Báo cáo hóa học: " Research Article On a Hilbert-Type Operator with a Class of Homogeneous Kernels" pptx

Volume 2009, Article ID 572176, 9 pagesdoi:10.1155/2009/572176 Research Article On a Hilbert-Type Operator with a Class of Homogeneous Kernels Bicheng Yang Department of Mathematics, Gua

Volume 2009, Article ID 572176, 9 pages

doi:10.1155/2009/572176

Research Article

On a Hilbert-Type Operator with a Class of

Homogeneous Kernels

Bicheng Yang

Department of Mathematics, Guangdong Education Institute, Guangzhou, Guangdong 510303, China

Correspondence should be addressed to Bicheng Yang,bcyang@pub.guangzhou.gd.cn

Received 15 September 2008; Accepted 20 February 2009

Recommended by Patricia J Y Wong

By using the way of weight coefficient and the theory of operators, we define a Hilbert-type operator with a class of homogeneous kernels and obtain its norm As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernels of−λ-degree

is established, and some particular cases are considered

Copyrightq 2009 Bicheng Yang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In 1908, Weyl published the well-known Hilbert’s inequality as the following If

{a n}∞n1 , {b n}∞n1 are real sequences, 0 <∞

n1 a2

n < ∞ and 0 <∞

n1 b2

n < ∞, then 1

∞



n1

∞



m1

a m b n

m  n < π

∞



n1

a2

n

∞



n1

b2

n

1/2

where the constant factor π is the best possible In 1925, Hardy gave an extension of 1.1 by introducing one pair of conjugate exponentsp, q1/p  1/q  1 as 2 If p > 1, a n , b n ≥ 0,

0 <∞

n1 a p n < ∞, and 0 <∞

n1 b q n < ∞, then

∞



n1

∞



m1

a m b n

m  n <

π

sinπ/p

∞



n1

a p n

1/p∞



n1

b q n

1/q

where the constant factor π/ sinπ/p is the best possible We named 1.2 Hardy-Hilbert’s inequality In 1934, Hardy et al.3 gave some applications of 1.1-1.2 and a basic theorem with the general kernelsee 3, Theorem 318

Theorem A Suppose that p > 1, 1/p  1/q  1, kx, y is a homogeneous function of −1-degree,

and k  ∞

0 ku, 1u −1/p du is a positive number If both ku, 1u −1/p and k1, uu −1/q are strictly decreasing functions for u > 0, a n , b n ≥ 0, 0 < a p  ∞n1 a p n1/p

< ∞, and 0 < b q 

∞n1 b q n1/q < ∞, then one has the following equivalent inequalities:

∞



n1

∞



m1

∞



n1

∞

m1

km, na m

p

< k p a p

where the constant factors k and k p are the best possible.

Note Hardy did not prove this theorem in3 In particular, we find some classical Hilbert-type inequalities as,

i for kx, y  1/x  y in 1.3, it reduces 1.2,

ii for kx, y  1/ max{x, y} in 1.3, it reduces to see 3, Theorem 341

∞



n1

∞



m1

a m b n

max{m, n} < pq

∞

n1

a p n

1/p∞

n1

b q n

1/q

iii for kx, y  lnx/y/x − y in 1.3, it reduces to see 3, Theorem 342

∞



n1

∞



m1

lnm/nam b n

m − n <



π

sinπ/p

2∞

n1

a p n

1/p∞

n1

b q n

1/q

Hardy also gave some multiple extensions of 1.3 see 3, Theorem 322 About introducing one pair of nonconjugate exponents p, q in 1.1, Hardy et al 3 gave that

if p, q > 1, 1/p  1/q ≥ 1, 0 < λ  2 − 1/p  1/q ≤ 1, then

∞



n1

∞



m1

a m b n

m  n λ ≤ Kp, q

∞

n1

a p n

1/p∞

n1

b q n

1/q

In 1951, Bonsall4 considered 1.7 in the case of general kernel; in 1991, Mitrinovi´c et al 5 summarized the above results

In 2001, Yang6 gave an extension of 1.1 as for 0 < λ ≤ 4,

∞



n1

∞



m1

a m b n

m  n λ < B

λ

2, λ

2

∞



n1

n1−λa2n

∞



n1

n1−λb n2

1/2

where the constant Bλ/2, λ/2,  is the best possible Bu, v is the Beta function For λ  1,

1.8 reduces to 1.1 And Yang 7 also gave an extension of 1.2 as

∞



n1

∞



m1

a m b n

m λ  n λ < π

λ sinπ/p

∞

n1

n p−11−λ a p n

1/p∞

n1

n q−11−λ b q n

1/q

, 1.9

where the constant factor π/λ sinπ/p 0 < λ ≤ 2 is the best possible.

In 2004, Yang8 published the dual form of 1.2 as follows:

∞



n1

∞



m1

a m b n

m  n <

π

sinπ/p

∞



n1

n p−2 a p n

1/p∞



n1

n q−2 b q n

1/q

, 1.10

where π/ sinπ/p is the best possible For p  q  2, both 1.10 and 1.2 reduce to 1.1 It means that there are more than two different best extensions of 1.1 In 2005, Yang 9 gave

an extension of1.8–1.10 with two pairs of conjugate exponents p, q, r, s p, r > 1, and two parameters α, λ > 0 αλ ≤ min{r, s} as

∞



n1

∞



m1

a m b n

m α  n αλ < k αλ r

∞



n1

n p1−αλ/r−1 a p n

1/p∞



n1

n q1−αλ/s−1 b n q

1/q

, 1.11

where the constant factor k αλ r  1/αBλ/r, λ/s is the best possible; Krni´c and Peˇcari´c

10 also considered 1.11 in the general homogeneous kernel, but the best possible property

of the constant factor was not proved by10

Note For A  B  α  β  1 in 10, inequality37, it reduces to the equivalent result of 3.1

in this paper

In 2006-2007, some authors also studied the operator expressing of1.3 and 1.4

Suppose that kx, y≥ 0 is a symmetric function with ky, x  kx, y, and k0p :

∞

0kx, yx/y 1/r dy r  p, q; x > 0 is a positive number independent of x Define an

operator T : l r → l r r  p, q as follows For a m ≥ 0, a  {a m}∞

m1 ∈ l p , there exists only

Ta  c  {c n}∞n1 ∈ l p , satisfying

Tan  c n:∞

m1

Then the formal inner product of Ta and b are defined as follows:

Ta, b ∞

n1

∞



m1

In 2007, Yang 11 proved that if for ε ≥ 0 small enough, kx, yx/y 1ε/r is strictly

decreasing for y > 0, the integral∞

0 kx, yx/y 1ε/r dy  k ε p is also a positive number independent of x > 0, k ε p  k0p  o1 ε → 0, and

∞



m1

1

m1ε

1

0

km, t

m t

1ε/r

dt  O1 ε −→ 0; r  p, q

, 1.14

then T p  k0p; in this case, if a m , b n ≥ 0, a  {a m}∞m1 ∈ l p , b  {b n}∞n1 ∈ l q , a p >

0, b q > 0, then we have two equivalent inequalities as

Ta, b < T p a p b q; Ta p < T p a p , 1.15

where the constant factorT p is the best possible In particular, for kx, y being −1-degree

homogeneous, inequalities1.15 reduce to 1.3-1.4 in the symmetric kernel Yang 12 also considered1.15 in the real space l2

In this paper, by using the way of weight coefficient and the theory of operators, we define a new Hilbert-type operator and obtain its norm As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernel of−λ-degree

is established; some particular cases are considered

2 On a New Hilbert-Type Operator and the Norm

If k λ x, y is a measurable function, satisfying for λ, u, x, y > 0, k λ ux, uy  u −λ k λ x, y, then

we call k λ x, y the homogeneous function of −λ-degree.

For k λ x, y ≥ 0, setting x  uy, we find k λ x, y1/x1−λ/r  1/y1λ/sk λ u, 1u λ/r−1

Hence, the following two words are equivalent: a k λ u, 1u λ/r−1 is decreasing in 0, ∞

and strictly decreasing in a subinterval of 0, ∞; b for any y > 0, k λ x, y1/x1−λ/r is

decreasing in x ∈ 0, ∞ and strictly decreasing in a subinterval of 0, ∞ The following two

words are also equivalent:ak λ 1, uu λ/s−1is decreasing in0, ∞ and strictly decreasing in

a subinterval of0, ∞; bfor any x > 0, k λ x, y1/y1−λ/s is decreasing in y ∈ 0, ∞ and

strictly decreasing in a subinterval of0, ∞.

Lemma 2.1 If fx≥ 0 is decreasing in 0, ∞ and strictly decreasing in a subinterval of 0, ∞,

and I0:∞0 fxdx < ∞, then

I1:

∞

1

fxdx ≤

∞



n1

Proof By the assumption, we findn1

n fxdx ≤ fn ≤n

n−1 fxdx n ∈ N, and there exists

n0− 1, n0 ⊂ 0, ∞, such that fn0 <n0

n0−1fxdx Hence,

I1∞

n1

n1

n

fxdx ≤

∞



n1

fn <

∞



n1

n

n−1

Lemma 2.2 If r > 1, 1/r  1/s  1, λ > 0, k λ x, y≥ 0 is a homogeneous function of −λ-degree,

and k λ r :∞0k λ u, 1u λ/r−1 du is a positive number, then i∞

0 k λ 1, uu λ/s−1 du  k λ r; ii

for x, y ∈ 0, ∞, setting the weight functions as

ω λ r, y :

∞

0

k λ x, y y λ/s

x1−λ/rdx,  λ s, x :

∞

0

k λ x, y x λ/r

then ω λ r, y   λ s, x  k λ r.

Proof i Setting v  1/u, by the assumption, we obtain ∞0k λ 1, uu λ/s−1 du  ∞

0k λ v,

1vλ/r−1 dv  k λ r ii Setting x  yu and y  xu in the integrals ω λ r, y and  λ s, x,

respectively, in view ofi, we still find that ω λ r, y   λ s, x  k λ r.

For p > 1, 1/p  1/q  1, we set φx  x p1−λ/r−1 , ψx  x q1−λ/s−1 , and ψ1−px 

x pλ/s−1 , x ∈ 0, ∞ Define the real space as l p φ: {a  {an}∞

n1;a p,φ: {∞

n1 φn|a n|p}1/p

<

∞}, and then we may also define the spaces l q

ψ and l p ψ1−p.

Lemma 2.3 As the assumption of Lemma 2.2, for a m ≥ 0, a  {a m}∞m1 ∈ l p

φ , setting

c n  ∞m1 k λ m, na m , if k λ u, 1u λ/r−1 and k λ 1, uu λ/s−1 are decreasing in 0, ∞ and strictly

decreasing in a subinterval of 0, ∞, then c  {c n}∞

n1 ∈ l p

ψ1−p Proof By H ¨older’s inequality13 and Lemmas2.1-2.2, we obtain

c n p

∞



m1

k λ m, n



m 1−λ/r/q

n 1−λ/s/p a m



n 1−λ/s/p

m 1−λ/r/q

p

≤

∞



m1

k λ m, n m 1−λ/rp/q

p m

 ∞



m1

k λ m, n n 1−λ/sq/p

m1−λ/r

p−1

≤ ω p−1

λ r, nn1−pλ/s∞

m1

k λ m, n m 1−λ/rp/q

p m

 k p−1

λ rn1−pλ/s∞

m1

k λ m, n m 1−λ/rp/q

p

m ,

c p,ψ1−p

∞



n1

n pλ/s−1 c n p

1/p



∞



n1

n pλ/s−1

∞



m1

k λ m, na m

p 1/p

≤ k 1/q

λ r

∞



n1

∞



m1

k λ m, n m 1−λ/rp/q

p m

1/p

 k 1/q

λ r

∞

m1

∞

n1

k λ m, n m λ/r

n1−λ/s



m p1−λ/r−1 a p m

1/p

< k 1/q λ r

∞

m1

 λ s, mm p1−λ/r−1 a p m

1/p

 k λ ra p,φ < ∞.

2.4

Therefore, c  {c n}∞n1 ∈ l p

ψ1−p

For a m ≥ 0, a  {a m}∞m1 ∈ l p

φ , define a Hilbert-type operator T : l p φ → l p

ψ1−p as Ta  c, satisfying c  {c n}∞n1 ,

Tan : c n∞

m1

k λ m, na m n ∈ N. 2.5

In view ofLemma 2.3, c ∈ l p ψ1−p and then T exists If there exists M > 0, such that for any

a ∈ l φ p , Ta p,ψ1−p ≤ Ma p,φ , then T is bounded and T  sup ap,φ1 Ta p,ψ1−p ≤ M Hence

by2.4, we find T ≤ k λ r and T is bounded.

Theorem 2.4 As the assumption of Lemma 2.3, it follows T  k λ r.

Proof For a m , b n ≥ 0, a  {a m}∞m1 ∈ l p

φ , b  {b n}∞n1 ∈ l q

ψ , a p,φ > 0, b q,ψ > 0, by H¨older’s

inequality12, we find

Ta, b ∞

n1



n λ/s−1/p

∞



m1

k λ m, na m





n −λ/s1/p b n



≤

∞

n1

n pλ/s−1

∞

m1

k λ m, na m

p 1/p

b q,ψ

2.6

Then by2.4, we obtain

Ta, b < k λ ra p,φ b q,ψ 2.7

For 0 < ε < min{pλ/r, qλ/s}, setting  a  { a n}∞

n1 , b  {b n}∞

n1asa n  n λ/r−ε/p−1 , b n 

n λ/s−ε/q−1 , for n ∈ N, if there exists a constant 0 < k ≤ k λ r, such that 2.7 is still valid when

we replace k λ r by k, then byLemma 2.1,

εT  a, b < εk a p,φ b q,ψ  εk



1∞

n2

1

n1ε



< εk



1

∞

1

1

y1εdy



 kε  1, 2.8

ε T  a, b

 ε∞

n1

∞

m1

k λ m, nm λ/r−1 m −ε/p



n λ/s−ε/q−1

≥ ε∞

n1

∞

1

k λ x, nx λ/r−ε/p−1 dx



n λ/s−ε/q−1

 ε

∞

1

∞

n1

k λ x, nn λ/s−ε/q−1



x λ/r−ε/p−1 dx

≥ ε

∞

1

∞

1

k λ x, yy λ/s−ε/q−1 x λ/r−ε/p−1 dy



dx.

2.9

In view of2.8 and 2.9, setting u  x/y, by Fubini’s theorem 13, it follows

kε  1 > ε

∞

1

x −1−ε

x

0

k λ u, 1u λ/rε/q−1 du dx



1

0

k λ u, 1u λ/rε/q−1 du  ε

∞

1

x −1−ε

x

1

k λ u, 1u λ/rε/q−1 du dx



1

0

k λ u, 1u λ/rε/q−1 du  ε

∞

1

∞

u

x −1−ε dx k λ u, 1u λ/rε/q−1 du



1

0

k λ u, 1u λ/rε/q−1 du 

∞

1

k λ u, 1u λ/r−ε/p−1 du.

2.10

Setting ε → 0in the above inequality, by Fatou’s lemma14, we find

k ≥ lim

ε → 0

1

0

k λ u, 1u λ/rε/q−1 du 

∞

1

k λ u, 1u λ/r−ε/p−1 du



≥

1

0

lim

ε → 0k λ u, 1u λ/rε/q−1 du 

∞

1

lim

ε → 0k λ u, 1u λ/r−ε/p−1 du



1

0

k λ u, 1u λ/r−1 du 

∞

1

k λ u, 1u λ/r−1 du  k λ r.

2.11

Hence k  k λ r is the best value of 2.7 We conform that k λ r is the best value of 2.4 Otherwise, we can get a contradiction by2.6 that the constant factor in 2.7 is not the best possible It follows thatT  k λ r.

3 An Extended Basic Theorem on Hilbert-Type Inequalities

Still setting φx  x p1−λ/r−1 , ψx  x q1−λ/s−1 , ψ1−px  x pλ/s−1 , x ∈ 0, ∞, and l φ p  {a  {a n}∞n1; a p,φ: {∞n1 φn|a n|p}1/p

< ∞}, we have the following theorem.

Theorem 3.1 Suppose that p, r > 1, 1/p  1/q  1, 1/r  1/s  1, λ > 0, k λ x, y≥ 0

is a homogeneous function of −λ-degree, k λ r  ∞0k λ u, 1u λ/r−1 du is a positive number, both

k λ u, 1u λ/r−1 and k λ 1, uu λ/s−1 are decreasing in 0, ∞ and strictly decreasing in a subinterval of

0, ∞ If a n , b n ≥ 0, a  {a n}∞n1 ∈ l p φ , b  {b n}∞n1 ∈ l q ψ , a p,φ > 0, b q,ψ > 0, then one has the equivalent inequalities as

Ta, b ∞

n1

∞



m1

k λ m, na m b n < k λ ra p,φ b q,ψ , 3.1

Ta p p,ψ1−p ∞

n1

n pλ/s−1

∞



m1

k λ m, na m

p

< k p λ ra p

p,φ , 3.2

where the constant factors k λ r and k p

λ r are the best possible.

Proof In view of2.7 and 2.4, we have 3.1 and 3.2 Based onTheorem 2.4, it follows that the constant factors in3.1 and 3.2 are the best possible

If 3.2 is valid, then by 2.6, we have 3.1 Suppose that 3.1 is valid By 2.4,

Ta p

p,ψ1−p < ∞ If Ta p p,ψ1−p  0, then 3.2 is naturally valid; if Ta p

p,ψ1−p > 0, setting

b n  n pλ/s−1∞m1 k λ m, na mp−1

, then 0 < b q q,ψ  Ta p

p,ψ1−p< ∞ By 3.1, we obtain

b q q,ψ  Ta p

p,ψ1−p Ta, b < k λ ra p,φ b q,ψ

b q−1 q,ψ  Ta p,ψ1−p< k λ ra p,φ ,

3.3

and we have3.2 Hence 3.1 and 3.2 are equivalent

Remark 3.2 a For λ  1, s  p, r  q, 3.1 and 3.2 reduce, respectively, to 1.6 and 1.7 Hence,Theorem 3.1is an extension of Theorem A

b Replacing the condition “k λ u, 1u λ/r−1 and k λ 1, uu λ/s−1are decreasing in0, ∞

and strictly decreasing in a subinterval of 0, ∞” by “for 0 < λ ≤ min{r, s}, k λ u, 1 and k λ 1, u are decreasing in 0, ∞ and strictly decreasing in a subinterval of 0, ∞,” the

theorem is still valid Then in particular,

i for k αλ x, y  1/x α  y αλ α, λ > 0, αλ ≤ min{r, s} in 3.1, we find

k αλ r 

∞

0

u αλ/r−1

u α 1λ du  1

α

∞

0

v λ/r−1

v  1 λ dv  1

α B

λ

r ,

λ

and then it deduces to1.11;

ii for k λ x, y  1/ max{x λ , y λ } 0 < λ ≤ min{r, s} in 3.1, we find

k λ r 

∞

0

1 max{uλ , 1} u

λ/r−1 du  rs

and then it deduces to the best extension of1.5 as

∞



n1

∞



m1

a m b n

max{m, n} λ < rs

λ a p,φ b q,ψ; 3.6

iii for k λ x, y  lnx/y/x λ − y λ  0 < λ ≤ min{r, s} in 3.1, we find 3

k λ r 

∞

0

ln u

u λ− 1u λ/r−1 du 



π

λ sinπ/r

2

andln u/u λ− 1< 0, and then it deduces to the best extension of 1.6 as

∞



n1

∞



m1

lnm/nam b n

m λ − n λ <



π

λ sinπ/r

2

a p,φ b q,ψ 3.8

1 H Weyl, Singulare Integralgleichungen mit besonderer Beriicksichtigung des Fourierschen Integraltheorems,

Inaugeral dissertation, University of G ¨ottingen, G ¨ottingen, Germany, 1908

2 G H Hardy, “Note on a theorem of Hilbert concerning series of positive terms,” Proceedings of the

London Mathematical Society, vol 23, no 2, pp 45–46, 1925.

3 G H Hardy, J E Littlewood, and G P´olya, Inequalities, Cambridge University Press, Cambridge, UK,

1934

4 F F Bonsall, “Inequalities with non-conjugate parameters,” The Quarterly Journal of Mathematics, vol.

2, no 1, pp 135–150, 1951

5 D S Mitrinovi´c, J E Peˇcari´c, and A M Fink, Inequalities Involving Functions and Their Integrals

and Derivatives, vol 53 of Mathematics and Its Applications (East European Series), Kluwer Academic

Publishers, Dordrecht, The Netherlands, 1991

6 B Yang, “A generalization of the Hilbert double series theorem,” Journal of Nanjing University

Mathematical Biquarterly, vol 18, no 1, pp 145–152, 2001.

7 B Yang, “An extension of Hardy-Hilbert’s inequality,” Chinese Annals of Mathematics Series A, vol 23,

no 2, pp 247–254, 2002

8 B Yang, “On new extensions of Hilbert’s inequality,” Acta Mathematica Hungarica, vol 104, no 4, pp.

291–299, 2004

9 B Yang, “On best extensions of Hardy-Hilbert’s inequality with two parameters,” Journal of

Inequalities in Pure and Applied Mathematics, vol 6, no 3, article 81, pp 1–15, 2005.

10 M Krni´c and J Peˇcari´c, “General Hilbert’s and Hardy’s inequalities,” Mathematical Inequalities &

Applications, vol 8, no 1, pp 28–51, 2005.

11 B Yang, “On the norm of a Hilbert’s type linear operator and applications,” Journal of Mathematical

Analysis and Applications, vol 325, no 1, pp 529–541, 2007.

12 B Yang, “On the norm of a self-adjoint operator and applications to the Hilbert’s type inequalities,”

Bulletin of the Belgian Mathematical Society, vol 13, no 4, pp 577–584, 2006.

13 J Kuang, Applied Inequalities, Shandong Science and Technology Press, Jinan, China, 2004.

14 J Kuang, Introduction to Real Analysis, Hunan Education Press, Changsha, China, 1996.

For a m ≥ 0, a  {a m}∞m1 ∈ l p

φ...