Abstract
In this paper, by the use of the way of weight coefficients, the transfer formula, and the technique of real analysis, we introduce some proper parameters and obtain a multidimensional Hilbert-type inequality with the following kernel:
and a best possible constant factor. The equivalent form, the operator expressions with the norm, and some particular cases are also considered. The lemmas and theorems provide an extensive account of this type of inequalities.
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1 Introduction
If \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f(x),g(y)\geq0\), \(f\in L^{p}(\mathbf{R}_{+})\), \(g\in L^{q}(\mathbf{R}_{+})\), \(\|f\|_{p}=(\int_{0}^{\infty }f^{p}(x)\,dx)^{\frac{1}{p}}>0\), \(\|g\|_{q}>0\), then we have the following Hardy-Hilbert’s integral inequality (cf. [1]):
where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is the best possible. Assuming that \(a_{m},b_{n}\geq0\), \(a=\{a_{m}\}_{m=1}^{\infty }\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\|a\|_{p}=(\sum_{m=1}^{ \infty}a_{m}^{p})^{\frac{1}{p}}>0\), \(\|b\|_{q}>0\), we have the following discrete Hardy-Hilbert’s inequality with the same best constant \(\frac {\pi}{\sin(\pi/p)}\):
Inequalities (1) and (2) are important in analysis and its applications (cf. [1–6]).
In 1998, by introducing an independent parameter \(\lambda\in(0,1]\), Yang [7] gave an extension of (1) at \(p=q=2\) with the kernel \(\frac{1}{(x+y)^{\lambda}}\). In recent years, Yang [3] and [4] gave some extensions of (1) and (2) as follows:
If \(\lambda_{1},\lambda_{2}\in\mathbf{R}\), \(\lambda_{1}+\lambda _{2}=\lambda\), \(k_{\lambda}(x,y)\) is a non-negative homogeneous function of degree −λ, with
\(\phi(x)=x^{p(1-\lambda_{1})-1}\), \(\psi(x)=x^{q(1-\lambda _{2})-1}\), \(f(x),g(y)\geq0\),
\(g\in L_{q,\psi}(\mathbf{R}_{+})\), \(\|f\|_{p,\phi},\|g\|_{q,\psi}>0\), then we have
where the constant factor \(k(\lambda_{1})\) is the best possible. Moreover, if \(k_{\lambda}(x,y)\) is finite and \(k_{\lambda}(x,y)x^{\lambda _{1}-1}(k_{\lambda}(x,y)y^{\lambda_{2}-1})\) is decreasing with respect to \(x>0\) (\(y>0\)), then for \(a_{m},b_{n}\geq0\),
\(b=\{b_{n}\}_{n=1}^{\infty}\in l_{q,\psi}\), \(\|a\|_{p,\phi },\|b\|_{q,\psi }>0\), we have the following inequality:
where the constant factor \(k(\lambda_{1})\) is still the best possible.
Clearly, for \(\lambda=1\), \(k_{1}(x,y)=\frac{1}{x+y}\), \(\lambda_{1}=\frac {1}{q}\), \(\lambda_{2}=\frac{1}{p}\), (3) reduces to (1), while (4) reduces to (2). Some other results including the multidimensional Hilbert-type integral, discrete, and half-discrete inequalities are provided by [8–26].
In this paper, by the use of the way of weight coefficients, the transfer formula and technique of real analysis, a multidimensional discrete Hilbert’s inequality with parameters and a best possible constant factor is given, which is an extension of (4) for
The equivalent form, the operator expressions with the norm, and some particular cases are also considered.
2 Some lemmas
If \(i_{0},j_{0}\in\mathbf{N}\) (N is the set of positive integers), \(\alpha ,\beta>0\), we put
Lemma 1
If \(g(t)\) (>0) is decreasing in \(\mathbf{R}_{+}\) and strictly decreasing in \([n_{0},\infty)\subset\mathbf{R}_{+}\) (\(n_{0}\in \mathbf{N}\)), satisfying \(\int_{0}^{\infty}g(t)\,dt\in\mathbf{R}_{+}\), then we have
Proof
Since by the assumption, we have
it follows that
In the same way, we still have
Hence, choosing plus for the above two inequalities, we have (6). □
Lemma 2
If \(s\in\mathbf{N}\), \(\gamma,M>0\), \(\Psi(u)\) is a non-negative measurable function in \((0,1]\), and
then we have the following transfer formula (cf. [27]):
Lemma 3
For \(s\in\mathbf{N}\), \(\gamma, \varepsilon>0\), we have
where \(\sum_{m}=\sum_{m_{s}=1}^{\infty}\cdots\) \(\sum_{m_{1}=1}^{\infty}\).
Proof
For \(M>s^{1/\gamma}\), we set
Then by Lemma 1 and (7), it follows that
By Lemma 1 and in the above way, we still find
For \(s=1\), \(0<\sum_{m=1}^{1}\|m\|_{\gamma}^{-1-\varepsilon}<\infty\); for \(s\geq2\),
and then
Then we have (8). □
Example 1
For \(s\in\mathbf{N}\), \(0< c_{1}\leq\cdots\leq c_{s}<\infty\), \(\lambda_{1},\lambda_{2}>-\gamma\), \(\lambda_{1}+\lambda _{2}=\lambda\), we set
(a) We find
If \(\lambda_{1}-\frac{i\lambda}{s}+(1-\frac{2i}{s})\gamma\neq0\), then
if there exists a \(i_{0}\in\{1,\ldots,s-1\}\), such that \(\lambda_{1}- \frac{i_{0}\lambda}{s}+(1-\frac{2i_{0}}{s})\gamma=0\), then we find
and we still indicate \(\ln(\frac{c_{i_{0}+1}}{c_{i_{0}}})\) by the following formal expression:
Hence, we may set
In particular, (i) for \(s=1\) (or \(c_{s}=\cdots=c_{1}\)), we have \(k_{\lambda }(x,y)=\frac{(\min\{x,c_{1}y\})^{\gamma}}{(\max\{x,c_{1}y\})^{\lambda +\gamma}}\) and
(ii) for \(s=2\), we have \(k_{\lambda}(x,y)=\frac{(\min\{x,c_{1}y\}\min \{x,c_{2}y\})^{\gamma/2}}{(\max\{x,c_{1}y\}\max\{x,c_{2}y\} )^{(\lambda +\gamma)/2}}\) and
(iii) for \(\gamma=0\), we have \(\lambda_{1},\lambda_{2}>0\), \(k_{\lambda }(x,y)=\frac{1}{\prod_{k=1}^{s}(\max\{x,c_{k}y\})^{\frac{\lambda }{s}}}\) and
(iv) for \(\gamma=-\lambda\), we have \(\lambda<\lambda_{1},\lambda_{2}<0\), \(k_{\lambda}(x,y)=\frac{1}{\prod_{k=1}^{s}(\min\{x,c_{k}y\})^{\frac{\lambda}{s}}}\) and
(v) for \(\lambda=0\), we have \(\lambda_{2}=-\lambda_{1}\), \(|\lambda _{1}|<\gamma\) (\(\gamma>0\)),
and
(b) Since for \(j_{0}\in\mathbf{N,}\) we find
for \(\lambda_{2}\leq j_{0}-\gamma\) (\(\lambda_{1}>-\gamma\)), \(k_{\lambda}(x,y)\frac{1}{y^{j_{0}-\lambda_{2}}}\) is decreasing for \(y>0\) and strictly decreasing for the large enough variable y. In the same way, for \(i_{0}\in\mathbf{N,}\) we find
then for \(\lambda_{1}\leq i_{0}-\gamma\) (\(\lambda_{2}>-\gamma\)), \(k_{\lambda}(x,y)\frac{1}{x^{i_{0}-\lambda_{1}}}\) is decreasing for \(x>0\) and strictly decreasing for the large enough variable x.
In view of the above results, for \(i_{0},j_{0}\in\mathbf{N}\), \(-\gamma <\lambda_{1}\leq i_{0}-\gamma\), \(-\gamma<\lambda_{2}\leq j_{0}-\gamma\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(k_{\lambda}(x,y)\frac{1}{y^{j_{0}-\lambda_{2}}}\) (\(k_{\lambda}(x,y)\frac{1}{x^{i_{0}-\lambda _{1}}}\)) is still decreasing for \(y>0\) (\(x>0\)) and strictly decreasing for the large enough variable \(y(x)\).
Definition 1
For \(s,i_{0},j_{0}\in\mathbf{N}\), \(0< c_{1}\leq \cdots\leq c_{s}<\infty\), \(-\gamma<\lambda_{1}\leq i_{0}-\gamma\), \(-\gamma <\lambda_{2}\leq j_{0}-\gamma\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(m=(m_{1},\ldots,m_{i_{0}})\in\mathbf{N}^{i_{0}}\), \(n=(n_{1},\ldots ,n_{j_{0}})\in\mathbf{N}^{j_{0}}\), define two weight coefficients \(w(\lambda_{1},n)\) and \(W(\lambda_{2},m)\) as follows:
where \(\sum_{m}=\sum_{m_{i_{0}}=1}^{\infty}\cdots\sum_{m_{1}=1}^{\infty}\) and \(\sum_{n}=\sum_{n_{j_{0}}=1}^{\infty}\cdots\sum_{n_{1}=1}^{\infty}\).
Lemma 4
As the assumptions of Definition 1, then (i) we have
where
(ii) for \(p>1\), \(0<\varepsilon<\frac{p}{2}(\lambda_{1}+\gamma)\), setting \(\widetilde{\lambda}_{1}=\lambda_{1}-\frac{\varepsilon}{p}\) (\(\in (-\gamma ,i_{0}-\gamma)\)), \(\widetilde{\lambda}_{2}=\lambda_{2}+\frac {\varepsilon}{p}\) (\(>{-}\gamma\)), we have
where
Proof
By Lemma 1, Example 1, and (7), it follows that
Hence, we have (18). In the same way, we have (19).
By Lemma 1, Example 1, and in the same way as obtaining (8), we have
For \(\|n\|_{\beta}\geq c_{1}^{-1}i_{0}^{1/\alpha}\), we find \(v\leq i_{0}^{1/\alpha}/\|n\|_{\beta}\leq c_{1}\leq c_{k}\) (\(k=1,\ldots,s\)) and
and then (22) follows. □
3 Main results
Setting \(\Phi(m):=\|m\|_{\alpha}^{p(i_{0}-\lambda_{1})-i_{0}}\) (\(m\in \mathbf{N}^{i_{0}}\)) and \(\Psi(n):=\|n\|_{\beta}^{q(j_{0}-\lambda _{2})-j_{0}}\) (\(n\in\mathbf{N}^{j_{0}}\)), we have the following.
Theorem 1
If \(s,i_{0},j_{0}\in\mathbf{N}\), \(0< c_{1}\leq\cdots \leq c_{s}<\infty\), \(-\gamma<\lambda_{1}\leq i_{0}-\gamma\), \(-\gamma <\lambda_{2}\leq j_{0}-\gamma\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(k_{s}(\lambda_{1})\) is indicated by (10), then for \(p>1\), \(\frac {1}{p}+\frac{1}{q}=1\), \(a_{m},b_{n}\geq0\), \(0<\|a\|_{p,\Phi},\|b\|_{q,\Psi }<\infty\), we have the following inequality:
where the constant factor
is the best possible. In particular, for \(s=1\) (or \(c_{s}=\cdots=c_{1}\)), we have the following inequality:
where
Proof
By Hölder’s inequality (cf. [28]), we have
Then by (18) and (19), we have (24).
For \(0<\varepsilon<\frac{p}{2}(\lambda_{1}+\gamma)\), \(\widetilde {\lambda}_{1}=\lambda_{1}-\frac{\varepsilon}{p}\), \(\widetilde{\lambda }_{2}=\lambda _{2}+\frac{\varepsilon}{p}\), we set
Then by (8) and (21), we obtain
If there exists a constant \(K\leq(K_{1}^{(s)})^{\frac {1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\), such that (24) is valid as we replace \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) by K, then using (28) and (29) we have
For \(\varepsilon\rightarrow0^{+}\), we find
and then \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\leq K\). Hence, \(K=(K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) is the best possible constant factor of (24). □
Theorem 2
As regards the assumptions of Theorem 1, for \(0<\|a\|_{p,\Phi }<\infty\), we have the following inequality with the best constant factor \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\):
which is equivalent to (24). In particular, for \(s=1\) (or \(c_{s}=\cdots=c_{1}\)), we have the following inequality equivalent to (26):
Proof
We set \(b_{n}\) as follows:
Then it follows that \(J^{p}=\|b\|_{q,\Psi}^{q}\). If \(J=0\), then (30) is trivially valid for \(0<\|a\|_{p,\Phi}<\infty\); if \(J=\infty\), then it is impossible since the right hand side of (30) is finite. Suppose that \(0< J<\infty\). Then by (24), we find
namely,
and then (30) follows.
On the other hand, assuming that (30) is valid, by Hölder’s inequality, we have
Then by (30), we have (24). Hence (30) and (24) are equivalent.
By the equivalency, the constant factor \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) in (30) is the best possible. Otherwise, we would reach a contradiction by (32) that the constant factor \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) in (24) is not the best possible. □
4 Operator expressions and some particular cases
For \(p>1\), we define two real weight normal discrete spaces \(l_{p,\varphi}\) and \(l_{q,\psi}\) as follows:
As regards the assumptions of Theorem 1, in view of \(J<(K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\|a\|_{p,\Phi}\), we give the following definition.
Definition 2
Define a multidimensional Hilbert-type operator \(T:l_{p,\Phi}\rightarrow l_{p,\Psi^{1-p}}\) as follows: For \(a\in l_{p,\Phi }\), there exists an unique representation \(Ta\in l_{p,\Psi^{1-p}}\), satisfying
For \(b\in l_{q,\Psi}\), we define the following formal inner product of Ta and b as follows:
Then by Theorem 1 and Theorem 2, for \(0<\|a\|_{p,\varphi},\|b\|_{q,\psi }<\infty\), we have the following equivalent inequalities:
It follows that T is bounded with
Since the constant factor \((K_{1}^{(s)})^{\frac {1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) in (36) is the best possible, we have the following.
Corollary 1
As regards the assumptions of Theorem 2, T is defined by Definition 2, it follows that
Remark 1
(i) For \(i_{0}=j_{0}=1\) in (24), we have the inequality
Hence, (24) is an extension of (4) for
(ii) For \(\gamma=0\) in (24) and (30), we have \(0<\lambda _{1}\leq i_{0}\), \(0<\lambda_{2}\leq j_{0}\) and the following equivalent inequalities:
where the best possible constant factor is defined by
and \(\widetilde{k}_{s}(\lambda_{1})\) is indicated by (13).
(iii) For \(\gamma=-\lambda\) in (24) and (30), we have \(\lambda<\lambda_{1}\leq i_{0}+\lambda\), \(\lambda<\lambda_{2}\leq j_{0}+\lambda\), \(\lambda_{1},\lambda_{2}<0\) and the following equivalent inequalities:
where the best possible constant factor is defined by
and \(\widehat{k}_{s}(\lambda_{1})\) is indicated by (14).
(iv) For \(\lambda=0\) in (24) and (30), we have \(\lambda _{2}=-\lambda_{1}\), \(0<\gamma+\lambda_{1}\leq i_{0}\), \(0<\gamma-\lambda _{1}\leq j_{0}\) (\(\gamma>0\)), and the following equivalent inequalities:
where the best possible constant factor is defined by
and \(k_{s}^{(0)}(\lambda_{1})\) is indicated by (15).
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Acknowledgements
This work is supported by Hunan Province Natural Science Foundation (No. 2015JJ4041), and Science Research General Foundation Item of Hunan Institution of Higher Learning College and University (No. 14C0938).
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BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. YS participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Shi, Y., Yang, B. On a multidimensional Hilbert-type inequality with parameters. J Inequal Appl 2015, 371 (2015). https://doi.org/10.1186/s13660-015-0898-7
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DOI: https://doi.org/10.1186/s13660-015-0898-7