A MULTIPLE HARDY-HILBERT’S INTEGRAL INEQUALITY WITH THE BEST CONSTANT FACTOR

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A MULTIPLE HARDY-HILBERT’S INTEGRAL INEQUALITY WITH THE BEST CONSTANT FACTOR

HONG Yong

Department of Mathematics,Guangdong University of Business Study , Guangzhou 510320,People’s Republic of China

November 14, 2006

Abstract

In this paper, by introducing the norm kxkα(x ∈ Rⁿ), we give a multiple Hardy-Hilbert’s integral inequality with a best constant factor and two parameters α,λ.

Key words: multiple Hardy-Hilbert’s integral inequality, the Γ-function, the best constant factor.

Mathematics subject classification (2000): 26D15

1 Introduction

If p >1, ¹_p +¹_q = 1,f ≥0, g ≥0, 0<R∞

0 f^p(x)dx <+∞, 0<R∞

0 g^q(x)dx <+∞, then the well known Hardy-Hilbert’s integral inequality is given by (see [1]):

Z ∞ 0

f(x)g(x)

x+y dxdy < π sin(^π_p)

Z ∞ 0

f^p(x)dx

¹_pZ ∞ 0

g^q(x)dx ¹_q

, (1)

where the constant factor _sin(^ππ

p) is the best possible. it’s equivalent form is:

Z ∞ 0

f(x) x+ydx

p

dy <

"

π sin(^π_p)

#p

Z ∞ 0

f^p(x)dx, (2)

where the constant factor

π sin(^π_p)

p

is also the best possible.

Hardy-Hilbert’t inequality is valuable in harmonic analysis, real analysis and operator theory. In recent years, many valuable results (see [2-5]) have been obtained in

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generalization and improvement of Hardy-Hilbert’s inequality. In 1999,Kuang [6] gave a generalization with a parameterλof (1) as follows:

Z ∞ 0

f(x)g(y)

x^λ+y^λ dxdy < hλ(p) Z ∞

0

x^1−λf^p(x)dx

¹_pZ ∞ 0

x^1−λg^q(x)dx ¹_q

, (3) where max{¹_p,¹_q} < λ≤ 1, hλ(p) =π[λsin¹^p(_pλ^π ) sin¹^q(_qλ^π)]⁻¹. Because of the constant factor h_λ(p) being not the best possible, Yang [7] gave a new generalization of (1) in 2002 as follows:

Z ∞ 0

f(x)g(y)

x^λ+y^λ dxdy < π λsin(^π_p)

× Z ∞

0

x^{(1−λ)(p−1)}f^p(x)dx

_p¹Z ∞ 0

x^{(1−λ)(q−1)}g^q(x)dx ¹_q

, (4) it’s equivalent form is:

Z ∞ 0

y^λ−1 Z ∞

0

f(x) x^λ+y^λdx

p

dy <

"

π λsin(^π_p)

#p

Z ∞ 0

x^{(1−λ)(p−1)}f^p(x)dx, (5) where the constant factors _λ_sin(^π π

p) in (4) and [_λ_sin(^π π

p)]^p in (5) are all the best possible.

At present, because of the requirement of higher-dimensional harmonic analysis and higher-dimensional operator theory, multiple Hardy-Hilbert’s integral inequality is re- searched. Hong [8] obtained: If a > 0, Pn

i=1 1

pi = 1, pi > 1, fi ≥ 0, ri = _p¹

i

Qn j=1pj, λ > ¹_a(n−1− _r¹

i),i= 1,2,· · ·, n, then Z ∞

α

· · · Z ∞

α

1 [Pn

i=1(x_i−α_i)^a]^λ

n

Y

i=1

fi(x)dx1· · ·dxn≤ Γⁿ⁻²(_α¹) αⁿ⁻¹Γ(λ)

×

n

Y

i=1

Γ

1 a(1− 1

ri

)

Γ

λ− 1

a(n−1− 1 ri

) Z ∞

α

(t−α)^n−1−αλf_i^pⁱdt ¹

pi .(6)

Afterwards, Bicheng Yang and Kuang Jichang etc. obtained some multiple Hardy- Hilbert’s integral inequalities (see [9-10]).

In this paper, by introducing the Γ-function, we generalize (3) and (4) into multiple Hardy-Hilbert’s integral inequalities with the best constant factors.

2 Some lemmas

First of all, we introduce the signs as:

Rⁿ₊={x= (x1,· · ·, xn) :x1, . . . , xn>0}, kxk_α = (x^α₁ +· · ·+x^α_n)^α¹, α >0.

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Lemma 2.1 (see [11]) If pi >0, ai >0, αi >0, i= 1,2,· · ·, n, Ψ(u) is a measurable function, then

Z . . .

Z

x1,···,xn>0;(^x_a¹

1)^α¹+···+(^xn

an)^αn≤1

Ψ

(x1

a₁)^α¹ +· · ·+ (xn

a_n)^αⁿ

×x^p₁¹⁻¹· · ·x^p_nⁿ⁻¹dx1· · ·dxn

= a^p₁¹· · ·a^pnⁿΓ(_α^p¹

1)· · ·Γ(_α^pⁿ

n) α₁· · ·α_nΓ(_α^p¹

1 +· · ·+_α^pⁿ

n) Z 1

0

Ψ(u)u

p1 α1+···+^pn

αn−1

du. (7)

where Γ(·) is the Γ-function.

Lemma 2.2 If p > 1, ¹_p + ¹_q = 1, n∈ Z₊, α >0, λ > 0, setting the weight function ω_α,λ(x, p, q) as:

ω_α,λ(x, p, q) = Z

Rⁿ₊

1 kxk^λ_α+kyk^λ_α



 kxk

1 q

α

kyk

1 p

α





(n−λ)p

kxk_α kyk_α

^λ_q dy,

then

ω_α,λ(x, p, q) =kxk^{(n−λ)(p−1)}_α πΓⁿ(_α¹)

sin(^π_p)λαⁿ⁻¹Γ(_αⁿ). (8)

Proof By lemma 2.1, we have

ω_α,λ(x, p, q) =kxk(n−λ)(p−1)+^λ α q

Z

Rⁿ₊

1

kxk^λ_α+kyk^λ_αkyk^−(n−λ+

λ q)

α dy

= kxk(n−λ)(p−1)+^λ

α q lim

R→+∞

Z

· · · Z

y1,···,yn>0;y^α₁+···+y^α_n<r^α

× h

r((^y_r¹)^α+· · ·+ (^y_rⁿ)^α)^α¹

i−(n−λ+^λ

q)

kxk^λ_α+h

r((^y_r¹)^α+· · ·+ (^y_rⁿ)^α)^α¹iλy¹⁻¹₁ · · ·y_n¹⁻¹dy1· · ·dyn

= kxk(n−λ)(p−1)+^λ_q

α lim

R→+∞

rⁿΓⁿ(_α¹) αⁿΓ(ⁿ_α)

Z 1 0

(ru¹^α)^−(n−λ+^λ^q⁾ kxk^λ_α+ (ru^α¹)^λ

uⁿ^α⁻¹du

= kxk(n−λ)(p−1)+^λ α q

Γⁿ(_α¹)

αⁿ⁻¹Γ(ⁿ_α) lim

R→+∞

Z _r

0

1

kxk^λ_α+u^λu^λ−^λ^q⁻¹du

= kxk(n−λ)(p−1)+^λ α q

Γⁿ(_α¹) αⁿ⁻¹Γ(ⁿ_α)

Z ∞ 0

1

kxk^λ_α+u^λu^λ−^λ^q⁻¹du

= kxk^{(n−λ)(p−1)}_α Γⁿ(_α¹) λαⁿ⁻¹Γ(ⁿ_α)

Z ∞ 0

1

1 +uu¹^p⁻¹du

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= kxk^{(n−λ)(p−1)}_α Γⁿ(_α¹) λαⁿ⁻¹Γ(ⁿ_α)Γ(1

p)Γ(1−1 p)

= kxk^{(n−λ)(p−1)}_α πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α), hence (8) is valid.

Lemma 2.3 If p > 1, ¹_p + ¹_q = 1, n ∈ Z+, α > 0, λ > 0, 0 < ε < λ(q−1), setting

˜

ω_α,λ(x, q, ε) as:

˜

ω_α,λ(x, q, ε) = Z

Rⁿ₊

1

kxk^λ_α+kyk^λ_αkyk⁻

(n−λ)(q−1)+n+ε q

α dy,

then we have

˜

ω_α,λ(x, q, ε) =kxk⁻

λ q−^ε α q

Γⁿ(_α¹) λαⁿ⁻¹Γ(ⁿ_α)Γ(1

p − ε λq)Γ(1

q + ε

λq). (9)

Proof By the same way of lemma 2.2, lemma 2.3 can be proved.

3 Main Results

Theorem 3.1 If p >1, ¹_p +¹_q = 1, n∈Z₊,α >0, λ >0, f ≥0, g≥0, and 0<

Z

Rⁿ₊

kxk^{(n−λ)(p−1)}_α f^p(x)dx <∞, 0<

Z

Rⁿ₊

kxk^{(n−λ)(q−1)}_α g^q(x)dx <∞, (10) then

Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y)

kxk^λ_α+kyk^λ_αdxdy < πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α)

× Z

Rⁿ₊

kxk^{(n−λ)(p−1)}_α f^p(x)dx

!¹_p Z

Rⁿ₊

kxk^{(n−λ)(q−1)}_α g^q(x)dx

!¹_q

; (11)

Z

Rⁿ₊

kyk^λ−n_α Z

Rⁿ₊

f(x) kxk^λ_α+kyk^λ_αdx

!p

dy <

"

πΓⁿ(¹_α) sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α)

#p

× Z

Rⁿ₊

kxk^{(n−λ)(p−1)}_α f^p(x)dx, (12)

where the constant factors ^πΓ

n(_α¹)

sin(^π_p)λαⁿ⁻¹Γ(_αⁿ) and[ ^πΓ

n(_α¹)

sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α)]^p are all the best possible.

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Proof By H¨older’s inequality, we have A:=

Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y) kxk^λ_α+kyk^λ_αdxdy

= Z

Rⁿ₊

Z

Rⁿ₊

f(x) (kxk^λ_α+kyk^λ_α)¹^p



 kxk

1 q

α

kyk

1 p

α





n−λ

kxk_α kyk_α

_pq^λ

× g(y)

(kxk^λ_α+kyk^λ_α)¹^q



 kyk

1

αp

kxk

1

αq





n−λ

kyk_α kxk_α

_pq^λ dxdy

≤





 Z

Rⁿ₊

Z

Rⁿ₊

f^p(x) kxk^λ_α+kyk^λ_α



 kxk

1

αq

kyk

1

αp





(n−λ)p

kxk_α kyk_α

^λ

q

dxdy







1 p

×





 Z

R₊ⁿ

Z

Rⁿ₊

g^q(x) kxk^λ_α+kyk^λ_α



 kyk

1 p

α

kxk

1 q

α





(n−λ)q

kyk_α kxk_α

^λ_p dxdy







1 q

=





 Z

Rⁿ₊

f^p(x)





 Z

Rⁿ₊



 kxk

1

αq

kyk

1

αp





(n−λ)p

kxk_α kyk_α

^λ_q dy





dx







1 p

×





 Z

Rⁿ₊

g^q(y)





 Z

Rⁿ₊



 kyk

1 p

α

kxk

1 q

α





(n−λ)q

kyk_α kxk_α

^λ

p

dx





dy







1 q

= Z

Rⁿ₊

f^p(x)ωα,λ(x, p, q)dx

!¹

p Z

R₊ⁿ

g^q(y)ωα,λ(y, q, p)dy

!¹

q

,

according to the condition of taking equality in H¨older’s inequality, if this inequality takes the form of an equality, then there exist constantsC1 and C2, such that they are not all zero, and

C₁f^p(x) kxk^λ_α+kyk^λ_α



 kxk

1

αq

kyk

1

αp





(n−λ)p

kxk_α kyk_α

^λ_q

= C₂g^q(y) kxk^λ_α+kyk^λ_α



 kyk

1 p

α

kxk

1 q

α





(n−λ)q

kyk_α kxk_α

^λ_p

, a.e. in Rⁿ₊×Rⁿ₊,

it follows that C1kxk(n−λ)(p−1)+n

α f^p(x) =C2kyk(n−λ)(q−1)+n

α g^q(y) =C(constant), a.e. in Rⁿ₊×Rⁿ₊,

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which contradicts (10), hence we have A <

Z

Rⁿ₊

f^p(x)ω_α,λ(x, p, q)dx

!¹_p Z

R₊ⁿ

g^q(y)ω_α,λ(y, q, p)dy

!¹_q .

By lemma 2.2 and sin(^π_p) = sin(^π_q), we have

A <

"

πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α)

#¹

p Z

Rⁿ₊

!¹

p

×

"

πΓⁿ(_α¹) sin(^π_q)λαⁿ⁻¹Γ(ⁿ_α)

#¹

q Z

Rⁿ₊

kyk^{(n−λ)(q−1)}_α g^q(y)dy

!¹

q

= πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α)

Z

Rⁿ₊

!¹

p Z

Rⁿ₊

kxk^{(n−λ)(q−1)}_α g^q(x)dx

!¹

q

.

Hence (11) is valid.

For 0< a < b <∞, setting g_a,b(y) =

(

kyk^λ−n_α R

Rⁿ₊

f(x)

kxk^λ_α+kyk^λ_αdxp−1

, a <kyk_α< b

0, 0<kyk_α ≤a or kyk_α≥b

˜

g(y) =kyk^λ−n_α Z

Rⁿ₊

!p−1

, y ∈Rⁿ₊,

by (10), for sufficiently smalla >0 and sufficiently largeb >0, we have 0<

Z

a<kykα<b

kyk^{(n−λ)(q−1)}_α g_a,b^q (y)dy <∞.

Hence by (11), we have Z

a<kykα<b

kyk^{(n−λ)(q−1)}_α g˜^q(y)dy= Z

a<kykα<b

kyk^λ−n_α Z

R₊ⁿ

!p

dy

= Z

a<kykα<b

kyk^λ−n_α Z

Rⁿ₊

!p−1

Z

R₊ⁿ

! dy

= Z

Rⁿ₊

Z

Rⁿ₊

f(x)g_a,b(y) kxk^λ_α+kyk^λ_αdxdy

< πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(_αⁿ)

Z

Rⁿ₊

!¹_p Z

Rⁿ₊

kyk^{(n−λ)(q−1)}_α g^q_a,b(y)dy

!¹_q

= πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(_αⁿ)

Z

Rⁿ₊

!¹_p Z

a<kykα<b

kyk^{(n−λ)(q−1)}_α ˜g^q(y)dy

!¹_q ,

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it follows that Z

a<kykα<b

kyk^{(n−λ)(q−1)}_α g˜^q(y)dy <

"

πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(_αⁿ)

#p

Z

Rⁿ₊

kxk^{(n−λ)(p−1)}_α f^p(x)dx.

Fora→0⁺,b→+∞, by (10), we obtain 0<

Z

Rⁿ₊

kyk^{(n−λ)(q−1)}_α ˜g^q(y)dy≤

"

#p

Z

Rⁿ₊

kxk^{(n−λ)(p−1)}_α f^p(x)dx <∞, hence by (11), we have

Z

Rⁿ₊

kyk^λ−n_α Z

Rⁿ₊

!p

dy= Z

Rⁿ₊

Z

R₊ⁿ

f(x)˜g(y) kxk^λ_α+kyk^λ_αdxdy

< πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α)

Z

Rⁿ₊

!_p¹ Z

Rⁿ₊

kyk^{(n−λ)(q−1)}_α g˜^q(y)dy

!¹_q

= πΓⁿ(_α¹) sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α)

Z

Rⁿ₊

!_p¹

×

"

Z

Rⁿ₊

kyk^λ−n_α Z

Rⁿ₊

!p

dy

#¹_q ,

it follows that

Z

Rⁿ₊

kyk^λ−n_α Z

Rⁿ₊

!p

dy

<

"

#p

Z

Rⁿ₊

kxk^{(n−λ)(p−1)}_α f^p(x)dx.

Hence (12) is valid.

Remark: By (12), we can also obtain (11), hence (12) and (11) are equivalent.

If the constant factor Cn,α(λ, p) := ^πΓ

n(¹_α)

sin(^π_p)λαⁿ⁻¹Γ(_αⁿ) in (11) is not the best possible, then there exists a positive constantK < Cn,α(λ, p), such that

Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y) kxk^λ_α+kyk^λ_αdxdy

< K Z

Rⁿ₊

!¹_p Z

Rⁿ₊

kyk^{(n−λ)(q−1)}_α g^q(y)dy

!¹_q

. (13)

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In particular, setting f(x) =kxk⁻

(n−λ)(p−1)+n+ε

α p , g(y) =kyk⁻

(n−λ)(q−1)+n+ε

α q , 0< ε < λ(q−1), (13) is still true. By the properties of limit, there exists a sufficiently smalla >0, such that

Z

kxkα>a

Z

R₊ⁿ

1

kxk^λ_α+kyk^λ_αkxk⁻

(n−λ)(p−1)+n+ε

α p kyk⁻

(n−λ)(q−1)+n+ε

α q dxdy

< K Z

kxk_α>a

kxk^{(n−λ)(p−1)}_α kxk−(n−λ)(p−1)−n−ε

α dx

!¹_p

× Z

kyk_α>a

kyk^{(n−λ)(q−1)}_α kyk−(n−λ)(q−1)−n−ε

α dy

!¹_q

= K

Z

kxkα>a

kxk^−n−ε_α dx.

On the other hand, by lemma 2.3, we have Z

kxk_α>a

Z

Rⁿ₊

1

kxk^λ_α+kyk^λ_αkxk⁻

(n−λ)(p−1)+n+ε

α p kyk⁻

(n−λ)(q−1)+n+ε

α q dxdy

= Z

kxkα>a

kxk⁻ⁿ⁺

λ q−^ε α p

Z

Rⁿ₊

1

kxk^λ_α+kyk^λ_αkyk⁻

(n−λ)(q−1)+n+ε

α q dydx

= Z

kxkα>a

kxk⁻ⁿ⁺

λ q−^ε

α pω˜_α,λ(x, q, ε)dx

= Γⁿ(_α¹) λαⁿ⁻¹Γ(ⁿ_α)Γ

1 p− ε

λq

Γ 1

q + ε λq

Z

kxkα>a

kxk^−n−ε_α dx, hence we obtain

Γⁿ(_α¹) λαⁿ⁻¹Γ(ⁿ_α)Γ

1 p− ε

λq

Γ 1

q + ε λq

< K.

forε→0⁺, we have

Cn,α(λ, p) = πΓⁿ(_α¹)

sin(^π_p)λαⁿ⁻¹Γ(ⁿ_α) ≤K,

this contradicts the fact that K < Cn,α(λ, p).Hence the constant factor in (11) is the best possible.

Since (12) and (11) are equivalent, the constant factor in (12) is also the best possible.

Corollary 3.1 If p >1, ¹_p +¹_q = 1, n∈Z₊, α >0, f ≥0, g≥0, and 0<

Z

Rⁿ₊

f^p(x)dx <∞, 0<

Z

Rⁿ₊

g^q(x)dx <∞, (14)

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then

Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y)

kxkⁿ_α+kykⁿ_αdxdy < πΓⁿ(_α¹) sin(^π_p)nαⁿ⁻¹Γ(ⁿ_α)

× Z

Rⁿ₊

f^p(x)dx

!¹_p Z

Rⁿ₊

g^q(x)dx

!¹_q

; (15)

Z

Rⁿ₊

Z

Rⁿ₊

f(x) kxkⁿ_α+kykⁿ_αdx

!p

dy <

"

πΓⁿ(¹_α) sin(^π_p)nαⁿ⁻¹Γ(ⁿ_α)

#p

Z

Rⁿ₊

f^p(x)dx, (16) where the constant factors in (15) and (16) are all the best possible.

Proof By takingλ=n in theorem 3.1, (15) and (16) can be obtained.

Corollary 3.2 If p >1, ¹_p +¹_q = 1, n∈Z+, f ≥0, g≥0, and 0<

Z

Rⁿ₊

f^p(x)dx <∞, 0<

Z

Rⁿ₊

g^q(x)dx <∞, (17) then

Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y) (Pn

i=1x_i)ⁿ+ (Pn

i=1y_i)ⁿdxdy < π n! sin(^π_p)

× Z

Rⁿ₊

f^p(x)dx

!¹_p Z

Rⁿ₊

g^q(x)dx

!¹_q

; (18)

Z

Rⁿ₊

Z

Rⁿ₊

f(x) (Pn

i=1xi)ⁿ+ (Pn

i=1yi)ⁿdx

!p

dy <

"

π n! sin(^π_p)

#p

Z

Rⁿ₊

f^p(x)dx, (19) where the constant factors in (18) and (19) are all the best possible.

Proof By takingα= 1 in corollary 3.1, (18) and (19) can be obtained.

References

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