On a Hilbert Type Operator with a Class of Homogeneous Kernels

Journal of Inequalities and Applications Volume 2009, Article ID 572176,9pages 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,[email protected]

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.

Copyrightq2009 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

{an}∞n1, {bn}∞n1are real sequences, 0<∞n1a2n<∞and 0<

_∞

n1b2n<∞,then1

∞

n1

∞

m1

ambn mn < π

_∞

n1

a2

n ∞

n1

b2

n

1/2

, 1.1

where the constant factorπis the best possible. In 1925, Hardy gave an extension of1.1by introducing one pair of conjugate exponentsp, q1/p1/q1as2. Ifp >1, an, bn ≥0, 0<∞n1apn<∞, and 0<∞n1bqn<∞,then

∞

n1

∞

m1

ambn mn <

π sinπ/p

_∞

n1

apn

1/p_∞

n1

bqn

1/q

where the constant factorπ/sinπ/pis the best possible. We named1.2Hardy-Hilbert’s inequality. In 1934, Hardy et al.3gave some applications of1.1-1.2and a basic theorem with the general kernelsee3, Theorem 318.

Theorem A. Suppose thatp >1, 1/p1/q1, kx, yis a homogeneous function of−1-degree, andk ∞₀ ku,1u−1/p_{duis a positive number. If bothku,1u}−1/p _{andk1, uu}−1/q_{are strictly} decreasing functions for u > 0,an, bn ≥ 0, 0 < ap

_∞

n1apn1/p < ∞, and 0 < bq

∞n1bqn1/q<∞,then one has the following equivalent inequalities:

∞

n1 ∞

m1

km, nambn< kapbq, 1.3

∞

n1

_∞

m1

km, nam

p

< kpapp, 1.4

where the constant factorskandkp_{are the best possible.}

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

iforkx, y 1/xyin1.3, it reduces1.2,

iiforkx, y 1/max{x, y}in1.3, it reduces tosee3, Theorem 341

∞

n1

∞

m1

ambn

max{m, n} < pq

_∞

n1

apn

1/p_∞

n1

bqn

1/q

, 1.5

iiiforkx, y lnx/y/x−yin1.3, it reduces tosee3, Theorem 342

∞

n1

∞

m1

lnm/nambn m−n <

π sinπ/p

2∞

n1

apn

1/p_∞

n1

bqn

1/q

. 1.6

Hardy also gave some multiple extensions of 1.3 see3, Theorem 322. About introducing one pair of nonconjugate exponents p, q in 1.1, Hardy et al.3 gave that ifp, q >1, 1/p1/q≥1, 0< λ2−1/p1/q≤1,then

∞

n1

∞

m1

ambn

mnλ ≤Kp, q

_∞

n1

apn

1/p_∞

n1

bqn

1/q

. 1.7

In 1951, Bonsall4considered1.7in the case of general kernel; in 1991, Mitrinovi´c et al.5 summarized the above results.

In 2001, Yang6gave an extension of1.1as for 0< λ≤4,

∞

n1

∞

m1

ambn

mnλ < B λ 2,

λ 2

∞

n1

n1−λa2n ∞

n1

n1−λbn2

1/2

where the constantBλ/2, λ/2,is the best possibleBu, vis the Beta function. Forλ 1,

1.8reduces to1.1. And Yang7also gave an extension of1.2as

∞

n1

∞

m1

ambn mλ_nλ <

π λsinπ/p

_∞

n1

np−11−λapn

1/p_∞

n1

nq−11−λbqn

1/q

, 1.9

where the constant factorπ/λsinπ/p 0< λ≤2is the best possible. In 2004, Yang8published the dual form of1.2as follows:

∞

n1

∞

m1

ambn mn<

π sinπ/p

_∞

n1

np−2apn

1/p_∞

n1

nq−2bqn

1/q

, 1.10

whereπ/sinπ/pis the best possible. Forpq2,both1.10and1.2reduce to1.1. It means that there are more than two different best extensions of1.1. In 2005, Yang9gave an extension of1.8–1.10with two pairs of conjugate exponentsp, q, r, s p, r >1, and two parametersα, λ >0 αλ≤min{r, s}as

∞

n1

∞

m1

ambn

mα_nαλ < kαλr

_∞

n1

np1−αλ/r−1apn

1/p_∞

n1

nq1−αλ/s−1bnq

1/q

, 1.11

where the constant factorkαλr 1/αBλ/r, λ/sis the best possible; Krni´c and Peˇcari´c

10also considered1.11in the general homogeneous kernel, but the best possible property of the constant factor was not proved by10.

Note. ForABαβ1 in10, inequality37, it reduces to the equivalent result of3.1 in this paper.

In 2006-2007, some authors also studied the operator expressing of1.3and1.4. Suppose thatkx, y≥0is a symmetric function withky, x kx, y,andk0p:

_∞

0kx, yx/y 1/r

dyr p, q;x > 0 is a positive number independent of x. Define an operatorT : lr _{→ l}r _{r p, qas follows. Fora}

m ≥ 0, a {am}∞m1 ∈ lp,there exists only

Tac{cn}∞n1∈lp,satisfying

Tan cn: ∞

m1

km, nam n∈N. 1.12

Then the formal inner product ofTaandbare defined as follows:

Ta, b ∞

n1

∞

m1

In 2007, Yang 11 proved that if for ε ≥ 0 small enough, kx, yx/y1ε/r is strictly decreasing fory >0,the integral∞₀ kx, yx/y1ε/rdy kεpis also a positive number independent ofx >0, kεp k0p o1 ε → 0,and

∞

m1

1 m1ε

1

0

km, t m t

1ε/r

dtO1 ε−→0; rp, q, 1.14

then Tp k0p; in this case, ifam, bn ≥ 0, a {am}m∞1 ∈ lp, b {bn}∞n1 ∈ lq, ap > 0, bq>0,then we have two equivalent inequalities as

Ta, b<Tpapbq; Tap<Tpap, 1.15

where the constant factorTpis the best possible. In particular, forkx, ybeing−1-degree homogeneous, inequalities1.15reduce to1.3-1.4 in the symmetric kernel. Yang12 also considered1.15in the real spacel2_.

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

Ifkλx, yis a measurable function, satisfying forλ, u, x, y >0, kλux, uy u−λkλx, y,then we callkλx, ythe homogeneous function of−λ-degree.

Forkλx, y≥0,settingxuy,we findkλx, y1/x1−λ/r 1/y1λ/skλu,1uλ/r−1. Hence, the following two words are equivalent: a kλu,1uλ/r−1 is decreasing in 0,∞ and strictly decreasing in a subinterval of 0,∞;bfor any y > 0, kλx, y1/x1−λ/r is decreasing inx∈0,∞and strictly decreasing in a subinterval of0,∞. The following two words are also equivalent:akλ1, uuλ/s−1is decreasing in0,∞and strictly decreasing in a subinterval of0,∞;bfor anyx >0,kλx, y1/y1−λ/sis decreasing iny∈0,∞and strictly decreasing in a subinterval of0,∞.

Lemma 2.1. Iffx≥0is decreasing in0,∞and strictly decreasing in a subinterval of0,∞, andI0:

_∞

0 fxdx <∞,then

I1:

_∞

1

fxdx≤ ∞

n1

fn< I0. 2.1

Proof. By the assumption, we findn_n1fxdx≤fn≤n_n−1fxdxn∈N,and there exists

n0−1, n0⊂0,∞,such thatfn0<

_n0

n0−1fxdx.Hence,

I1

∞

n1

_n1

n

fxdx≤ ∞

n1

fn< ∞

n1

_n

n−1

Lemma 2.2. Ifr >1, 1/r1/s1, λ >0, kλx, y≥0is a homogeneous function of−λ-degree, andkλr :

_∞

0kλu,1uλ/r−1duis a positive number, theni

_∞

0 kλ1, uuλ/s−1du kλr; ii

forx, y∈0,∞,setting the weight functions as

ωλr, y:

_∞

0

kλx, y y λ/s

x1−λ/rdx, λs, x:

_∞

0

kλx, y x λ/r

y1−λ/sdy, 2.3

thenωλr, y λs, x kλr.

Proof. i Setting v 1/u, by the assumption, we obtain ₀∞kλ1, uuλ/s−1du

_∞

0kλv,

1vλ/r−1_{dv k}

λr.ii Settingx yu andy xuin the integralsωλr, yand λs, x, respectively, in view ofi, we still find thatωλr, y λs, x kλr.

Forp >1, 1/p1/q 1,we setφx xp1−λ/r−1_{, ψx x}q1−λ/s−1_,andψ1−p_x xpλ/s−1_{, x∈0,∞.Define the real space asl}p

φ:{a{an}∞n1;ap,φ:{∞n1φn|an|p}1/p<

∞},and then we may also define the spaceslψq andlp_ψ1−p.

Lemma 2.3. As the assumption of Lemma 2.2, for am ≥ 0, a {am}∞m1 ∈ lpφ, setting cn ∞m1kλm, nam, if kλu,1uλ/r−1 andkλ1, uuλ/s−1 are decreasing in 0,∞ and strictly decreasing in a subinterval of0,∞, thenc{cn}∞n1∈l

p ψ1−p.

Proof. By H ¨older’s inequality13and Lemmas2.1-2.2, we obtain

cnp

_∞

m1

kλm, n

m1−λ/r/q n1−λ/s/pam

n1−λ/s/p m1−λ/r/q

p

≤ _∞

m1

kλm, nm

1−λ/rp/q

n1−λ/s a p m

_∞

m1

kλm, nn

1−λ/sq/p

m1−λ/r

p−1

≤ωp_λ−1r, nn1−pλ/s∞ m1

kλm, nm

1−λ/rp/q

n1−λ/s a p m

kp_λ−1rn1−pλ/s ∞

m1

kλm, nm

1−λ/rp/q

n1−λ/s a p m,

cp,ψ1−p

_∞

n1

npλ/s−1cnp

1/p

_∞

n1

npλ/s−1

_∞

m1

kλm, nam

p1/p

≤k1_λ/qr

_∞

n1

∞

m1

kλm, nm

1−λ/rp/q

n1−λ/s a p m

1/p

k1_λ/qr

_∞

m1

_∞

n1

kλm, nm λ/r

n1−λ/s

mp1−λ/r−1apm

1/p

< k1_λ/qr

_∞

m1

λs, mmp1−λ/r−1apm

1/p

kλrap,φ<∞.

2.4

Foram ≥0, a{am}m∞1 ∈lpφ,define a Hilbert-type operatorT :l p φ → l

p

ψ1−p asTac,

satisfyingc{cn}∞n1,

Tan:cn ∞

m1

kλm, nam n∈N. 2.5

In view ofLemma 2.3,c ∈ lp_ψ1−p and thenT exists. If there exists M > 0,such that for any

a∈l_φp, Tap,ψ1−p ≤ Ma_p,φ,thenT is bounded andT sup_a

p,φ1Tap,ψ1−p ≤M.Hence

by2.4, we findT ≤kλrandT is bounded.

Theorem 2.4. As the assumption ofLemma 2.3, it followsTkλr.

Proof. Foram, bn ≥0, a{am}∞m1∈lpφ, b{bn}∞n1∈lqψ, ap,φ >0, bq,ψ >0,by H ¨older’s inequality12, we find

Ta, b ∞

n1

nλ/s−1/p ∞

m1

kλm, nam

n−λ/s1/pbn

≤ _∞

n1

npλ/s−1

_∞

m1

kλm, nam

p1/p

bq,ψ.

2.6

Then by2.4, we obtain

Ta, b< kλrap,φbq,ψ. 2.7

For 0< ε <min{pλ/r, qλ/s},settinga {an}∞n1,b{bn}∞n1asan nλ/r−ε/p−1, bn nλ/s−ε/q−1_{,forn∈N,if there exists a constant 0< k≤k}

λr,such that2.7is still valid when we replacekλrbyk,then byLemma 2.1,

εTa, b< εkap,φbq,ψεk

1 ∞

n2

1 n1ε

< εk

1

_∞

1

1 y1εdy

kε1, 2.8

εTa, bε ∞

n1

_∞

m1

kλm, nmλ/r−1m−ε/p

nλ/s−ε/q−1

≥ε ∞

n1

∞

1

kλx, nxλ/r−ε/p−1dx

nλ/s−ε/q−1

ε

_∞

1

_∞

n1

kλx, nnλ/s−ε/q−1

xλ/r−ε/p−1dx

≥ε

_∞

1

_∞

1

kλx, yyλ/s−ε/q−1xλ/r−ε/p−1dy

dx.

In view of2.8and2.9, settingux/y, by Fubini’s theorem13, it follows

kε1> ε

_∞

1

x−1−ε x

0

kλu,1uλ/rε/q−1du

dx

1

0

kλu,1uλ/rε/q−1duε

_∞

1

x−1−ε x

1

kλu,1uλ/rε/q−1du

dx

₁

0

kλu,1uλ/rε/q−1duε

_∞

1

∞

u

x−1−εdx

kλu,1uλ/rε/q−1du

1

0

kλu,1uλ/rε/q−1du

_∞

1

kλu,1uλ/r−ε/p−1du.

2.10

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

k≥ lim ε→0

1

0

kλu,1uλ/rε/q−1du

_∞

1

kλu,1uλ/r−ε/p−1du

≥ 1

0

lim

ε→0kλu,1u

λ/rε/q−1_du

_∞

1

lim

ε→0kλu,1u

λ/r−ε/p−1_du

₁

0

kλu,1uλ/r−1du

_∞

1

kλu,1uλ/r−1dukλr.

2.11

Hencek kλris the best value of2.7. We conform thatkλris the best value of2.4. Otherwise, we can get a contradiction by2.6that the constant factor in2.7is not the best possible. It follows thatTkλr.

3. An Extended Basic Theorem on Hilbert-Type Inequalities

Still settingφx xp1−λ/r−1_{, ψx x}q1−λ/s−1_{, ψ}1−p_{x x}pλ/s−1_{, x∈0,∞, andl}p

φ {a

{an}∞n1; ap,φ:{

_∞

n1φn|an|p}1/p<∞},we have the following theorem.

Theorem 3.1. Suppose that p, r > 1, 1/p1/q 1, 1/r 1/s 1, λ > 0, kλx, y≥ 0 is a homogeneous function of −λ-degree, kλr

_∞

0kλu,1uλ/r−1du is a positive number, both

kλu,1uλ/r−1andkλ1, uuλ/s−1are decreasing in0,∞and strictly decreasing in a subinterval of

0,∞. Ifan, bn ≥0, a{an}∞n1 ∈lpφ, b{bn}∞n1 ∈lqψ,ap,φ >0, bq,ψ >0,then one has the equivalent inequalities as

Ta, b ∞

n1

∞

m1

kλm, nambn< kλrap,φbq,ψ, 3.1

Tap_p,ψ1−p

∞

n1

npλ/s−1

_∞

m1

kλm, nam

p

< kp_λrap_p,φ, 3.2

Proof. In view of2.7and2.4, we have3.1and3.2. Based onTheorem 2.4, it follows that the constant factors in3.1and3.2are the best possible.

If 3.2 is valid, then by 2.6, we have 3.1. Suppose that 3.1 is valid. By2.4,

Tap_p,ψ1−p < ∞. If Ta p

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

p,ψ1−p > 0, setting

bnnpλ/s−1

_∞

m1kλm, namp−1,then 0<bqq,ψTap_p,ψ1−p <∞.By3.1, we obtain

bqq,ψTap_p,ψ1−p Ta, b< kλrap,φbq,ψ

bqq,ψ−1 Tap,ψ1−p < k_λra_p,φ,

3.3

and we have3.2. Hence3.1and3.2are equivalent.

Remark 3.2. aForλ1, sp, rq,3.1and3.2reduce, respectively, to1.6and1.7. Hence,Theorem 3.1is an extension of Theorem A.

bReplacing the condition “kλu,1uλ/r−1andkλ1, uuλ/s−1are decreasing in0,∞ and strictly decreasing in a subinterval of 0,∞” by “for 0 < λ ≤ min{r, s}, kλu,1 andkλ1, uare decreasing in 0,∞and strictly decreasing in a subinterval of0,∞,” the theorem is still valid. Then in particular,

iforkαλx, y 1/xαyαλα, λ >0, αλ≤min{r, s}in3.1, we find

kαλr

_∞

0

uαλ/r−1

uα₁λdu 1 α

_∞

0

vλ/r−1

v1λdv 1 αB

λ r,

λ s

, 3.4

and then it deduces to1.11;

iiforkλx, y 1/max{xλ, yλ} 0< λ≤min{r, s}in3.1, we find

kλr

_∞

0

1 max{uλ_,1}u

λ/r−1_du rs

λ, 3.5

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

∞

n1

∞

m1

ambn

max{m, n}λ < rs

λap,φbq,ψ; 3.6

iiiforkλx, y lnx/y/xλ−yλ 0< λ≤min{r, s}in3.1, we find3

kλr

_∞

0

lnu uλ₋₁u

λ/r−1_du

π λsinπ/r

2

, 3.7

andlnu/uλ_{−1<0, and then it deduces to the best extension of1.6as}

∞

n1

∞

m1

lnm/nambn mλ_−nλ <

π λsinπ/r

2

References

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 ofMathematics 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.