Abstract
Let \(K(x_{1},\ldots,x_{n})\) satisfy
for \(t>0\). With this integral kernel, by using the method and technique of weight coefficients, the equivalent conditions and the best constant factors for the validity of Hilbert-type integral inequalities involving multiple functions are discussed. Finally, the applications of the integral inequalities are considered.
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1 Introduction
Let \(x=(x_{1},\ldots,x_{n}), \mathbf{R}_{+}^{n}=\{x=(x_{1},\ldots ,x_{n}):x_{i}>0\ (i=1,\ldots,n)\},r>1,f(t)\geq0\), and α be a constant. Set
If \(\sum_{i=1}^{n}\frac{1}{p_{i}}=1\ (p_{i}>1,i=1,\ldots,n),\alpha _{i}\in \mathbf{R},f_{i}(x_{i})\in L_{\alpha_{i}}^{p_{i}}(0,+\infty)\) \((i=1,\ldots ,n),K(x_{1},\ldots,x_{n})\geq0\), M is a constant, then we name the following inequality a Hilbert-type integral inequality:
An integral kernel \(K(x_{1},\ldots,x_{n})\) is said to be a quasi-homogeneous function with parameters \((\lambda,\lambda _{1},\ldots ,\lambda_{n})\) if, for \(t>0\),
Obviously, \(K(x_{1},\ldots,x_{n})\) becomes a homogeneous function of order \(\lambda\lambda_{0}\) when \(\lambda_{1}=\lambda_{2}=\cdots=\lambda _{n}=\lambda_{0}\).
So far, many good results have been obtained in the study of Hilbert-type inequalities (cf. [1–24]). What are the necessary and sufficient conditions for the validity of a Hilbert-type inequality? What is the best constant factor when the inequality holds? The research on such problems is undoubtedly of great significance to the study and applications of Hilbert-type inequality theory, but unfortunately, the research on this type of problems is rarely seen.
In this paper, we focus on the quasi-homogeneous integral kernels, discuss the equivalent conditions for the validity of Hilbert-type integral inequalities involving multiple functions, and obtain the expressions of the best constant factors when the inequalities are established. Finally, we discuss their applications.
2 Some lemmas
Lemma 1
Let integer \(n\geq2,\sum_{i=1}^{n}\frac{1}{p_{i}}=1\ (p_{i}>1,i=1,\ldots,n),\lambda\in\mathbf{R},\alpha_{i}\in\mathbf {R},\lambda_{i}>0\) (or \(\lambda_{i}<0\)) \((i=1,\ldots,n)\), and \(K(x_{1},\ldots,x_{n})\) be a nonnegative measurable function with parameters \((\lambda,\lambda_{1},\ldots,\lambda_{n}),\sum_{i=1}^{n}\frac{\alpha _{i}+1}{\lambda_{i}p_{i}}=\lambda+\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\). Set
Then \(\frac{1}{\lambda_{1}}W_{1}=\frac{1}{\lambda_{2}}W_{2}=\cdots =\frac{1}{\lambda n}W_{n}\), and
where \(j=1,\ldots,n\).
Proof
When \(j\geq2\), by \(\sum_{i=1}^{n}\frac{\alpha_{i}+1}{\lambda_{i}p_{i}}=\lambda+\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\),
Therefore \(\frac{1}{\lambda_{j}}W_{j}=\frac{1}{\lambda _{1}}W_{1}\ (j\geq2)\). When \(j=1,\ldots,n\), we also get
□
Lemma 2
([25])
Let \(p_{i}>0,a_{i}>0,\alpha_{i}>0\) \((i=1,\ldots,n),\psi(u)\) be measurable. Then
where Γ represents the gamma function.
3 Main results and their proofs
Theorem 1
Suppose that \(n\geq2,\sum_{i=1}^{n}\frac{1}{p_{i}}=1\ (p_{i}>1),\lambda\in\mathbf{R},\lambda_{i}>0\) (or \(\lambda _{i}<0\)), \(\alpha_{i}\in\mathbf{R}\) \((i=1,\ldots,n)\). If \(K(x_{1},\ldots ,x_{n})\) is a quasi-homogeneous positive function with parameters (\(\lambda ,\lambda_{1},\ldots,\lambda_{n}\)), and
is convergent, then
-
(i)
the inequality
$$ \int_{\mathbf{R}_{+}^{n}}K(x_{1},\ldots,x_{n})\prod _{i=1}^{n}f_{i}(x_{i})\,dx_{1} \cdots \,dx_{n}\leq M\prod_{i=1}^{n} \Vert f_{i} \Vert _{p_{i},\alpha_{i}} $$(1)holds for some constant \(M>0\) if and only if \(\sum_{i=1}^{n}\frac{\alpha _{i}+1}{\lambda_{i}p_{i}}=\lambda+\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\), where \(f_{i}(x_{i})\in L_{\alpha_{i}}^{p_{i}}(0,+\infty)\) \((i=1,\ldots ,n)\);
-
(ii)
if (1) holds, then its best constant factor is \(\inf M=\frac {W_{1}}{|\lambda_{1}|}\prod_{i=1}^{n}|\lambda_{i}|^{1/p_{i}}\).
Proof
(i) Sufficiency. Assume that \(\sum_{i=1}^{n}\frac{\alpha _{i}+1}{\lambda_{i}p_{i}}=\lambda+\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\). Since
by Hölder’s inequality and Lemma 1, we obtain
thus (1) holds when taking any constant \(M\geq\frac {W_{1}}{|\lambda _{1}|}\prod_{i=1}^{n}|\lambda_{i}|^{1/p_{i}}\).
Necessity. Assume that (1) holds. Set \(c=\sum_{i=1}^{n}\frac {\alpha _{i}+1}{\lambda_{i}p_{i}}-\lambda-\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\). Next we will prove \(c=0\).
First consider the case of \(\lambda_{i}>0\ (i=1,\ldots,n)\). If \(c>0\), for \(0<\varepsilon<c\), take
where \(i=1,\ldots,n\). Then
It follows from (1), (2), and (3) that
Since \(-1-\lambda_{1}c+\lambda_{1}\varepsilon<-1\), \(\int_{0}^{1}x_{1}^{-1-\lambda_{1}c+\lambda_{1}\varepsilon}\, {d}x_{1}\) diverges to +∞. Whence it is a contradiction to (4). In other words, it is not valid for \(c>0\).
If \(c<0\), for \(0<\varepsilon<-c\), take
where \(i=1,\ldots,n\). Similarly, we get
Since \(-1-\lambda_{1}c+ \lambda_{1}\varepsilon>-1\) and \(\int_{1}^{+\infty }x_{1}^{-1-\lambda_{1}c-\lambda_{1}\varepsilon} \,dx_{1}\) diverges to +∞, which contradicts the above inequality, hence it does not hold for \(c<0\).
To sum up, we have \(c=0\) for \(\lambda_{i}>0\ ( i=1,\ldots,n ) \).
Now let us consider the case of \(\lambda_{i}<0\ ( i=1,\ldots,n ) \). If \(c>0\), for \(0< \varepsilon<c\), take
where \(i=1,\ldots,n\). Consequently,
It follows from (1), (6), and (7) that
Since \(-1-\lambda_{1}c+\lambda_{1}\varepsilon>-1\), \(\int_{1}^{+\infty }x_{1}^{-1-\lambda_{1}c+\lambda_{1}\varepsilon}\,dx_{1}\) diverges to +∞. Thus it is a contradiction to the above inequality. That is, it does not hold for \(c>0\).
If \(c<0\), for \(0<\varepsilon<-c\), take
where \(i=1,\ldots,n\). Similarly, one can get
Since \(-1-\lambda_{1}c- \lambda_{1}\varepsilon<-1\), \(\int_{0}^{1}x_{1}^{-1-\lambda_{1}c-\lambda_{1}\varepsilon}\, {d}x_{1}\) diverges to +∞, which also contradicts the above inequality. It does not hold for \(c<0\).
To sum up, we also get \(c=0\) for \(\lambda_{i}<0\) \((i=1,\ldots,n)\).
(ii) Suppose that (1) holds. If the constant factor \(\inf M\neq\frac{W_{1}}{|\lambda_{1}|} \prod_{i=1}^{n}|\lambda_{i}|^{1/p_{i}}\), then there exists a constant \(M_{0}<\frac{W_{1}}{|\lambda_{1}|}\prod_{i=1}^{n}|\lambda_{i}|^{1/p_{i}}\) such that
For sufficiently small \(\varepsilon>0\) and \(\delta>0\), take
For \(i=2,3,\ldots,n\), take
Therefore,
It follows from (1), (10), and (11) that
Consequently,
as \(\varepsilon\rightarrow0^{+}\). And then let \(\delta\rightarrow 0^{+}\), we eventually get
this is a contradiction. Hence \(\inf M=\frac{W_{1}}{|\lambda_{1}|}\prod_{i=1}^{n}|\lambda_{i}|^{1/p_{i}}\), i.e., the constant factor \(\frac{W_{1}}{|\lambda_{1}|}\prod_{i=1}^{n}|\lambda_{i}|^{1/p_{i}}\) is the best. □
4 Applications
Theorem 2
Suppose that \(n\geq2,\sum_{i=1}^{n}\frac {1}{p_{i}}=1\ (p_{i}>1),\lambda_{i}>0\) (or \(\lambda_{i}<0\)), \(f_{i}(x_{i})\in L_{p_{i}-1}^{p_{i}}(0,+\infty), i=1,\ldots,n\). Then
where the constant factor is the best.
Proof
Set \(\alpha_{i}=p_{i}-1,\lambda=0\), then \(\sum_{i=1}^{n} \frac{\alpha_{i}+1}{\lambda_{i}p_{i}}=\lambda+\sum_{i=1}^{n}\frac{1}{ \lambda_{i}}\). Take
then \(K(x_{1},\ldots,x_{n})\) is a quasi-homogeneous positive function with parameters \((\lambda,\lambda_{1},\ldots,\lambda_{n})\), and
In view of [1], we get
it follows that
According to Theorem 1, we know that Theorem 2 holds. □
Theorem 3
Suppose that \(n\geq2,\sum_{i=1}^{n}\frac {1}{p_{i}}=1\ (p_{i}>1),a>0,\lambda_{i}>0\), \(\alpha_{i}\in\mathbf {R},p_{i}>1+\alpha _{i}\). Then
-
(i)
the inequality
$$ \int_{\mathbf{R}_{+}^{n}}\frac{1}{ ( x_{1}^{\lambda _{1}}+x_{2}^{\lambda _{2}}+\cdots+x_{n}^{\lambda_{n}} ) ^{a}}\prod_{i=1}^{n}x_{i}^{-\frac{\alpha_{i}+1}{p_{i}}} \,dx_{1}\cdots\,dx_{n}\leq M\prod _{i=1}^{n} \Vert f_{i} \Vert _{p_{i},\alpha_{i}} $$(12)holds for some constant \(M>0\) if and only if \(\sum_{i=1}^{n}\frac{\alpha _{i}+1}{\lambda_{i}p_{i}}=-a+\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\).
-
(ii)
if (12) holds, then its best constant factor is
$$\inf M=\frac{1}{\Gamma(a)}\prod_{i=1}^{n} \lambda_{i}{}^{\frac {1}{p_{i}}-1}\prod_{i=1}^{n} \Gamma \biggl( \frac{1}{\lambda_{i}} \biggl( 1-\frac {\alpha _{i}+1}{p_{i}} \biggr) \biggr). $$
Proof
Set \(K(x_{1},\ldots,x_{n})=1/ ( x_{1}^{\lambda _{1}}+x_{2}^{\lambda_{2}}+\cdots+x_{n}^{\lambda_{n}} ) ^{a}\), then \(K(x_{1},\ldots,x_{n})\) is a quasi-homogeneous positive funct+ion with parameters \((-a,\lambda_{1},\ldots,\lambda_{n})\). By Lemma 2,
Based on this, we can obtain
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Authors’ information
Yong Hong (1959-), Professor, the major field of interest is in the area of inequality and harmonic analysis. Junfei Cao, email: jfcaomath@163.com. Bing He, email: hzs314@163.com. Bicheng Yang, email: bcyang@gdei.edu.cn.
Funding
This work is supported by the Guangdong Province Natural Science Foundation (No. 2015A030313896), the Characteristic Innovation Project (Natural Science) of Guangdong Province (Nos. 2015KTSCX097, 2016KTSCX094), the Science and Technology Program Project of Guangzhou (No. 201707010230).
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JC carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. BH, YH, and BY participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Cao, J., He, B., Hong, Y. et al. Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels. J Inequal Appl 2018, 206 (2018). https://doi.org/10.1186/s13660-018-1797-5
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DOI: https://doi.org/10.1186/s13660-018-1797-5