Hong et al. Journal of Inequalities and Applications (2017) 2017:316 https://doi.org/10.1186/s13660-017-1592-8
RESEARCH
Open Access
The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications Yong Hong1 , Qiliang Huang2* , Bicheng Yang2 and Jianquan Liao2 *
Correspondence:
[email protected] 2 Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China Full list of author information is available at the end of the article
Abstract
ρ ρ For x = (x1 , . . . , xn ), u(x) = ( ni=1 ai xi )1/ρ , v(y) = ( ni=1 bi yi )1/ρ , by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel K(u(x), v(y)) = G(uλ1 (x)vλ2 (y)) and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered. MSC: 26D15; 47A07 Keywords: Hilbert-type inequality; non-homogeneous kernel; sufficient and necessary conditions; best possible constant factor; bounded operator; operator norm
1 Introduction For n ≥ 1, Rn+ = {x = (x1 , . . . , xn ) : xi > 0, i = 1, . . . , n}, ai , bi > 0 (i = 1, . . . , n), ω(x) > 0 (x ∈ Rn+ ), and ρ > 0, we set u(x) =
n i=1
ρ1 ai xρi
,
v(y) =
Lpω Rn+ := f (x) ≥ 0 : f p,ω =
n
ρ1 bi yρi
,
i=1
Rn+
ω(x)f p (x) dx
1/p
< +∞ .
If p > 1, p1 + q1 = 1, K(u, v) ≥ 0 (u, v > 0), then the Hilbert-type multiple integral inequality is of the form
Rn+
Rn+
K u(x), v(y) f (x)g(y) dx dy ≤ Mf p,uα gq,vβ .
(1)
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Hong et al. Journal of Inequalities and Applications (2017) 2017:316
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Define a singular integral operator T:
T(f )(y) :=
Rn+
K u(x), v(y) f (x) dx,
y ∈ Rn+ ,
(2)
then (1) may be rewritten as follows:
Rn+
T(f )(y)g(y) dy ≤ Mf p,uα gq,vβ .
It is easy to prove that (1) is equivalent to the following inequality:
T(f ) γ ≤ Mf p,uα gq,vβ , p,v
(3)
where γ = β(1 – p). When the operator T satisfies (3), T is called bounded operator from p p Luα (Rn+ ) to Lvγ (Rn+ ). At present, there are lots of research results on Hilbert-type single integral inequality (cf. [1–14]). But there are relatively few studies on Hilbert-type multiple integral inequality. In particular, there are fewer studies on the necessary and sufficient conditions for the existence of the multiple integral inequality. In this article, by using the methods and techniques of real analysis, we give the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the non-homogeneous kernel K u(x), v(y) = G uλ1 (x)vλ2 (y) , and calculate the best possible constant factor. Furthermore, its application in the operator theory is considered.
2 Some lemmas Lemma 1 Suppose that p > 1, p1 + q1 = 1, n ≥ 1, ρ > 0, λ1 λ2 > 0, ai , bi > 0 (i = 1, . . . , n), ρ 1 ρ 1 u(x) = ( ni=1 ai xi ) ρ , v(y) = ( ni=1 bi yi ) ρ . If K(u(x), v(y)) = G(uλ1 (x)vλ2 (y)) is a non-negative measurable function, setting
W1 :=
W2 :=
– β+n v(t) q K 1, v(t) dt,
Rn+
– α+n u(t) p K u(t), 1 dt,
Rn+
we have the following:
ω1 (x) :=
Rn+
ω2 (x) :=
Rn+
v(y)
– β+n λ1 ( β+n –n) q K u(x), v(y) dy = u(x) λ2 q W1 ,
u(x)
– α+n λ2 ( α+n –n) p K u(x), v(y) dx = v(y) λ1 p W2 .
Hong et al. Journal of Inequalities and Applications (2017) 2017:316
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λ1
λ1
Proof Since v(ay) = av(y) (a > 0), in view of K(tu, v) = K(u, t λ2 v), setting t = u λ2 (x)y, we find dy = u
–nλ1 λ2
(x) dt and
ω1 (x) =
Rn+
=
Rn+
v(y)
λ1 – β+n q K 1, u λ2 (x)v(y) dy
λ 2
– λ1
u
(x)v(t)
– β+n –nλ1 q K 1, v(t) u λ2 (x) dt
λ1 ( β+n –n) W1 . = u(x) λ2 q In the same way, we have λ2 ( α+n –n) W2 . ω2 (x) = v(y) λ1 p
The lemma is proved.
Lemma 2 (cf. [15]) If pi > 0, ai > 0, αi > 0 (i = 1, . . . , n) and ψ(t) is a measurable function, then we have the following:
···
x {xi >0; ni=1 ( ai )αi ≤1}
ψ
n x i αi i=1
i
=
ai
p a1 1
p · · · ann ( αp11 ) · · · ( αpnn ) α1 · · · αn ( ni=1 αpii )
p –1
x1 1
· · · xpnn –1 dx1 · · · dxn
n
1
ψ(t)t
pi i=1 αi –1
dt,
0
where (t) is the gamma function. In particular, for αi = ρ, pi = 1, bi = have
···
ρ {xi >0; ni=1 bi xi ≤1}
n =
– ρ1 n 1 i=1 bi ( ρ ) ρ n ( ρn )
ψ
n
1 ρ ai
(i = 1, . . . , n), we
bi xρi
dx1 · · · dxn
i=1
1
n
ψ(t)t ρ –1 dt.
0
3 Main results We set
(a < b) = x = (x1 , . . . , xn ); a < u(x) < b ,
(a < b) = x = (x1 , . . . , xn ); a < v(y) < b . Theorem 1 Suppose that n ≥ 1, p > 1, p1 + q1 = 1, ρ > 0, α, β ∈ R, λ1 λ2 > 0, ai > 0, bi > 0 ρ 1/ρ ρ 1/ρ (i = 1, . . . , n), u(x) = ( ∞ , v(y) = ( ∞ , K(u(x), v(y)) = G(uλ1 (x)vλ2 (y)) is a i=1 ai xi ) i=1 bi yi ) non-negative measurable function,
0 < W1 =
0 < W2 =
Rn+
Rn+
v(t)
– β+n q K 1, v(t) dt < ∞,
u(t)
– α+n p K u(t), 1 dt < ∞,
Hong et al. Journal of Inequalities and Applications (2017) 2017:316
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and for a = 0, b = 1 (or a = 1, b = +∞),
v(t)
– β+n q K 1, v(t) dt > 0,
(a 0, if c > 0, putting ε > 0 small enough and
⎧ ⎨(u(x))(–α–n–λ1 ε)/p , u(x) > 1, f (x) = ⎩0, 0 < u(x) ≤ 1,
⎧ ⎨(v(y))(–β–n+λ2 ε)/q , 0 < v(y) < 1, g(y) = ⎩0, v(y) ≥ 1, by Lemma 2, we have f p,uρ gq,vρ
1/p
–n–λ1 ε u(x) = dx
(0
