z b

x

u(s)ds

=V^(1+αp0)pq0(t) Z b

t

u(s)ds− lim

x→b−V^(1+αp0)pq0(x) Z b

x

u(s)ds +(1 +αp⁰)q

p⁰

Z b t

Z b x

u(s)ds

V^{(1+αp0)pq0−1}(x)dV(x)

≤V^αq(t)A^q(t) + (1 +αp⁰)q p⁰

Z b t

A^q(x)V^αq−1(x)dV(x)

≤A^q_∗

V^αq(t) + (1 +αp⁰)q p⁰

Z b t

V^αq−1(x)dV(x)

≤ − 1

αp⁰A^q_∗V^αq(t).

Since J₁(t, α) =A^q(t, α)V^αq(t) due to the definition of A(t, α), we finally obtain that

A(t, α)≥(−αp⁰)^−1qA_∗, i.e.

A^∗(α)≥(−αp⁰)^−1qA∗, and from (2.4) it follows that

J^1q ≥ (−αp⁰)^−1q (1 +αp⁰)^p10

A_∗ Z b

a

f^p(t)dt p¹

.

For the best constant C we have sup

α∈(0,−¹

p0)

(−αp⁰)^−1q (1 +αp⁰)^p10

A∗ =

1 + p⁰ q

¹_q 1 + q

p^{0 1}_q

A∗ ≤C.

The sufficiency part is proved.

(Necessity) From inequality (1.2) we get that C ≤

Z b a

Z x a

f(t)v(t)dt q

u(x)dx

¹q Z b a

f(t)^pdt ⁻¹p

≤ Z b

τ

Z x a

f(t)^pdt −^q

pZ x

a

f(t)v(t)dt q

u(x)dx

!¹_q . If we choose

f(t) =v^p0−1(t),

then we get

C ≤ Z b

τ

Z x a

v(t)^p0dt _p^q0

u(x)dx

!¹_q

≤ Z τ

a

v(t)^p0dt

_p¹0 Z b τ

u(x)dx

1 q

=A(τ), and consequently,

A(τ)≥C,

which proves the necessity of the condition.

Proposition 2.3. Let the assumptions of Theorem 2.2 be satisfied. Then the best constant C of inequality (1.2) satisfies

sup

α∈(0,−¹

p0)

A^∗(α) (1 +αp⁰)^p10

≤C.

Proof. The proof follows from (2.4).

Let us denote

B(t) :=















supess

(a, t)

v(x) Rt

au(x)dx¹_q

if p= 1,

Rt

av^p0(x)dx_p¹0 Rt

au(x)dx¹_q

if p > 1.

Then we can formulate the following theorem:

Theorem 2.4. Let p ≥ 1 and q < 0. Then inequality (1.1) holds if and only if there exists τ ∈(a, b) such that

B(τ)<∞.

Moreover, i) if

0< B:= inf

(a,b)

B(t)<∞,

and C is the best constant of inequality (1.1) then C ≤B;

ii) if

B = 0

then the best constant of the inequality does not exist, more precisely, the left hand side of (1.1) is zero for all nonnegative functions f.

Proof. Let τ ∈(a, b) be arbitrary. Then J :=

Z b a

Z x a

f(t)v(t)dt q

u(x)dx≥ Z τ

0

Z x a

f(t)v(t)dt q

u(x)dx

≥ Z τ

0

Z τ a

f(t)v(t)dt q

u(x)dx= Z τ

0

u(x)dx Z τ

a

f(t)v(t)dt q

.

We estimate the second integral in the last expression as follows:

If p= 1 then Z τ

a

f(t)v(t)dt ≤supess

(a, τ)

v(t) Z τ

a

f(t)dt.

If p >1 then we apply the H¨older inequality Z τ

a

f(t)v(t)dt ≤ Z τ

a

v(t)^p0dt

_p¹0 Z τ a

f(t)^pdt ¹_p

. Consequently, we have that

Z b a

Z x a

f(t)v(t)dt q

u(x)dx≥B(τ)^q Z b

a

f(t)^pdt

q p

, i.e.

Z b a

Z x a

f(t)v(t)dt q

u(x)dx ¹q

≤B(τ) Z b

a

f(t)^pdt ¹p

.

The rest of the proof follows analogously as in the proof of Theorem 2.1.

Remark 2.5. In Theorem 2.2 we supposed that f >0, which is important, since we can construct a nonnegative function f for which inequality (1.2) does not hold.

Remark 2.6. If we denote A(t) :=

Z b t

v^p0(x)dx

_p¹0Z t a

u(x)dx ¹_q

and

B(t) :=















supess

(t, b)

v(x) Rb

t u(x)dx¹_q

if p= 1,

Rb

t v^p0(x)dx_p¹0 Rb

t u(x)dx¹_q

if p > 1

then we are able to formulate results analogous to Theorems 2.1, 2.2 and 2.4 for inequalities (1.3) and (1.4). The formulation and the proofs are left to the reader.

Acknowledgement. The first author was supported by the Institutional Re- search Plan No. AV0Z10190503, the other authors were supported by the Re- search Plan MSM4977751301 of the Ministry of Education, Youth and Sports of the Czech Republic.

References

1. P.R. Beesack and H. P. Heinig, Hardy’s inequalities with indices less than 1, Proc. Amer.

Math. Soc.,83(3)(1981), 532–536.

2. A. Kufner and K. Kuliev,The Hardy inequality with ”negative powers”, Adv. Algebra Anal., 1(2006), 219–228.

3. A. Kufner, L. Maligranda and L.-E. Persson,The Hardy Inequality - About its History and Some Related Results, Pilsen, 2007.

4. A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, World Scientific Pub- lishing Co, Singapore/ New Jersey/ London/ Hong Kong, 2003.

5. B. Opic and A. Kufner, Hardy-Type Inequalities,Pitman Research Notes in Mathematics Series 219,Longman Scientific and Technical, Harlow, 1990.

6. D.V. Prokhorov, Weighted Hardy inequalities for negative indices, J. Pub. Mat.,48(2004), 423–443.

1 Mathematical Institute, Academy of Sciences of the Czech Republic, ˇZitna 25, 11567 Praha 1, Czech Republic

E-mail address: [email protected]

2 Department of Mathematics, University of West Bohemia, Univerzitn´ı 22, 30614 Pilsen, Czech Republic

E-mail address: [email protected]

3 Department of Mathematics, University of West Bohemia, Univerzitn´ı 22, 30614 Pilsen, Czech Republic

E-mail address: [email protected]