Volume 2010, Article ID 812636, 10 pagesdoi:10.1155/2010/812636 Research Article On a New Hilbert-Hardy-Type Integral Operator and Applications 1 Department of Mathematics, Zhaoqing Univ
Trang 1Volume 2010, Article ID 812636, 10 pages
doi:10.1155/2010/812636
Research Article
On a New Hilbert-Hardy-Type Integral Operator and Applications
1 Department of Mathematics, Zhaoqing University, Guangdong, Zhaoqing 526061, China
2 Department of Mathematics, Guangdong Institute of Education, Guangdong, Guangzhou 510303, China
Correspondence should be addressed to Bicheng Yang,bcyang@pub.guangzhou.gd.cn
Received 7 September 2010; Accepted 26 October 2010
Academic Editor: Sin E Takahasi
Copyrightq 2010 X Liu and B 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
By applying the way of weight functions and a Hardy’s integral inequality, a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained As applications, a new Hilbert-Hardy-type inequality similar to Hilbert-type integral inequality is given, and two equivalent inequalities with the best constant factors as well as some particular examples are considered
1 Introduction
Theorem A If kx, y ≥ 0 is a homogeneous function of degree −1 in 0, ∞ × 0, ∞, p > 1,
1/p 1/q 1, and k p 0∞k u, 1u −1/p du ∈ 0, ∞, then for fx, gy ≥ 0, 0 < f p : {0∞f p xdx} 1/p < ∞, and 0 < g q < ∞, one has
0
k
x, y
f xgy
dx dy < k pf
pg
where the constant factor k p is the best possible.
Trang 2Theorem B If p > 1, ρ / 1, fx ≥ 0, and Fx :x
0 f tdtρ > 1; Fx :x∞f tdtρ < 1,
0 <∞
0 x p −ρ f p xdx < ∞, then one has
0
x −ρ F p xdx <
p
ρ− 1
p∞
0
where the constant factor p/|ρ − 1| p is the best possible (cf [ 1 , Theorem 330]).
Theorem C If p > 1, 1/p 1/q 1, λ > 0, k λ x, y ≥ 0 is a homogeneous function of degree −λ
in 0, ∞×0, ∞, and for any r > 11/r 1/s 1, 0 < k λ r :0∞k λ u, 1u λ/r−1du < ∞, then for
f x, gy ≥ 0, ϕx : x p 1−λ/r−1 , ψ y : y q 1−λ/s−1 , 0 < f p,ϕ: {0∞ϕ x|fx| p dx}1/p <∞
and 0 < g q,ψ < ∞, we have
0
k λ
x, y
f xgy
dx dy < k λ rf
p,ϕg
where the constant factor k λ r is the best possible.
For λ 1, r q, 1.3 reduces to 1.1 We name of 1.1 and 1.3 Hilbert-type integral inequalities Inequalities1.1, 1.2 and 1.3 are important in analysis and its applications
cf 4 6
Setting k1x, y 1/xy γ x R −1/q y S −1/p R, S > 0, RS γ, Fx x
0f tdt, Gy
y
similar to Pachpatte’s inequalitycf 7,8 as follows:
0
x R −1/q y S −1/p
x yγ
F x
x
G
y
y dx dy < pqB R, Sf
pg
1/max{x, y} λ x β/q1y α/p1α, β > −1, p λ − α − 1 > 1, q λ − β − 1 > 1 But he cannot
show that the constant factor in the new inequality is the best possible
a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained
equivalent inequalities with a best constant factor as well as some particular examples are considered
Trang 32 A Lemma and Two Equivalent Inequalities
Lemma 2.1 If λ < 2, k λ x, y is a nonnegative homogeneous function of degree −λ in 0, ∞ ×
0, ∞ with k λ ux, uy u −λ k x, yu, x, y > 0, and for any α ∈ λ − 1, 1, 0 < kα :
∞
0 k λ 1, uu α−1du < ∞, then0∞k λ u, 1u λ −α−1 du kα and
0 <
0
k λ 1, uu α−1|ln u|du
0
Proof Setting v 1/u, we find
0
k λ u, 1u λ −α−1 du
0
There exists β > 0, satisfying α ± β ∈ λ − 1, 1 and 0 < kα ± β < ∞ Since we find
lim
u→ 0
ln u
u β u −β lim
u→ ∞
ln u
there exists M > 0, such that | ln u| ≤ Mu β u −β u ∈ 0, ∞, and then
0 <
0
k λ u, 1u λ −α−1 |ln u|du
0
k λ 1, uu α−1|ln u|du
≤ M
0
k λ 1, uu α−1 u β u −β
du
M k
α β kα − β < ∞.
2.4
The lemma is proved
Theorem 2.2 If p > 1, 1/p 1/q 1, λ1 λ2 λ < 2, k λ x, y≥ 0 is a homogeneous function of
degree −λ in 0, ∞ × 0, ∞, and for any λ1∈ λ − 1, 1, 0 < kλ1 0∞k u, 1u λ1 −1du < ∞, then
for f x, gy ≥ 0, ϕx : x p 2−λ−λ1 −1, ψ y : y q 1−λ2 −1,
F λ x :
x
1
t λ f tdt, Gλ
y :
y
1
0 < f p, ϕ < ∞, and 0 < G λq,ψ < ∞, one has the following equivalent inequalities:
I :
0
k λ
x, y F λ x G λ
y
dx dy < k λ1
1− λ1
f
p, ϕ G
λ
J :
0
ψ1−py
0
k λ x, y F λ xdx
p
dy
1/p
< k λ1
1− λ1
f
Trang 4Proof Setting the weight functions ω λ1, y 2, x as follows:
ω
λ1, y :
0
k λ
x, y y λ2dx
x1−λ1 , λ2, x :
0
k λ
x, y x λ1dy
ω
λ1, yu x/y
0
k u, 1u λ1 −1du kλ1,
λ2, xu y/x
0
k 1, uu λ2 −1du kλ1.
2.9
By H ¨older’s inequalitycf 11 and 2.8, 2.9, we obtain
0
k λ
x, y F λ xdx ∞
0
k λ
x, yx 1−λ1/q
y 1−λ2/p F λ x
y 1−λ2/p
x 1−λ1/q
dx
≤
0
k λ x, y x 1−λ1p−1
y1−λ2 F p
1/p
×
y q 1−λ2 −1∞
0
k λ x, y y λ2dx
x1−λ1
1/q
k 1/q λ1y 1/p−λ2
0
k λ x, y x 1−λ1p−1
y1−λ2 F p
1/p
.
2.10
Then by Fubini theoremcf 12, it follows:
J p ≤ k p−1λ1
0
k λ
x, y x 1−λ1p−1
y1−λ2 F p
k p−1λ1
0
0
k λ
x, y x 1−λ1p−1
y1−λ2 dy
F p
k p λ1
0
x −pλ1 −11F p
λ xdx.
2.11
Since λ1< 1, ρ pλ1− 1 1 < 1, then by 1.2 for ρ < 1, we have
0
x −pλ1 −11F p
λ xdx <
1
1− λ1
p∞
0
x p −pλ1 −11f x
x λ
p
dx
1
1− λ1
p∞
0
x p 2−λ−λ1 −1f p xdx.
2.12
Trang 5Hence by2.11, we have 2.7 Still by H¨older’s inequality, we find
I
0
ψ −1/q
y ∞ 0
k λ
x, y F λ xdxψ 1/q
y G λ
y
dy ≤ J G
λ
Then by2.7, we have 2.6
On the other-hand, supposing that2.6 is valid, by 2.11 and 1.2 for ρ < 1, it follows J < ∞ If J 0, then 2.7 is naturally valid; if 0 < J < ∞, setting
G λ
y
ψ1−p
y∞ 0
k λ x, y F λ xdx
p−1
then by2.6, we find
G λq q,ψ J p I < k λ1
1− λ1
f
p, ϕ G
λ
q,ψ ,
G λq−1
q,ψ J < k λ1
1− λ1
f
p, ϕ
2.15
Hence, we have2.7, which is equivalent to 2.6
3 A Hilbert-Hardy-Type Integral Operator and Applications
Setting a real function space as follows:
L p ϕ 0, ∞ :
f;f
p, ϕ
0 ϕxf xp
dx
1/p
<∞
for f ≥ 0 ∈ L p
ϕ 0, ∞, F λ x x∞ft/t λ dt, define an integral operator T : L p
L p ψ1−p0, ∞ as follows:
Tf
y :
0
k λ
Then, by2.7, Tf ∈ L p
f / θ∈L p
0,∞
Tf
p,ψ1−p
f
p, ϕ
≤ k λ1
Trang 6Theorem 3.1 Let the assumptions of Theorem 2.2 be fulfilled, and additionally setting ψy :
y q 2−λ−λ2 −1 Then one has
0
k λ
x, y F λ x G λ
y
dx dy < k λ1
1 − λ11 − λ2f
p, ϕg
where the constant factor k λ1/1 − λ11 − λ2 is the best possible Moreover the constant factor in
2.6 and 2.7 is the best possible and then
T k λ1
1− λ1
Proof Since λ2< 1, by1.2, for ρ qλ2− 1 1 < 1, it follows:
G λ
q,ψ
0
y −qλ2 −11Gq
λ
y
dy
1/q
< q
1− q λ2− 1 1
0
y q −qλ2 −11
g y
y λ
q
dy
1/q
1− λ2
0
y q 2−λ−λ2 −1g q ydy
1/q
1− λ2
g
q, ψ
3.6
Then, by2.6, we have 3.4
For T > 2, setting f x, gy as follows:
⎧
⎪
⎪
x λ λ1 −2, 1 ≤ x ≤ T,
0, 0 < x < 1; x > T,
gy
⎧
⎪
⎪
y λ λ2 −2, 1 ≤ y ≤ T,
0, 0 < y < 1; y > T,
3.7
Trang 7then for 1≤ x, y ≤ T, we find
F λ x
x
f t
t λ dt
T
x
t λ1 −2dt 1
1− λ1 x λ1 −1− T λ1 −1
,
G λ
y
y
gt
t λ dt 1
1− λ2
y λ2 −1− T λ2 −1
,
I :T
1
k λ
x, y F λ x G λ
y
dx dy 1 − λ 1
11 − λ2
×
T
1
T
1
k λ
x, y x λ1 −1− T λ1 −1 y λ2 −1− T λ2 −1
dy
dx
11 − λ2I1− I2− I3,
3.8
where I1, I2, and I3are indicated as follows;
I1:
T
1
T
1
k λ
x, y
x λ1 −1y λ2 −1dy
dx,
I2: Tλ1 −1T
1
T
1
k λ
x, y
y λ2 −1dy
dx,
I3: Tλ2 −1T
1
T
1
k λ
x, y
x λ1 −1dx
dy.
3.9
If there exists a positive constant k ≤ kλ1, such that 3.4 is still valid as we replace
k λ1 by k, then in particular, we find
1 − λ11 − λ2 f
p, ϕ g q, ψ
1 − λ11 − λ2
1
x p 2−λ−λ/r−1 x p λλ/r−2 dx
1/p
×
1
y q 2−λ−λ/s−1 y q λλ/s−2 dy
1/q
1 − λ k ln T
11 − λ2.
3.10
By3.8 and 3.10, we find
1
ln T I1− 1
Trang 8Since by Fubini theorem, we obtain
I1
T
1
1
x
T/x
1/x
k λ 1, uu λ2 −1du dx
1
0
1/u
1
x dx
k λ 1, uu λ2 −1du
T
1
1
1
x dx
k λ 1, uu λ2 −1du
ln T
1
0
k λ 1, uu λ2 −1du 1
ln T
1
0
k λ 1, uln uu λ2 −1du
T
1
k λ 1, uu λ2 −1du− 1
ln T
T
1
k λ 1, uln uu λ2 −1du
,
0≤ I2 T λ1 −1T
1
1
x λ1
T/x
1/x
k λ 1, uu λ2 −1dudx
T λ1 −11
0
1/u
1
x λ1dx
k λ 1, uu λ2 −1du
T
1
1
1
x λ1dx
k λ 1, uu λ2 −1du
1− λ1
1
0
T
1−λ1
k λ 1, uu λ2 −1du
T
1
T
1−λ1
k λ 1, uu λ2 −1du
1− λ1
2
1
0
k λ 1, uu λ2 −1du
1
k λ 1, uu λ2 −1du
< ∞,
1− λ2
2
1
0
k λ u, 1u λ1 −1du
1
k λ u, 1u λ1 −1du
< ∞,
3.12
then for T → ∞ in 3.10, byLemma 2.1, we obtain kλ1 0∞k 1, uu λ2 −1du ≤ k Hence
k kλ1, and then kλ1/1 − λ11 − λ2 is the best value of 3.4
get a contradiction by1.2 that the constant factor in 3.4 is not the best possible By the
3.3, we have 3.5
Trang 9Corollary 3.2 For λ 1, λ1 1/q, λ2 1/p, F1x :x∞1/tftdt, G1y :y∞1/tgtdt,
in2.6, 2.7 and 3.4, one has the following basic Hilbert-Hardy-type integral inequalities with the
best constant factors:
0
k1
x, y F1x G1
y
dx dy < pk pf
p G
1
0
0
k1x, y F1xdx
p
dy
1/p
< pk pf
0
k1
x, y F1x G1
y
dx dy < pqk pf
pg
where k p k1/q 0∞k λ u, 1u −1/p du, and3.13 is equivalent to 3.14.
Example 3.3 For p > 1, r > 1, 1/p 1/q 1/r 1/s 1, λ1 λ/r, and λ2 λ/s in3.4,
a if 0 < λ < max{r, s}, k λ x, y 1/x y λ , 1/max{x, y} λand lnx/y/xλ − y λ, then we obtain the following integral inequalities:
0
F λ x G λ
y
x yλ dx dy < rsB λ/r, λ/s
r − λs − λf
p, ϕg
q, ψ ,
0
F λ x G λ
y
x, y λ dx dy < r
2s2
λ r − λs − λf
p, ϕg
q, ψ ,
0
ln
x/y F λ x G λ
y
x λ − y λ dx dy < rs πcscπ/r2
λ2r − λs − λf
p, ϕg
q, ψ;
3.16
b if 0 < λ < 1, k λ x, y 1/|x − y| λ, then we have
0
F λ x G λ
y
x − yλ dx dy < rs B1 − λ, λ/r B1 − λ, λ/s
p, ϕg
c if λ < 0, k λ x, y min{x, y} −λ, then we find
0
F λ x G λ
y
x, y λ dx dy < −r2s2
λ r − λs − λf
p, ϕg
where the constant factors in the above inequalities are the best possible
Trang 10This work is supported by the Emphases Natural Science Foundation of Guangdong
References
1 G H Hardy, J E Littlewood, and G Polya, Inequalities, Cambridge University Press, Cambridge, UK,
1934
2 G H Hardy, “Note on some points in the integral calculus LXIV,” Messenger of Math, vol 57, pp.
12–16, 1928
3 B Yang, “A survey of the study of Hilbert-type inequalities with parameters,” Advances in Math, vol.
38, no 3, pp 257–268, 2009
4 D S Mintrinovic, J E Pecaric, and A M Kink, Inequalities Involving Functions and their Integrals and
Derivertives, Kluwer Academic Publishers, Boston, Mass, USA, 1991.
5 B Yang, The Norm of Operator and Hilbert-Type Inequalities, Science, Beijin, China, 2009.
6 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.
7 B G Pachpatte, “On some new inequalities similar to Hilbert’s inequality,” Journal of Mathematical
Analysis and Applications, vol 226, no 3, pp 166–179, 1998.
8 B G Pachpatte, “Inequalities similar to certain extensions of Hilbert’s inequality,” Journal of
Mathematical Analysis and Applications, vol 243, no 2, pp 217–227, 2000.
9 N Das and S Sahoo, “New inequalities similar to Hardy-Hilbert’s inequality,” Turkish Journal of
Mathematics, vol 33, pp 1–13, 2009.
10 W T Sulaiman, “On three inequalities similar to Hardy-Hilbert’s integral inequality,” Acta
Mathematica Universitatis Comenianae, vol 76, no 2, pp 273–278, 2007.
11 J Kuang, Applied Inequalities, Shangdong Science Technic, Jinan, China, 2004.
12 J Kuang, Introduction to Real Analysis, Hunan Education, Changsha, China, 1996.
... Trang 10This work is supported by the Emphases Natural Science Foundation of Guangdong
References...
p, ϕ
≤ k λ1
Trang 6Theorem... 1
Trang 8Since by Fubini theorem, we obtain
I1
T