Hindawi Publishing Corporation
Journal of Inequalities and Applications
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.
Copyright q 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
∞ 2
∞ 2
∞
{an }∞
n1 , {bn }n1 are real sequences, 0 <
n1 an < ∞ and 0 <
n1 bn < ∞, then 1
1/2
∞
∞ ∞
∞
am bn
2
2
<π
an bn
,
mn
n1 m1
n1
n1
1.1
where the constant factor π is the best possible. In 1925, Hardy gave an extension of 1.1 by
introducing one pair of conjugate exponents p, q1/p 1/q 1 as 2. If p > 1, an , bn ≥ 0,
∞ q
p
0< ∞
n1 an < ∞, and 0 <
n1 bn < ∞, then
∞
∞ π
am bn
<
m n sinπ/p
n1 m1
∞
n1
p
an
1/p ∞
n1
1/q
q
bn
,
1.2
2
Journal of Inequalities and Applications
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 kernel see 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
p 1/p
< ∞, and 0 < bq decreasing functions for u > 0, an , bn ≥ 0, 0 < ap ∞
n1 an ∞ q 1/q
n1 bn < ∞, then one has the following equivalent inequalities:
∞
∞ km, nam bn < kap bq ,
n1 m1
∞
∞
n1
1.3
p
p
1.4
< kp ap ,
km, nam
m1
where the constant factors k and kp are the best possible.
Note. Hardy did not prove this theorem in 3. In particular, we find some classical Hilberttype 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
1/p 1/q
∞
∞
am bn
p
q
< pq
an
bn
,
max{m, n}
n1 m1
n1
n1
∞
∞ 1.5
iii for kx, y lnx/y/x − y in 1.3, it reduces to see 3, Theorem 342
1/p 1/q
2 ∞ ∞
∞
∞
π
lnm/nam bn
p
q
<
an
bn
.
m−n
sinπ/p
n1 m1
n1
n1
1.6
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
∞
∞ am bn
n1 m1
m nλ
≤ Kp, q
∞
p
an
1/p ∞
n1
1/q
q
bn
1.7
.
n1
In 1951, Bonsall 4 considered 1.7 in the case of general kernel; in 1991, Mitrinović et al. 5
summarized the above results.
In 2001, Yang 6 gave an extension of 1.1 as for 0 < λ ≤ 4,
∞
∞ λ λ
,
<B
λ
2 2
n1 m1 m n
am bn
∞
n1
n1−λ a2n
∞
n1
1/2
n1−λ bn2
,
1.8
Journal of Inequalities and Applications
3
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
1/p 1/q
∞
∞ ∞
∞
am bn
π
p−11−λ p
q−11−λ q
<
n
an
n
bn
,
λ
λ
λ sinπ/p n1
n1 m1 m n
n1
1.9
where the constant factor π/λ sinπ/p 0 < λ ≤ 2 is the best possible.
In 2004, Yang 8 published the dual form of 1.2 as follows:
∞
∞ π
am bn
<
m
n
sinπ/p
n1 m1
∞
p
np−2 an
1/p ∞
n1
1/q
q
nq−2 bn
1.10
,
n1
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 of 1.8–1.10 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
am bn
mα nα
λ < kαλ r
∞
p
np1−αλ/r−1 an
1/p ∞
n1
1/q
q
nq1−αλ/s−1 bn
,
1.11
n1
where the constant factor kαλ r 1/αBλ/r, λ/s is the best possible; Krnić and Pečarić
10 also considered 1.11 in the general homogeneous kernel, but the best possible property
of the constant factor was not proved by 10.
Note. For A B α β 1 in 10, inequality 37, it reduces to the equivalent result of 3.1
in this paper.
In 2006-2007, some authors also studied the operator expressing of 1.3 and 1.4.
Suppose that kx, y≥ 0 is a symmetric function with ky, x kx, y, and k0 p :
∞
kx,
yx/y1/r dy r p, q; x > 0 is a positive number independent of x. Define an
0
p
operator T : lr → lr r p, q as follows. For am ≥ 0, a {am }∞
m1 ∈ l , there exists only
∞
p
T a c {cn }n1 ∈ l , satisfying
T an cn :
∞
km, nam
n ∈ N.
1.12
m1
Then the formal inner product of T a and b are defined as follows:
T a, b ∞
∞ n1 m1
km, nam bn .
1.13
4
Journal of Inequalities and Applications
In 2007, Yang 11 proved that if for ε ≥ 0 small enough, kx, yx/y1ε/r is strictly
∞
decreasing for y > 0, the integral 0 kx, yx/y1ε/r dy kε p is also a positive number
independent of x > 0, kε p k0 p o1 ε → 0 , and
∞
1
1ε
m
m1
1
km, t
0
m
t
1ε/r
dt O1
ε −→ 0 ; r p, q ,
1.14
∞
p
q
then T p k0 p; in this case, if am , bn ≥ 0, a {am }∞
m1 ∈ l , b {bn }n1 ∈ l , ap >
0, bq > 0, then we have two equivalent inequalities as
T a, b < T p ap bq ;
T ap < T p ap ,
1.15
where the constant factor T p is the best possible. In particular, for kx, y being −1-degree
homogeneous, inequalities 1.15 reduce to 1.3-1.4 in the symmetric kernel. Yang 12
also considered 1.15 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 kλ x, y is a measurable function, satisfying for λ, u, x, y > 0, kλ ux, uy u−λ kλ x, y, then
we call kλ x, y the homogeneous function of −λ-degree.
For kλ x, y ≥ 0, setting x uy, we find kλ x, y1/x1−λ/r 1/y1λ/s kλ u, 1uλ/r−1 .
Hence, the following two words are equivalent: a kλ u, 1uλ/r−1 is decreasing in 0, ∞
and strictly decreasing in a subinterval of 0, ∞; b for any y > 0, kλ 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: a kλ 1, uuλ/s−1 is decreasing in 0, ∞ and strictly decreasing in
a subinterval of 0, ∞; b for any x > 0, kλ x, y1/y1−λ/s is decreasing in y ∈ 0, ∞ and
strictly decreasing in a subinterval of 0, ∞.
Lemma 2.1.
∞ If fx≥ 0 is decreasing in 0, ∞ and strictly decreasing in a subinterval of 0, ∞,
and I0 : 0 fxdx < ∞, then
∞
I1 :
fxdx ≤
1
∞
fn < I0 .
2.1
n1
n1
n
Proof. By the assumption, we find n fxdx ≤ fn ≤ n−1 fxdx n ∈ N, and there exists
n
n0 − 1, n0 ⊂ 0, ∞, such that fn0 < n00 −1 fxdx. Hence,
I1 ∞ n1
n1
n
fxdx ≤
∞
n1
fn <
∞ n
n1
n−1
fxdx I0 .
2.2
Journal of Inequalities and Applications
5
function of −λ-degree,
Lemma 2.2. If r > 1, 1/r 1/s 1, λ > 0, kλ x, y≥ 0 is a homogeneous
∞
∞
and kλ r : 0 kλ u, 1uλ/r−1 du is a positive number, then i 0 kλ 1, uuλ/s−1 du kλ r; ii
for x, y ∈ 0, ∞, setting the weight functions as
ωλ r, y :
∞
0
yλ/s
kλ x, y 1−λ/r dx,
x
λ s, x :
∞
kλ x, y
0
xλ/r
dy,
y1−λ/s
2.3
then ωλ r, y λ s, x kλ r.
∞
∞
Proof. i Setting v 1/u, by the assumption, we obtain 0 kλ 1, uuλ/s−1 du 0 kλ v,
1vλ/r−1 dv kλ r. ii Setting x yu and y xu in the integrals ωλ r, y and λ s, x,
respectively, in view of i, we still find that ωλ r, y λ s, x kλ r.
x
pλ/s−1
For p > 1, 1/p 1/q 1, we set φx xp1−λ/r−1 , ψx xq1−λ/s−1 , and ψ 1−p x ∞
p
p 1/p
, x ∈ 0, ∞. Define the real space as lφ : {a {an }∞
<
n1 ; ap,φ : { n1 φn|an | }
q
p
∞}, and then we may also define the spaces lψ and lψ 1−p .
Lemma 2.3. As the assumption of Lemma 2.2, for am ≥ 0, a {am }∞
m1 ∈ lφ , setting
∞
λ/r−1
λ/s−1
cn m1 kλ m, nam , if kλ u, 1u
and kλ 1, uu
are decreasing in 0, ∞ and strictly
p
∈
l
.
decreasing in a subinterval of 0, ∞, then c {cn }∞
n1
ψ 1−p
p
Proof. By Hölder’s inequality 13 and Lemmas 2.1-2.2, we obtain
p
cn
cp,ψ 1−p
p
n1−λ/s/p
m1−λ/r/q
kλ m, n 1−λ/s/p am
n
m1−λ/r/q
m1
p−1
∞
∞
m1−λ/rp/q p
n1−λ/sq/p
≤
kλ m, n
am
kλ m, n
n1−λ/s
m1−λ/r
m1
m1
∞
m1−λ/rp/q p
p−1
≤ ωλ r, nn1−pλ/s
kλ m, n
am
n1−λ/s
m1
∞
m1−λ/rp/q p
p−1
kλ rn1−pλ/s
kλ m, n
am ,
n1−λ/s
m1
1/p p 1/p
∞
∞
∞
pλ/s−1 p
pλ/s−1
n
cn
n
kλ m, nam
n1
≤
∞
1/q
kλ r
∞
∞ n1
kλ m, n
m
m1
1−λ/rp/q
p
am
1/p
n1−λ/s
1/p
∞ ∞
mλ/r
1/q
p1−λ/r−1 p
kλ r
kλ m, n 1−λ/s m
am
n
m1 n1
1/p
∞
1/q
p1−λ/r−1 p
< kλ r
λ s, mm
am
kλ rap,φ < ∞.
n1 m1
m1
Therefore, c {cn }∞
n1 ∈ lψ 1−p .
p
2.4
6
Journal of Inequalities and Applications
For am ≥ 0, a {am }∞
m1 ∈ lφ , define a Hilbert-type operator T : lφ → lψ 1−p as T a c,
∞
satisfying c {cn }n1 ,
p
p
T an : cn ∞
kλ m, nam
p
n ∈ N.
2.5
m1
p
In view of Lemma 2.3, c ∈ lψ 1−p and then T exists. If there exists M > 0, such that for any
p
a ∈ lφ , T ap,ψ 1−p ≤ Map,φ , then T is bounded and T supap,φ 1 T ap,ψ 1−p ≤ M. Hence
by 2.4, we find T ≤ kλ r and T is bounded.
Theorem 2.4. As the assumption of Lemma 2.3, it follows T kλ r.
∞
Proof. For am , bn ≥ 0, a {am }∞
m1 ∈ lφ , b {bn }n1 ∈ lψ , ap,φ > 0, bq,ψ > 0, by Hölder’s
inequality 12, we find
p
T a, b ∞
q
λ/s−1/p
n
∞
n1
m1
≤
∞
npλ/s−1
n1
kλ m, nam
∞
n−λ/s1/p bn
2.6
p 1/p
kλ m, nam
bq,ψ .
m1
Then by 2.4, we obtain
T a, b < kλ rap,φ bq,ψ .
2.7
∞
For 0 < ε < min{pλ/r, qλ/s}, setting a {
an }∞
n nλ/r−ε/p−1 , bn n1 , b {bn }n1 as a
, for n ∈ N, if there exists a constant 0 < k ≤ kλ r, such that 2.7 is still valid when
n
we replace kλ r by k, then by Lemma 2.1,
λ/s−ε/q−1
∞
∞
1
1
< εk 1 dy kε 1,
εT a, b < εk
ap,φ bq,ψ εk 1 1ε
1ε
1 y
n2 n
2.8
∞ ∞
λ/r−1
−ε/p
nλ/s−ε/q−1
kλ m, nm
m
ε T a, b ε
n1 m1
≥ε
∞ ∞
n1
ε
1
∞ ∞
1
≥ε
kλ x, nxλ/r−ε/p−1 dx nλ/s−ε/q−1
kλ x, nnλ/s−ε/q−1 xλ/r−ε/p−1 dx
n1
∞ ∞
1
1
2.9
kλ x, yy
λ/s−ε/q−1 λ/r−ε/p−1
x
dy dx.
Journal of Inequalities and Applications
7
In view of 2.8 and 2.9, setting u x/y, by Fubini’s theorem 13, it follows
kε 1 > ε
∞
1
x−1−ε
x
1
kλ u, 1uλ/rε/q−1 du dx
0
kλ u, 1u
du ε
λ/rε/q−1
0
1
x
−1−ε
x
1
kλ u, 1u
du ε
λ/rε/q−1
0
1
∞
kλ u, 1u
du λ/rε/q−1
x
∞
0
kλ u, 1u
du dx
1
∞ ∞
1
λ/rε/q−1
−1−ε
2.10
dx kλ u, 1uλ/rε/q−1 du
u
kλ u, 1uλ/r−ε/p−1 du.
1
Setting ε → 0 in the above inequality, by Fatou’s lemma 14, we find
k ≥ lim
ε → 0
≥
1
1
kλ u, 1u
λ/rε/q−1
0
kλ u, 1uλ/r−ε/p−1 du
1
lim kλ u, 1uλ/rε/q−1 du 0 ε→0
1
du ∞
kλ u, 1uλ/r−1 du 0
∞
∞
2.11
lim kλ u, 1uλ/r−ε/p−1 du
1 ε→0
kλ u, 1uλ/r−1 du kλ r.
1
Hence k kλ r is the best value of 2.7. We conform that kλ r is the best value of 2.4.
Otherwise, we can get a contradiction by 2.6 that the constant factor in 2.7 is not the best
possible. It follows that T kλ r.
3. An Extended Basic Theorem on Hilbert-Type Inequalities
p
Still setting φx xp1−λ/r−1 , ψx xq1−λ/s−1 , ψ 1−p x xpλ/s−1 , x ∈ 0, ∞, and lφ {a ∞
p 1/p
< ∞}, we have the following theorem.
{an }∞
n1 ; ap,φ : { n1 φn|an | }
Theorem 3.1. Suppose that p, r > 1, 1/p 1/q 1, 1/r 1/s 1, λ > 0, kλ x, y≥ 0
∞
is a homogeneous function of −λ-degree, kλ r 0 kλ u, 1uλ/r−1 du is a positive number, both
λ/r−1
λ/s−1
and kλ 1, uu
are decreasing in 0, ∞ and strictly decreasing in a subinterval of
kλ u, 1u
p
q
∞
0, ∞. If an , bn ≥ 0, a {an }∞
n1 ∈ lφ , b {bn }n1 ∈ lψ , ap,φ > 0, bq,ψ > 0, then one has the
equivalent inequalities as
T a, b ∞
∞ kλ m, nam bn < kλ rap,φ bq,ψ ,
n1 m1
p
T ap,ψ 1−p
∞
pλ/s−1
n
n1
∞
p
kλ m, nam
m1
p
3.1
where the constant factors kλ r and kλ r are the best possible.
p
p
< kλ rap,φ ,
3.2
8
Journal of Inequalities and Applications
Proof. In view of 2.7 and 2.4, we have 3.1 and 3.2. Based on Theorem 2.4, it follows that
the constant factors in 3.1 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,
p
p
p
T ap,ψ 1−p < ∞. If T ap,ψ 1−p 0, then 3.2 is naturally valid; if T ap,ψ 1−p > 0, setting
q
p
p−1
bn npλ/s−1 ∞
, then 0 < bq,ψ T ap,ψ 1−p < ∞. By 3.1, we obtain
m1 kλ m, nam q
p
bq,ψ T ap,ψ 1−p T a, b < kλ rap,φ bq,ψ
q−1
bq,ψ
3.3
T ap,ψ 1−p < kλ rap,φ ,
and we have 3.2. 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 3.1 is an extension of Theorem A.
b Replacing the condition “kλ u, 1uλ/r−1 and kλ 1, uuλ/s−1 are decreasing in 0, ∞
and strictly decreasing in a subinterval of 0, ∞” by “for 0 < λ ≤ min{r, s}, kλ u, 1
and kλ 1, u are decreasing in 0, ∞ and strictly decreasing in a subinterval of 0, ∞,” the
theorem is still valid. Then in particular,
i for kαλ x, y 1/xα yα λ α, λ > 0, αλ ≤ min{r, s} in 3.1, we find
kαλ r ∞
0
uαλ/r−1
1
λ du α
α
u 1
∞
0
λ λ
1
,
,
dv B
α
r s
v 1λ
vλ/r−1
3.4
and then it deduces to 1.11;
ii for kλ x, y 1/ max{xλ , yλ } 0 < λ ≤ min{r, s} in 3.1, we find
kλ r ∞
0
1
rs
uλ/r−1 du ,
λ
max{uλ , 1}
3.5
and then it deduces to the best extension of 1.5 as
∞
∞ n1 m1
am bn
max{m, n}
λ
<
rs
ap,φ bq,ψ ;
λ
3.6
iii for kλ x, y lnx/y/xλ − yλ 0 < λ ≤ min{r, s} in 3.1, we find 3
kλ r ∞
0
2
ln u λ/r−1
π
u
du ,
λ sinπ/r
uλ − 1
3.7
and ln u/uλ − 1 < 0, and then it deduces to the best extension of 1.6 as
2
∞
∞ lnm/nam bn
π
<
ap,φ bq,ψ .
λ sinπ/r
mλ − nλ
n1 m1
3.8
Journal of Inequalities and Applications
9
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