Abstract
In this chapter, by the use of the methods of weight functions and techniques of Real Analysis, we provide a general multidimensional Hilbert-type integral inequality with a non-homogeneous kernel and a best possible constant factor. The equivalent forms, the reverses and some Hardy-type inequalities are obtained. Furthermore, we consider the operator expressions with the norm, some particular inequalities with the homogeneous kernel and a large number of particular examples.
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Keywords
- Multidimensional Hilbert-type integral inequality
- Weight function
- Equivalent form
- Hilbert-type integral operator
Mathematics Subject Classification
1 Introduction
Suppose that \(p > 1, \frac{1} {p} + \frac{1} {q} = 1,f(x),g(y) \geq 0,f \in L^{p}(\mathbf{R}_{ +}),g \in L^{q}(\mathbf{R}_{ +})\),
| | g | | q > 0. We have the following Hardy–Hilbert’s integral inequality (cf. [1]):
where the constant factor \(\frac{\pi }{\sin (\pi /p)}\) is the best possible. If \(a_{m},b_{n} \geq 0,a =\{ a_{m}\}_{m=1}^{\infty }\in l^{p},b =\{ b_{n}\}_{n=1}^{\infty }\in l^{q}\),
| | b | | q > 0, then 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 λ ∈ (0, 1], Yang [7] gave an extension of (1) for p = q = 2. In 2009 and 2011, Yang [3, 4] gave some extensions of (1) and (2) as follows: If \(\lambda _{1},\lambda _{2} \in \mathbf{R = (-\infty,\infty )},\lambda _{1} +\lambda _{2} =\lambda,k_{\lambda }(x,y)\) is a nonnegative homogeneous function of degree −λ in \(\mathbf{R}_{+}^{2}\), with
f(x), g(y) ≥ 0, satisfying
\(g \in L_{q,\psi }(\mathbf{R}_{+}),\vert \vert f\vert \vert _{p,\phi },\vert \vert g\vert \vert _{q,\psi } > 0\), then we have
where the constant factor k(λ 1) is the best possible. Moreover, if k λ (x, y) is finite and \(k_{\lambda }(x,y)x^{\lambda _{1}-1}(k_{\lambda }(x,y)y^{\lambda _{2}-1})\) is strict decreasing with respect to x > 0(y > 0), then for a m, b n ≥ 0,
\(b =\{ b_{n}\}_{n=1}^{\infty }\in l_{q,\psi }\), \(\vert \vert a\vert \vert _{p,\phi },\vert \vert b\vert \vert _{q,\psi } > 0\), we have
where the constant factor k(λ 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 multidimensional Hilbert-type integral inequalities are provided by Yang et al. [8], Krnić and Pečarić [9], Yang and Rassias [10, 11], Azar [12], Arpad and Choonghong [13], Kuang and Debnath [14], Zhong [15], Hong [16], Zhong and Yang [17], Yang and Krnić [18], and Li and He [19].
In this chapter, by the use of the methods of weight functions and techniques of real analysis, we give a general multidimensional Hilbert-type integral inequality with a nonhomogeneous kernel and a best possible constant factor. The equivalent forms, the reverses and some Hardy-type inequalities are obtained. Furthermore, we consider the operator expressions with the norm, some particular inequalities with the homogeneous kernel and a large number of particular examples.
2 Some Lemmas
If \(i_{0},j_{0} \in \mathbf{N(N}\) is the set of positive integers), α, β > 0, we put
Lemma 1.
If \(s \in \mathbf{N,}\gamma,M > 0,\varPsi (u)\) is a nonnegative measurable function in (0,1], and
then we have the following expression (cf. [6]):
In view of (5) and the conditions, it follows that
-
(i)
for
$$\displaystyle{\mathbf{R}_{+}^{s} = \left \{x \in \mathbf{R}_{ +}^{s};0 < u =\sum _{ i=1}^{s}\left ( \frac{x_{i}} {M}\right )^{\gamma } \leq 1(M \rightarrow \infty )\right \}\!\!,}$$we have
$$\displaystyle\begin{array}{rcl} & \int \cdots \int _{\mathbf{R}_{+}^{s}}\varPsi \left (\sum _{i=1}^{s}\left (\frac{x_{i}} {M}\right )^{\gamma }\right )dx_{ 1}\cdots dx_{s} & \\ & =\lim _{M\rightarrow \infty }\frac{M^{s}\varGamma ^{s}\big(\frac{1} {\gamma } \big)} {\gamma ^{s}\varGamma \big(\frac{s} {\gamma } \big)} \int _{0}^{1}\varPsi (u)u^{\frac{s} {\gamma } -1}du;& {}\end{array}$$(6) -
(ii)
for
$$\displaystyle\begin{array}{rcl} \mathbf{\{}x& \in &\mathbf{R}_{+}^{s};\vert \vert x\vert \vert _{\gamma }\geq 1\} {}\\ & =& \left \{x \in \mathbf{R}_{+}^{s}; \frac{1} {M^{\gamma }} < u =\sum _{ i=1}^{s}\left ( \frac{x_{i}} {M}\right )^{\gamma } \leq 1(M \rightarrow \infty )\right \}\!\!, {}\\ \end{array}$$setting \(\varPsi (u) = 0\big(u \in \big (0, \frac{1} {M^{\gamma }}\big)\big)\), we have
$$\displaystyle\begin{array}{rcl} & \int \cdots \int _{\mathbf{\{}x\in \mathbf{R}_{+}^{s};\vert \vert x\vert \vert _{\gamma }\geq 1\}}\varPsi \left (\sum _{i=1}^{s}\left (\frac{x_{i}} {M}\right )^{\gamma }\right )dx_{ 1}\cdots dx_{s}& \\ & =\lim _{M\rightarrow \infty }\frac{M^{s}\varGamma ^{s}\big(\frac{1} {\gamma } \big)} {\gamma ^{s}\varGamma \big(\frac{s} {\gamma } \big)} \int _{ \frac{1} {M^{\gamma }} }^{1}\varPsi (u)u^{\frac{s} {\gamma } -1}du; & {}\end{array}$$(7) -
(iii)
for
$$\displaystyle\begin{array}{rcl} \mathbf{\{}x& \in &\mathbf{R}_{+}^{s};\vert \vert x\vert \vert _{\gamma }\leq 1\} {}\\ & =& \left \{x \in \mathbf{R}_{+}^{s};0 < u =\sum _{ i=1}^{s}\left ( \frac{x_{i}} {M}\right )^{\gamma } \leq \frac{1} {M^{\gamma }}\right \}\!\!, {}\\ \end{array}$$setting \(\varPsi (u) = 0\big(u \in \big ( \frac{1} {M^{\gamma }},\infty \big)\big)\), we have
$$\displaystyle\begin{array}{rcl} & \int \cdots \int _{\mathbf{\{}x\in \mathbf{R}_{+}^{s};\vert \vert x\vert \vert _{\gamma }\leq 1\}}\varPsi \left (\sum _{i=1}^{s}\left (\frac{x_{i}} {M}\right )^{\gamma }\right )dx_{ 1}\cdots dx_{s}& \\ & = \frac{M^{s}\varGamma ^{s}\big(\frac{1} {\gamma } \big)} {\gamma ^{s}\varGamma \big(\frac{s} {\gamma } \big)} \int _{0}^{ \frac{1} {M^{\gamma }} }\varPsi (u)u^{\frac{s} {\gamma } -1}du. & {}\end{array}$$(8)
Lemma 2.
For s ∈ N, γ > 0, \(\varepsilon > 0\) , we have
Proof.
By (7), it follows
By (8), we find
Hence, we have (9) and (10). The lemma is proved.
Note.
By (9) and (10), for δ = ±1, we have the following unified expression:
Definition 1.
If \(x = (x_{1},\ldots,x_{i_{0}}) \in \mathbf{R}_{+}^{i_{0}},y = (y_{1},\ldots,y_{j_{ 0}}) \in \mathbf{R}_{+}^{j_{0}}\), h(u) is a nonnegative measurable function in R +, σ ∈ R, δ ∈ {−1, 1}, then we define two weight functions ω δ (σ, y) and \(\varpi _{\delta }(\sigma,x)\) as follows:
By (6), we find
Setting \(v = M^{\delta }u^{\frac{\delta }{\alpha }}\vert \vert y\vert \vert _{ \beta }\) in the above integral, in view of δ = ±1, we obtain
where \(k(\sigma ) =\int _{ 0}^{\infty }h(v)v^{\sigma -1}dv\).
By (6), setting \(v = M\vert \vert x\vert \vert _{\alpha }^{\delta }u^{\frac{1} {\beta } }\), we find
Lemma 3.
As the assumptions of Definition 1 , for k(σ) ∈ R +,\(p \in \mathbf{R}\setminus \{0,1\}, \frac{1} {p} + \frac{1} {q} = 1\) , setting
then we have
where \(\tilde{K}(\sigma ):= L(\alpha,\beta )k(\sigma )\mathbf{,}\)
Moreover, if there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0}),k(\tilde{\sigma }) \in \mathbf{R}\), then we have
Proof.
For \(\varepsilon > 0\), setting \(\tilde{\sigma }=\sigma + \frac{\varepsilon }{q}\) and
in view of (16), it follows
Putting
by (8), we find
Setting \(v = \vert \vert x\vert \vert _{\alpha }^{\delta }Mu^{\frac{1} {\beta } }\) in the above, it follows
Putting \(\varPsi (u) = M^{-\delta \varepsilon -i_{0}}u^{\frac{1} {\alpha } (-\delta \varepsilon -i_{0})}H(M^{\delta }u^{\frac{\delta }{\alpha }})\), for δ = 1, by (7), we obtain
for δ = −1, by (8), we still find that
Hence, we find
By Fatou lemma (cf. [20]), it follows
and then (17) follows.
Moreover, for \(0 <\varepsilon <\delta _{0}\min \{\vert p\vert,\vert q\vert \},\tilde{\sigma }\in \big (\sigma -\frac{1} {2}\delta _{0},\sigma +\frac{1} {2}\delta _{0}\big)\), since
by Lebesgue control convergence theorem (cf. [20]), it follows that
Then by (20), (19) follows. The lemma is proved.
Lemma 4.
As the assumptions of Definition 1 , if \(p \in \mathbf{R}\setminus \{0,1\}, \frac{1} {p} + \frac{1} {q} = 1\),\(f(x) = f(x_{1},\ldots,x_{i_{0}}) \geq 0\),\(g(y) = g(y_{1},\ldots,y_{j_{0}}) \geq 0\) , then
-
(i)
for p > 1, we have the following inequality:
$$\displaystyle\begin{array}{rcl} J_{1}&:=& \left \{\int _{\mathbf{R}_{+}^{j_{0}}} \frac{\vert \vert y\vert \vert _{\beta }^{p\sigma -j_{0}}} {\left [\omega _{\delta }(\sigma,y)\right ]^{p-1}}\left (\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)dx\right )^{p}dy\right \}^{\frac{1} {p} } \\ & \leq &\left \{\int _{\mathbf{R}_{+}^{i_{0}}}\varpi _{\delta }(\sigma,x)\vert \vert x\vert \vert _{\alpha }^{p\left (i_{0}-\delta \sigma \right )-i_{0} }f^{p}(x)dx\right \}^{\frac{1} {p} }; {}\end{array}$$(21) -
(ii)
for 0 < p < 1, or p < 0, we have the reverse of (21) .
Proof.
-
(i)
For p > 1, by Hölder’s inequality with weight (cf. [21]), it follows
$$\displaystyle\begin{array}{rcl} & & \int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)dx \\ & & \quad =\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })\left [\frac{\vert \vert x\vert \vert _{\alpha }^{(i_{0}-\delta \sigma )/q}f(x)} {\vert \vert y\vert \vert _{\beta }^{(j_{0}-\sigma )/p}} \right ]\left [\frac{\vert \vert y\vert \vert _{\beta }^{(j_{0}-\sigma )/p}} {\vert \vert x\vert \vert _{\alpha }^{(i_{0}-\delta \sigma )/q}}\right ]dx \\ & & \quad \leq \left \{\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })\frac{\vert \vert x\vert \vert _{\alpha }^{(i_{0}-\delta \sigma )(p-1)}} {\vert \vert y\vert \vert _{\beta }^{j_{0}-\sigma }} f^{p}(x)dx\right \}^{\frac{1} {p} } \\ & & \qquad \times \left \{\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })\frac{\vert \vert y\vert \vert _{\beta }^{(j_{0}-\sigma )(q-1)}} {\vert \vert x\vert \vert _{\alpha }^{i_{0}-\delta \sigma }} dx\right \}^{\frac{1} {q} } \\ & & \quad = [\omega _{\delta }(\sigma,y)]^{\frac{1} {q} }\vert \vert y\vert \vert _{\beta }^{\frac{j_{0}} {p} -\sigma } \\ &&\qquad \times \left \{\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })\frac{\vert \vert x\vert \vert _{\alpha }^{(i_{0}-\delta \sigma )(p-1)}} {\vert \vert y\vert \vert _{\beta }^{j_{0}-\sigma }} f^{p}(x)dx\right \}^{\frac{1} {p} }. {}\end{array}$$(22)Then by Fubini theorem (cf. [20]), we have
$$\displaystyle\begin{array}{rcl} J_{1}& \leq &\left \{\int _{\mathbf{R}_{+}^{j_{0}}}\left [\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })\frac{\vert \vert x\vert \vert _{\alpha }^{(i_{0}-\delta \sigma )(p-1)}} {\vert \vert y\vert \vert _{\beta }^{j_{0}-\sigma }} f^{p}(x)dx\right ]dy\right \}^{\frac{1} {p} } \\ & =& \left \{\int _{\mathbf{R}_{+}^{i_{0}}}\left [\int _{\mathbf{R}_{+}^{j_{0}}}h(\vert \vert x\vert \vert _{\alpha }\vert \vert y\vert \vert _{\beta })\frac{\vert \vert x\vert \vert _{\alpha }^{(i_{0}-\delta \sigma )(p-1)}} {\vert \vert y\vert \vert _{\beta }^{j_{0}-\sigma }} dy\right ]f^{p}(x)dx\right \}^{\frac{1} {p} } \\ & =& \left \{\int _{\mathbf{R}_{+}^{i_{0}}}\varpi _{\delta }(\sigma,x)\vert \vert x\vert \vert _{\alpha }^{p(i_{0}-\delta \sigma )-i_{0} }f^{p}(x)dx\right \}^{\frac{1} {p} }. {}\end{array}$$(23)Hence, (21) follows.
-
(ii)
For 0 < p < 1, or p < 0, by the reverse Hölder’s inequality with weight (cf. [21]), we obtain the reverse of (22). Then by Fubini theorem, we still can obtain the reverse of (21). The lemma is proved.
Lemma 5.
As the assumptions of Lemma 4 , then
-
(i)
for p > 1, we have the following inequality equivalent to (21) :
$$\displaystyle\begin{array}{rcl} I&:=& \int _{\mathbf{R}_{+}^{j_{0}}}\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)g(y)dxdy \\ & \leq &\left \{\int _{\mathbf{R}_{+}^{i_{0}}}\varpi _{\delta }(\sigma,x)\vert \vert x\vert \vert _{\alpha }^{p(i_{0}-\delta \sigma )-i_{0} }f^{p}(x)dx\right \}^{\frac{1} {p} } \\ & & \times \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\omega _{\delta }(\sigma,y)\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )-j_{0} }g^{q}(y)dy\right \}^{\frac{1} {q} }; {}\end{array}$$(24) -
(ii)
for 0 < p < 1, or p < 0, we have the reverse of (24) equivalent to the reverse of (21) .
Proof.
-
(i)
For p > 1, by Hölder’s inequality (cf. [21]), it follows
$$\displaystyle\begin{array}{rcl} I& =& \int _{\mathbf{R}_{+}^{j_{0}}} \frac{\vert \vert y\vert \vert _{\beta }^{\frac{j_{0}} {q} -(j_{0}-\sigma )}} {\left [\omega _{\delta }(\sigma,y)\right ]^{\frac{1} {q} }} \left [\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)dx\right ] {}\\ & & \times \left [[\omega _{\delta }(\sigma,y)]^{\frac{1} {q} }\vert \vert y\vert \vert _{\beta }^{(j_{0}-\sigma )-\frac{j_{0}} {q} }g(y)\right ]dy {}\\ \end{array}$$$$\displaystyle{ \leq J_{1}\left \{\int _{\mathbf{R}_{+}^{j_{0}}}\omega _{\delta }(\sigma,y)\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )-j_{0} }g^{q}(y)dy\right \}^{\frac{1} {q} }. }$$(25)On the other hand, assuming that (24) is valid, we set
$$\displaystyle{g(y):= \frac{\vert \vert y\vert \vert _{\beta }^{p\sigma -j_{0}}} {[\omega _{\delta }(\sigma,y)]^{p-1}}\left (\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)dx\right )^{p-1},y \in \mathbf{R}_{ +}^{j_{0} }.}$$Then it follows
$$\displaystyle{J_{1}^{p} =\int _{\mathbf{ R}_{+}^{j_{0}}}\omega _{\delta }(\sigma,y)\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )-j_{0} }g^{q}(y)dy.}$$If J 1 = 0, then (21) is trivially valid; if J 1 = ∞, then by (23), (21) keeps the form of equality ( = ∞). Suppose that 0 < J 1 < ∞. By (24), we have
$$\displaystyle\begin{array}{rcl} 0& <& \int _{\mathbf{R}_{+}^{j_{0}}}\omega _{\delta }(\sigma,y)\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )-j_{0} }g^{q}(y)dy = J_{ 1}^{p} = I {}\\ & \leq &\left \{\int _{\mathbf{R}_{+}^{i_{0}}}\varpi _{\delta }(\sigma,x)\vert \vert x\vert \vert _{\alpha }^{p(i_{0}-\delta \sigma )-i_{0} }f^{p}(x)dx\right \}^{\frac{1} {p} } {}\\ & & \times \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\omega _{\delta }(\sigma,y)\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )-j_{0} }g^{q}(y)dy\right \}^{\frac{1} {q} } < \infty. {}\\ \end{array}$$It follows
$$\displaystyle\begin{array}{rcl} J_{1}& =& \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\omega _{\delta }(\sigma,y)\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )-j_{0} }g^{q}(y)dy\right \}^{\frac{1} {p} } {}\\ & \leq &\left \{\int _{\mathbf{R}_{+}^{i_{0}}}\varpi _{\delta }(\sigma,x)\vert \vert x\vert \vert _{\alpha }^{p(i_{0}-\delta \sigma )-i_{0} }f^{p}(x)dx\right \}^{\frac{1} {p} }, {}\\ \end{array}$$ -
(ii)
For 0 < p < 1, or p < 0, by the same way, we can obtain the reverse of (24) equivalent to the reverse of (21). The lemma is proved.
3 Main Results and Operator Expressions
Setting
by Lemmas 3– 5, it follows
Theorem 1.
Suppose that α,β > 0, σ ∈ R ,h(v) ≥ 0,
\(\delta \in \{-1,1\},p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1\),\(\ f(x) = f(x_{1},\ldots,x_{i_{0}}) \geq 0\),\(g(y) = g(y_{1},\ldots,y_{j_{0}})\) ≥ 0,
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle{ I =\int _{\mathbf{R}_{+}^{j_{0}}}\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)g(y)dxdy < K(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert g\vert \vert _{q,\varPsi }, }$$(26)$$\displaystyle\begin{array}{rcl} J&:=& \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\vert \vert y\vert \vert _{\beta }^{p\sigma -j_{0} }\left (\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)dx\right )^{p}dy\right \}^{\frac{1} {p} } \\ & & < K(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}; {}\end{array}$$(27) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (26) and (27) with the same best constant factor K(σ).
Proof.
-
(i)
For p > 1, by the conditions, we can prove that (22) takes the form of strict inequality for a.e. \(y \in \mathbf{R}_{+}^{j_{0}}\). Otherwise, if (22) takes the form of equality for a \(y \in \mathbf{R}_{+}^{j_{0}}\), then there exist constants A and B, which are not all zero, such that
$$\displaystyle{ A\frac{\vert \vert x\vert \vert _{\alpha }^{(i_{0}-\delta \sigma )(p-1)}} {\vert \vert y\vert \vert _{\beta }^{j_{0}-\sigma }} f^{p}(x) = B\frac{\vert \vert y\vert \vert _{\beta }^{(j_{0}-\sigma )(q-1)}} {\vert \vert x\vert \vert _{\alpha }^{i_{0}-\delta \sigma }} \mathrm{\,\, a.e.\,\, in \,\,}x \in \mathbf{R}_{+}^{i_{0} }. }$$(28)If A = 0, then B = 0, which is impossible; if A ≠ 0, then (28) reduces to
$$\displaystyle{\vert \vert x\vert \vert _{\alpha }^{p(i_{0}-\delta \sigma )-i_{0} }f^{p}(x) = \frac{B\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )}} {A\vert \vert x\vert \vert _{\alpha }^{i_{0}}} \mathrm{\,\,a.e.\,\, in \,\,}x \in \mathbf{R}_{+}^{i_{0} },}$$which contradicts the fact that \(0 < \vert \vert f\vert \vert _{p,\varPhi _{\delta }} < \infty \). In fact, by (9) (for \(\varepsilon \rightarrow 0^{+})\), it follows
$$\displaystyle{\int _{\mathbf{R}_{+}^{i_{0}}}\vert \vert x\vert \vert _{\alpha }^{-i_{0} }dx \geq \int _{\left \{x\in \mathbf{R}_{+}^{i_{0}};\vert \vert x\vert \vert _{\alpha }\geq 1\right \}}\vert \vert x\vert \vert _{\alpha }^{-i_{0} }dx = \infty.}$$Hence (22) still takes the form of strict inequality. By (14) and (15), we obtain (27).
Similarly to (25), we still have
$$\displaystyle{ I \leq J\left \{\int _{\mathbf{R}_{+}^{j_{0}}}\vert \vert y\vert \vert _{\beta }^{q(j_{0}-\sigma )-j_{0} }g^{q}(y)dy\right \}^{\frac{1} {q} }. }$$(29)Then by (29) and (27), we have (26). It is evident that by Lemma 5 and the assumptions, inequalities (27) and ( 26) are also equivalent.
For \(\varepsilon > 0\), we set \(\tilde{f}(x),\tilde{g}(y)\) as follows:
$$\displaystyle\begin{array}{rcl} & \tilde{f}(x):= \left \{\begin{array}{l} 0,\quad 0 < \vert \vert x\vert \vert _{\alpha }^{\delta } < 1, \\ \vert \vert x\vert \vert _{\alpha }^{\delta \big(\sigma -\frac{\varepsilon }{p}\big)-i_{0} },\quad \vert \vert x\vert \vert _{\alpha }^{\delta }\geq 1,\end{array} \right.& {}\\ & \tilde{g}(y):= \left \{\begin{array}{l} \vert \vert y\vert \vert _{\beta }^{\sigma + \frac{\varepsilon }{q}-j_{0} },\quad 0 < \vert \vert y\vert \vert _{\beta }\leq 1, \\ 0,\quad \vert \vert y\vert \vert _{\beta }\geq 1.\end{array} \right.& {}\\ \end{array}$$In view of (11) and (10), it follows
$$\displaystyle\begin{array}{rcl} & & \vert \vert \tilde{f}\vert \vert _{p,\varPhi _{\delta }}\vert \vert \tilde{g}\vert \vert _{q,\varPsi } {}\\ & & \quad = \left \{\int _{\left \{x\in \mathbf{R}_{+}^{i_{0}};\vert \vert x\vert \vert _{\alpha }^{\delta }\geq 1\right \}}\vert \vert x\vert \vert _{\alpha }^{-i_{0}-\delta \varepsilon }dx\right \}^{\frac{1} {p} }\left \{\int _{ \left \{y\in \mathbf{R}_{+}^{j_{0}};\vert \vert y\vert \vert _{\beta }\leq 1\right \}}\vert \vert y\vert \vert _{\beta }^{-j_{0}+\varepsilon }dy\right \}^{\frac{1} {q} } {}\\ & & \quad = \frac{1} {\varepsilon } \left \{ \frac{\varGamma ^{i_{0}}\left (\frac{1} {\alpha } \right )} {\alpha ^{i_{0}-1}\varGamma \left (\frac{i_{0}} {\alpha } \right )}\right \}^{\frac{1} {p} }\left \{ \frac{\varGamma ^{j_{0}}\left (\frac{1} {\beta } \right )} {\beta ^{j_{0}-1}\varGamma \left (\frac{j_{0}} {\beta } \right )}\right \}^{\frac{1} {q} }. {}\\ \end{array}$$If there exists a constant K ≤ K(σ), such that (26) is valid when replacing K(σ) by K, then in particular, by (16) and ( 17), we have
$$\displaystyle\begin{array}{rcl} & & \tilde{K}(\sigma ) + o(1) \leq \varepsilon \tilde{ I} {}\\ & & \quad =\varepsilon \int _{\mathbf{R}_{+}^{j_{0}}}\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })\tilde{f}(x)\tilde{g}(y)dxdy {}\\ & & \quad <\varepsilon K\vert \vert \tilde{f}\vert \vert _{p,\varPhi _{\delta }}\vert \vert \tilde{g}\vert \vert _{q,\varPsi } {}\\ & & \quad = K\left \{ \frac{\varGamma ^{i_{0}}\left (\frac{1} {\alpha } \right )} {\alpha ^{i_{0}-1}\varGamma \left (\frac{i_{0}} {\alpha } \right )}\right \}^{\frac{1} {p} }\left \{ \frac{\varGamma ^{j_{0}}\left (\frac{1} {\beta } \right )} {\beta ^{j_{0}-1}\varGamma \left (\frac{j_{0}} {\beta } \right )}\right \}^{\frac{1} {q} }, {}\\ \end{array}$$and then we find \(K(\sigma ) \leq K(\varepsilon \rightarrow 0^{+})\). Hence K = K(σ) is the best possible constant factor of (26).
By the equivalency, we can prove that the constant factor K(σ) in (27) is the best possible. Otherwise, we would reach a contradiction by (29) that the constant factor K(σ) in (26) is not the best possible.
-
(ii)
For 0 < p < 1, or p < 0, by the same way, we still can obtain the equivalent reverses of (26) and (27). For \(\varepsilon > 0\), we set \(\tilde{f}(x),\tilde{g}(y)\) as the case of p > 1. If there exists a constant K ≥ K(σ), such that the reverse of (26) is valid when replacing K(σ) by K, then in particular, by (16) and (19), we have
$$\displaystyle\begin{array}{rcl} & & \tilde{K}(\sigma ) + o(1) =\varepsilon \tilde{ I} {}\\ & & \quad =\varepsilon \int _{\mathbf{R}_{+}^{j_{0}}}\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })\tilde{f}(x)\tilde{g}(y)dxdy {}\\ & & \quad >\varepsilon K\vert \vert \tilde{f}\vert \vert _{p,\varPhi _{\delta }}\vert \vert \tilde{g}\vert \vert _{q,\varPsi } {}\\ & & \quad = K\left \{ \frac{\varGamma ^{i_{0}}\big(\frac{1} {\alpha } \big)} {\alpha ^{i_{0}-1}\varGamma \left (\frac{i_{0}} {\alpha } \right )}\right \}^{\frac{1} {p} }\left \{ \frac{\varGamma ^{j_{0}}\left (\frac{1} {\beta } \right )} {\beta ^{j_{0}-1}\varGamma \left (\frac{j_{0}} {\beta } \right )}\right \}^{\frac{1} {q} }, {}\\ \end{array}$$and then we find \(K(\sigma ) \geq K(\varepsilon \rightarrow 0^{+})\). Hence K = K(σ) is the best possible constant factor of the reverse of (26). By the equivalency, we can prove that the constant factor K(σ) in the reverse of (27) is the best possible. Otherwise, we would reach a contradiction by the reverse of (29) that the constant factor K(σ) in the reverse of (26) is not the best possible. The theorem is proved.
In particular, for δ = 1 in Theorem 1, we have
Corollary 1.
Suppose that α,β > 0, σ ∈ R ,h(v) ≥ 0,
\(p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1\),\(\ f(x) = f(x_{1},\ldots,x_{i_{0}}) \geq 0\),\(g(y) = g(y_{1},\ldots,y_{j_{0}}) \geq 0\),
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle{ I =\int _{\mathbf{R}_{+}^{j_{0}}}\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }\vert \vert y\vert \vert _{\beta })f(x)g(y)dxdy < K(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{1}}\vert \vert g\vert \vert _{q,\varPsi }, }$$(30)$$\displaystyle\begin{array}{rcl} J&:=& \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\vert \vert y\vert \vert _{\beta }^{p\sigma -j_{0} }\left (\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert x\vert \vert _{\alpha }\vert \vert y\vert \vert _{\beta })f(x)dx\right )^{p}dy\right \}^{\frac{1} {p} } \\ & & < K(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{1}}; {}\end{array}$$(31) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (30) and (31) with the same best constant factor K(σ).
For i 0 = j 0 = α = β = 1 in Corollary 1, we have
Corollary 2.
Assuming that \(\sigma \in \mathbf{R},k(\sigma ) \in \mathbf{R}_{+},p \in \mathbf{R}\setminus \{0,1\}, \frac{1} {p} + \frac{1} {q} = 1\) , we set
If f(x) ≥ 0, g(y) ≥ 0,
then (i) for p > 1, we have the following equivalent inequalities with the best possible constant factor k(σ):
(ii) for 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k(\tilde{\sigma }) \in \mathbf{R}\) , we have the equivalent reverses of (32) and (33) with the same best constant factor.
As the assumptions of Theorem 1, for p > 1, in view of \(J < K(\sigma )\vert \vert f\vert \vert _{\varPhi _{\delta }}\), we can give the following definition:
Definition 2.
Define a multidimensional Hilbert-type integral operator
as follows: For \(f \in \mathbf{L}_{p,\varPhi _{\delta }}(\mathbf{R}_{+}^{i_{0}}),\) there exists a unique representation
satisfying
For \(g \in \mathbf{L}_{q,\varPsi }(\mathbf{R}_{+}^{j_{0}})\), we define the following formal inner product of Tf and g as follows:
Then by Theorem 1, for \(p > 1,0 < \vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert g\vert \vert _{q,\varPsi } < \infty \), we have the following equivalent inequalities:
It follows that T is bounded with
Since the constant factor K(σ) in (38) is the best possible, we have
4 A Corollary for δ = −1
Corollary 3.
Suppose that α,β > 0, μ,σ ∈ R ,μ + σ = λ,k λ (x,y) ≥ 0 is a homogeneous function of degree −λ,
\(p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1,\varPhi (x):= x^{p(i_{0}-\mu )-i_{0}},F(x) = F(x_{ 1},\ldots,x_{i_{0}}) \geq 0\),\(g(y) = g(y_{1},\ldots,y_{j_{0}}) \geq 0\),
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle\begin{array}{rcl} \int _{\mathbf{R}_{+}^{j_{0}}}\int _{\mathbf{R}_{+}^{i_{0}}}k_{\lambda }(\vert \vert x\vert \vert _{\alpha },\vert \vert y\vert \vert _{\beta })F(x)g(y)dxdy < K_{\lambda }(\sigma )\vert \vert F\vert \vert _{p,\varPhi }\vert \vert g\vert \vert _{q,\varPsi },& & {}\end{array}$$(40)$$\displaystyle\begin{array}{rcl} \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\vert \vert y\vert \vert _{\beta }^{p\sigma -j_{0} }\left (\int _{\mathbf{R}_{+}^{i_{0}}}k_{\lambda }(\vert \vert x\vert \vert _{\alpha },\vert \vert y\vert \vert _{\beta })F(x)dx\right )^{p}dy\right \}^{\frac{1} {p} } < K_{\lambda }(\sigma )\vert \vert F\vert \vert _{p,\varPhi };& & {}\end{array}$$(41) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{\lambda }(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (40) and (41) with the same best constant factor K λ (σ).
In particular, for \(i_{0} = j_{0} =\alpha =\beta = 1,\varphi _{1}(x):= x^{p(1-\mu )-1}\), if F(x) ≥ 0, g(y) ≥ 0,
then (i) for p > 1, we have the following equivalent inequalities with the best possible constant factor k λ (σ):
-
(ii)
for 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\), \(k_{\lambda }(\tilde{\sigma }) \in \mathbf{R}\), we have the equivalent reverses of (42) and (43) with the same best constant factor k λ (σ).
Proof.
For δ = −1 in Theorem 1, setting h(u) = k λ (1, u) and \(\vert \vert x\vert \vert _{\alpha }^{\lambda }f(x) = F(x)\), since μ = λ −σ, by simplifications, we can obtain (40) and (41) (for p > 1). It is evident that (40) and (41) are equivalent with the same best constant factor K λ (σ). By the same way, we can show the cases in 0 < p < 1 or p < 0. The corollary is proved.
Remark 1.
Inequality (42), (43) is equivalent to (32), (33). In fact, Setting \(x = \frac{1} {X},h(u) = k_{\lambda }(1,u)\) in (32), (33), replacing \(X^{\lambda }f( \frac{1} {X})\) by F(X), by simplification, we obtain (42), (43). On the other hand, by (42), (43), we can deduce (32), (33).
5 Two Classes of Hardy-Type Inequalities
If h(v) = 0(v > 1), then
by Theorem 1, we have the following first class of Hardy-type inequalities:
Corollary 4.
Suppose that \(\alpha,\beta > 0,\ \sigma \in \mathbf{R},h(v) \geq 0\),
\(\delta \in \{-1,1\},p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1\),\(f(x) = f(x_{1},\ldots,x_{i_{0}}) \geq 0\),\(g(y) = g(y_{1},\ldots,y_{j_{0}}) \geq 0\),
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor H 1 (σ):
$$\displaystyle\begin{array}{rcl} & \int _{\mathbf{R}_{+}^{j_{0}}}\left [\int _{\left \{x\in \mathbf{R}_{+}^{i_{0}};\vert \vert x\vert \vert _{\alpha }^{\delta }\leq \vert \vert y\vert \vert _{\beta }^{-1}\right \}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)dx\right ]g(y)dy& \\ & < H_{1}(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert g\vert \vert _{q,\varPsi }, & {}\end{array}$$(44)$$\displaystyle\begin{array}{rcl} & \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\vert \vert y\vert \vert _{\beta }^{p\sigma -j_{0}}\left (\int _{\left \{x\in \mathbf{R}_{ +}^{i_{0}};\vert \vert x\vert \vert _{\alpha }^{\delta }\leq \vert \vert y\vert \vert _{\beta }^{-1}\right \}}h\left (\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta }\right )f(x)dx\right )^{p}dy\right \}^{\frac{1} {p} }& \\ & < H_{1}(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}; & {}\end{array}$$(45) -
(ii)
If 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{1}(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (44) and (45) with the same best constant factor H 1 (σ).
For i 0 = j 0 = α = β = 1, δ = 1 in Corollary 4, we have
Corollary 5.
Assuming that \(\sigma \in \mathbf{R},k_{1}(\sigma ) \in \mathbf{R}_{+},p \in \mathbf{R}\setminus \{0,1\}, \frac{1} {p} + \frac{1} {q} = 1\) , we set
If f(x) ≥ 0, g(y) ≥ 0,
then (i) for p > 1, we have the following equivalent inequalities with the best possible constant factor k 1 (σ):
(ii) for 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{1}(\tilde{\sigma }) \in \mathbf{R}\) , we have the equivalent reverses of (46) and (47) with the same best constant factor k 1 (σ).
If k λ (x, y) = 0(x < y), by (42) and (43), we have
Corollary 6.
Assuming that μ,σ ∈ R ,μ + σ = λ,
\(p \in \mathbf{R}\setminus \{0,1\}, \frac{1} {p} + \frac{1} {q} = 1\),\(\varphi _{1}(x):= x^{p(1-\mu )-1}\) , if F(x) ≥ 0, g(y) ≥ 0,
then (i) for p > 1, we have the following equivalent inequalities with the best possible constant factor \(k_{\lambda }^{(1)}(\sigma ):\)
(ii) for 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{\lambda }^{(1)}(\tilde{\sigma }) \in \mathbf{R}\) , we have the equivalent reverses of (48) and (49) with the same best constant factor \(k_{\lambda }^{(1)}(\sigma )\) .
If h(v) = 0(0 < v < 1), then
by Theorem 1, we have the following second class of Hardy-type inequalities:
Corollary 7.
Suppose that \(\alpha,\beta > 0,\ \sigma \in \mathbf{R},h(v) \geq 0\),
\(\delta \in \{-1,1\},p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1\),\(f(x) = f(x_{1},\ldots,x_{i_{0}}) \geq 0\),\(g(y) = g(y_{1},\ldots,y_{j_{0}}) \geq 0\),
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor H 2 (σ):
$$\displaystyle\begin{array}{rcl} & \int _{\mathbf{R}_{+}^{j_{0}}}\left [\int _{\left \{x\in \mathbf{R}_{+}^{i_{0}};\vert \vert x\vert \vert _{\alpha }^{\delta }\geq \vert \vert y\vert \vert _{\beta }^{-1}\right \}}h\left (\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta }\right )f(x)dx\right ]g(y)dy& \\ & < H_{2}(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert g\vert \vert _{q,\varPsi }, & {}\end{array}$$(50)$$\displaystyle\begin{array}{rcl} & \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\vert \vert y\vert \vert _{\beta }^{p\sigma -j_{0}}\left (\int _{\left \{x\in \mathbf{R}_{ +}^{i_{0}};\vert \vert x\vert \vert _{\alpha }^{\delta }\geq \vert \vert y\vert \vert _{\beta }^{-1}\right \}}h(\vert \vert x\vert \vert _{\alpha }^{\delta }\vert \vert y\vert \vert _{\beta })f(x)dx\right )^{p}dy\right \}^{\frac{1} {p} }& \\ & < H_{2}(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}; & {}\end{array}$$(51) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{2}(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (50) and (51) with the same best constant factor H 2 (σ).
For \(i_{0} = j_{0} =\alpha =\beta = 1,\delta = 1\) in Corollary 7, we have
Corollary 8.
Assuming that \(\sigma \in \mathbf{R},k_{2}(\sigma ) \in \mathbf{R}_{+},p \in \mathbf{R}\setminus \{0,1\}, \frac{1} {p} + \frac{1} {q} = 1\) , we set
If f(x) ≥ 0, g(y) ≥ 0,
then (i) for p > 1, we have the following equivalent inequalities with the best possible constant factor k 2 (σ):
(ii) for 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{2}(\tilde{\sigma }) \in \mathbf{R}\) , we have the equivalent reverses of (52) and (53) with the same best constant factor k 2 (σ).
If k λ (x, y) = 0(x > y), by (42) and (43), we have
Corollary 9.
Assuming that μ,σ ∈ R ,μ + σ = λ,
\(p \in \mathbf{R}\setminus \{0,1\}, \frac{1} {p} + \frac{1} {q} = 1\),\(\varphi _{1}(x):= x^{p(1-\mu )-1}\) , if F(x) ≥ 0, g(y) ≥ 0,
then (i) for p > 1, we have the following equivalent inequalities with the best possible constant factor \(k_{\lambda }^{(2)}(\sigma ):\)
(ii) for 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{\lambda }^{(2)}(\tilde{\sigma }) \in \mathbf{R}\) , we have the equivalent reverses of (54) and (55) with the same best constant factor \(k_{\lambda }^{(2)}(\sigma )\) .
6 Multidimensional Hilbert-Type Inequalities with Two Variables
Suppose that \(u_{i}(s_{i}),u_{i}^{{\prime}}(s_{i}) > 0,u_{i}(a_{i}^{+}) = 0,u_{i}(b_{i}^{-}) = \infty (-\infty \leq a_{i} < b_{i} \leq \infty,i = 1,\ldots,i_{0})\), \(u(s) = (u_{1}(s_{1}),\ldots,u_{i_{0}}(s_{i_{0}})),v_{j}(t_{j}),v_{j}^{{\prime}}(t_{j}) > 0,v_{j}(c_{j}^{+}) = 0,v_{j}(d_{j}^{-}) = \infty \,\,\) \(\,\,(-\infty \leq c_{j} < d_{j} \leq \infty,j = 1,\ldots,j_{0})\),
\(v(t) = (v_{1}(t_{1}),\ldots,v_{j_{0}}(t_{j_{0}}))\),
Setting x = u(s), y = v(t) in Theorem 1, for
we have
Theorem 2.
Suppose that \(\alpha,\beta > 0,\ \sigma \in \mathbf{R},h(v) \geq 0\),
\(\delta \in \{-1,1\},p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1\),\(F(s) = F(s_{1},\ldots,s_{i_{0}}) \geq 0\),\(G(t) = G(t_{1},\ldots,t_{j_{0}})\) ≥ 0,
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle\begin{array}{rcl} & \int _{\left \{t\in \mathbf{R}^{j_{0}};c_{j}<t_{j}<d_{j}\right \}}\int _{\left \{s\in \mathbf{R}^{i_{0}};a_{i}<s_{i}<b_{i}\right \}}h(\vert \vert u(s)\vert \vert _{\alpha }^{\delta }\vert \vert v(t)\vert \vert _{\beta })F(s)G(t)dsdt& \\ & < K(\sigma )\vert \vert F\vert \vert _{p,\tilde{\varPhi }_{\delta }}\vert \vert g\vert \vert _{q,\tilde{\varPsi }}, & {}\end{array}$$(56)$$\displaystyle\begin{array}{rcl} & \left \{\int _{\{t\in \mathbf{R}^{j_{0}};c_{j}<t_{j}<d_{j}\}}\vert \vert v(t)\vert \vert _{\beta }^{p\sigma -j_{0}}\varPi _{j=1}^{j_{0}}v_{j}^{{\prime}}(t_{j})\left (\int _{\{s\in \mathbf{R}^{i_{ 0}};a_{i}<s_{i}<b_{i}\}}h(\vert \vert u(s)\vert \vert _{\alpha }^{\delta }\vert \vert v(t)\vert \vert _{\beta })\right.\right.& \\ & \left.\left.F(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < K(\sigma )\vert \vert F\vert \vert _{p,\tilde{\varPhi }_{\delta }}; & {}\end{array}$$(57) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (56) and (57) with the same best constant factor K(σ).
In particular, for i 0 = j 0 = α = β = 1,
-
(i)
if p > 1, then we have the following equivalent inequalities with the best possible constant factor k(σ):
$$\displaystyle\begin{array}{rcl} \int _{c}^{d}\int _{ a}^{b}h(u^{\delta }(s)v(t))F(s)G(t)dsdt < k(\sigma )\vert \vert F\vert \vert _{ p,\tilde{\phi }_{\delta }}\vert \vert G\vert \vert _{q,\tilde{\psi }},& & {}\end{array}$$(58)$$\displaystyle\begin{array}{rcl} \left \{\int _{c}^{d}(v(t))^{p\sigma -1}v^{{\prime}}(t)\left (\int _{ a}^{b}h(u^{\delta }(s)v(t))F(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < k(\sigma )\vert \vert F\vert \vert _{p,\tilde{\phi }_{\delta }};& & {}\end{array}$$(59) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\), \(k(\tilde{\sigma }) \in \mathbf{R}\), then we still have the equivalent reverses of (58) and (59) with the same best constant factor k(σ).
In particular, for \(\gamma,\eta > 0,u_{i}(s_{i}) = s_{i}^{\gamma },u_{i}^{{\prime}}(s_{i}) =\gamma s_{i}^{\gamma -1},u_{i}(0^{+}) = 0,u_{i}(\infty ) = \infty (a_{i} = 0,b_{i} = \infty,i = 1,\ldots,i_{0})\), \(\hat{u}(s) = (s_{1}^{\gamma },\ldots,s_{i_{0}}^{\gamma }),v_{ j}(t_{j}) = t_{j}^{\eta },v_{ j}^{{\prime}}(t_{ j}) =\eta t_{j}^{\eta -1},v_{ j}(0^{+}) = 0,v_{ j}(\infty ) = \infty (c_{j} = 0,d_{j} = \infty,j = 1,\ldots,j_{0})\), \(\hat{v}(t) = (t_{1}^{\eta },\ldots,t_{j_{0}}^{\eta })\), and
in Theorem 2, we have
Corollary 10.
Suppose that \(\alpha,\beta,\gamma,\eta > 0,\ \sigma \in \mathbf{R},h(v) \geq 0\),
\(\delta \in \{-1,1\},p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1\),\(F(s) = F(s_{1},\ldots,s_{i_{0}}) \geq 0\),\(G(t) = G(t_{1},\ldots,t_{j_{0}})\) ≥ 0,
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor \(\frac{1} {\gamma ^{i_{0}/q}\eta ^{j_{0}/p}}K(\sigma ):\)
$$\displaystyle\begin{array}{rcl} & \int _{\mathbf{R}_{+}^{j_{0}}}\int _{\mathbf{R}_{+}^{i_{0}}}h(\vert \vert \hat{u}(s)\vert \vert _{\alpha }^{\delta }\vert \vert \hat{v}(t)\vert \vert _{\beta })F(s)G(t)dsdt& \\ & < \frac{1} {\gamma ^{i_{0}/q}\eta ^{j_{0}/p}}K(\sigma )\vert \vert F\vert \vert _{p,\hat{\varPhi }_{\delta }}\vert \vert G\vert \vert _{q,\hat{\varPsi }}, & {}\end{array}$$(60)$$\displaystyle\begin{array}{rcl} & \left \{\int _{\mathbf{R}_{+}^{j_{0}}}\vert \vert \hat{v}(t)\vert \vert _{\beta }^{p\sigma -j_{0}}\varPi _{j=1}^{j_{0}}t_{j}^{\eta -1}\left (\int _{\mathbf{R}_{ +}^{i_{0}}}h(\vert \vert \hat{u}(s)\vert \vert _{\alpha }^{\delta }\vert \vert \hat{v}(t)\vert \vert _{\beta })\right.\right.& \\ & \left.\left.F(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < \frac{1} {\gamma ^{i_{0}/q}\eta ^{j_{0}/p}}K(\sigma )\vert \vert F\vert \vert _{p,\hat{\varPhi }_{\delta }}; & {}\end{array}$$(61) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (60) and (61) with the same best constant factor \(\frac{1} {\gamma ^{i_{0}/q}\eta ^{j_{0}/p}}K(\sigma )\) .
In particular, for i 0 = j 0 = α = β = 1,
-
(i)
if p > 1, then we have the following equivalent inequalities with the best possible constant factor \(\frac{1} {\gamma ^{1/q}\eta ^{1/p}}k(\sigma ):\)
$$\displaystyle\begin{array}{rcl} \int _{0}^{\infty }\int _{ 0}^{\infty }h(s^{\gamma \delta }t^{\eta })F(s)G(t)dsdt < \frac{1} {\gamma ^{1/q}\eta ^{1/p}}k(\sigma )\vert \vert F\vert \vert _{p,\hat{\phi }_{\delta }}\vert \vert G\vert \vert _{q,\hat{\psi }},& & {}\end{array}$$(62)$$\displaystyle\begin{array}{rcl} \left \{\int _{0}^{\infty }t^{p\eta \sigma -1}\left (\int _{ 0}^{\infty }h(s^{\gamma \delta }t^{\eta })F(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < \frac{1} {\gamma ^{1/q}\eta ^{1/p}}k(\sigma )\vert \vert F\vert \vert _{p,\hat{\phi }_{\delta }};& & {}\end{array}$$(63) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\), \(k(\tilde{\sigma }) \in \mathbf{R}\), then we still have the equivalent reverses of (62) and (63) with the same best constant factor \(\frac{1} {\gamma ^{1/q}\eta ^{1/p}}k(\sigma )\).
For \(\delta = -1,h(u) = k_{\lambda }(1,u)\), \(\vert \vert u(s)\vert \vert _{\alpha }^{\lambda }F(s) = f(s)\), μ = λ −σ and
in Theorem 2, by simplifications, we have
Corollary 11.
Suppose that \(\alpha,\beta > 0,\ \lambda,\mu,\sigma \in \mathbf{R},\mu +\sigma =\lambda,k_{\lambda }(x,y)(\geq 0)\) is a homogeneous function of degree −λ in \(\mathbf{R}_{+}^{2}\) , with
p ∈ R ∖{0,1}, \(\frac{1} {p} + \frac{1} {q} = 1\),\(f(s) = f(s_{1},\ldots,s_{i_{0}}) \geq 0\),\(G(t) = G(t_{1},\ldots,t_{j_{0}}) \geq 0\),
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle\begin{array}{rcl} & \int _{\left \{t\in \mathbf{R}^{j_{0}};c_{j}<t_{j}<d_{j}\right \}}\int _{\left \{s\in \mathbf{R}^{i_{0}};a_{i}<s_{i}<b_{i}\right \}}k_{\lambda }(\vert \vert u(s)\vert \vert _{\alpha },\vert \vert v(t)\vert \vert _{\beta })f(s)G(t)dsdt& \\ & < K_{\lambda }(\sigma )\vert \vert f\vert \vert _{p,\tilde{\varPhi }}\vert \vert G\vert \vert _{q,\tilde{\varPsi }}, & {}\end{array}$$(64)$$\displaystyle\begin{array}{rcl} & \left \{\int _{\left \{t\in \mathbf{R}^{j_{0}};c_{j}<t_{j}<d_{j}\right \}}\vert \vert v(t)\vert \vert _{\beta }^{p\sigma -j_{0}}\varPi _{j=1}^{j_{0}}v_{j}^{{\prime}}(t_{j})\left (\int _{\left \{s\in \mathbf{R}^{i_{ 0}};a_{i}<s_{i}<b_{i}\right \}}k_{\lambda }(\vert \vert u(s)\vert \vert _{\alpha },\vert \vert v(t)\vert \vert _{\beta })\right.\right.& \\ & \times \left.\left.f(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < K_{\lambda }(\sigma )\vert \vert f\vert \vert _{p,\tilde{\varPhi }}; & {}\end{array}$$(65) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{\lambda }(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (64) and (65) with the same best constant factor K λ (σ).
In particular, for i 0 = j 0 = α = β = 1,
-
(i)
if p > 1, then we have the following equivalent inequalities with the best possible constant factor k λ (σ):
$$\displaystyle\begin{array}{rcl} \int _{c}^{d}\int _{ a}^{b}k_{\lambda }(u(s),v(t))f(s)G(t)dsdt < k_{\lambda }(\sigma )\vert \vert f\vert \vert _{ p,\tilde{\phi }}\vert \vert G\vert \vert _{q,\tilde{\psi }},& & {}\end{array}$$(66)$$\displaystyle\begin{array}{rcl} \left \{\int _{c}^{d}(v(t))^{p\sigma -1}v^{{\prime}}(t)\left (\int _{ a}^{b}k_{\lambda }(u(s),v(t))f(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < k_{\lambda }(\sigma )\vert \vert f\vert \vert _{p,\tilde{\phi }};& & {}\end{array}$$(67) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\), \(k_{\lambda }(\tilde{\sigma }) \in \mathbf{R}\), then we still have the equivalent reverses of (66) and (67) with the same best constant factor k λ (σ).
In particular, for \(u_{i}(s_{i}) =\ln s_{i},u_{i}^{{\prime}}(s_{i}) = s_{i}^{-1},u_{i}(1^{+}) = 0,u_{i}(\infty ) = \infty (a_{i} = 1,b_{i} = \infty,i = 1,\ldots,i_{0})\), \(U(s) = (\ln s_{1},\ldots,\ln s_{i_{0}}),v_{j}(t_{j}) =\ln t_{j},v_{j}^{{\prime}}(t_{j}) = t_{j}^{-1},v_{j}(1^{+}) = 0,v_{j}(\infty )\) \(= \infty (c_{j} = 1,d_{j} = \infty,j = 1,\ldots,j_{0})\), \(V (t) = (\ln t_{1},\ldots,\ln t_{j_{0}})\), and
in Corollary 10, we have
Corollary 12.
Suppose that \(\alpha,\beta > 0,\ \lambda,\mu,\sigma \in \mathbf{R},\mu +\sigma =\lambda,k_{\lambda }(x,y)(\geq 0)\) is a homogeneous function of degree −λ in \(\mathbf{R}_{+}^{2}\) , with
\(p \in \mathbf{R}\setminus \{0,1\}\),\(\frac{1} {p} + \frac{1} {q} = 1\),\(f(s) = f(s_{1},\ldots,s_{i_{0}}) \geq 0\),\(G(t) = G(t_{1},\ldots,t_{j_{0}}) \geq 0\),
-
(i)
If p > 1, then we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle\begin{array}{rcl} & \int _{\left \{t\in \mathbf{R}^{j_{0}};1<t_{j}<\infty \right \}}\int _{\left \{s\in \mathbf{R}^{i_{0}};1<s_{i}<\infty \right \}}k_{\lambda }(\vert \vert U(s)\vert \vert _{\alpha },\vert \vert V (t)\vert \vert _{\beta })f(s)G(t)dsdt& \\ & < K_{\lambda }(\sigma )\vert \vert f\vert \vert _{p,\hat{\varPhi }}\vert \vert G\vert \vert _{q,\hat{\varPsi }}, & {}\end{array}$$(68)$$\displaystyle\begin{array}{rcl} & \left \{\int _{\left \{t\in \mathbf{R}^{j_{0}};1<t_{j}<\infty \right \}}\vert \vert V (t)\vert \vert _{\beta }^{p\sigma -j_{0}}\varPi _{j=1}^{j_{0}}t_{j}^{-1}\left (\int _{\left \{s\in \mathbf{R}^{i_{ 0}};1<s_{i}<\infty \right \}}k_{\lambda }(\vert \vert U(s)\vert \vert _{\alpha },\vert \vert V (t)\vert \vert _{\beta })\right.\right.& \\ & \times \left.\left.f(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < K_{\lambda }(\sigma )\vert \vert f\vert \vert _{p,\hat{\varPhi }}; & {}\end{array}$$(69) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\),\(k_{\lambda }(\tilde{\sigma }) \in \mathbf{R}\) , then we still have the equivalent reverses of (68) and (69) with the same best constant factor K λ (σ).
In particular, for i 0 = j 0 = α = β = 1,
-
(i)
if p > 1, then we have the following equivalent inequalities with the best possible constant factor k λ (σ):
$$\displaystyle\begin{array}{rcl} \int _{1}^{\infty }\int _{ 1}^{\infty }k_{\lambda }(\ln s,\ln t)f(s)G(t)dsdt < k_{\lambda }(\sigma )\vert \vert f\vert \vert _{ p,\hat{\phi }}\vert \vert G\vert \vert _{q,\hat{\psi }},& & {}\end{array}$$(70)$$\displaystyle\begin{array}{rcl} \left \{\int _{1}^{\infty }(\ln t)^{p\sigma -1}\frac{1} {t}\left (\int _{1}^{\infty }k_{\lambda }(\ln s,\ln t)f(s)ds\right )^{p}dt\right \}^{\frac{1} {p} } < k_{\lambda }(\sigma )\vert \vert f\vert \vert _{p,\hat{\phi }};& & {}\end{array}$$(71) -
(ii)
if 0 < p < 1, or p < 0, there exists a constant δ 0 > 0, such that for any \(\tilde{\sigma }\in (\sigma -\delta _{0},\sigma +\delta _{0})\), \(k_{\lambda }(\tilde{\sigma }) \in \mathbf{R}\), then we still have the equivalent reverses of (70) and (71) with the same best constant factor k λ (σ).
7 Some Particular Examples on the Norm
Example 1.
For \(h(v) = \frac{\vert \ln v\vert ^{\gamma }} {(1+v)^{\lambda }}(\gamma \geq 0,\mu,\sigma > 0,\mu +\sigma =\lambda )\), we have
Since \(\frac{\vert \ln v\vert ^{\gamma }} {(1+v)^{\lambda /2}} v^{ \frac{\sigma }{ 2} } \rightarrow 0(v \rightarrow 0^{+}\) or v → ∞), there exists a constant number L > 0, such that
Then it follows that
and \(k_{\gamma }(\sigma ) \in \mathbf{R}_{+}\). We find
For γ ≥ 0, we obtain
Setting t = −lnv, we find
In view of Theorem 1 and (39), we have
Example 2.
For \(h(v) = \frac{\vert \ln v\vert ^{\gamma }} {1+v^{\lambda }}(\gamma \geq 0,\mu,\sigma > 0,\mu +\sigma =\lambda )\), we have
Since \(\frac{\vert \ln v\vert ^{\gamma }} {(1+v^{\lambda })^{1/2}} v^{ \frac{\sigma }{ 2} } \rightarrow 0(v \rightarrow 0^{+}\) or v → ∞), there exists a constant number L > 0, such that
Then it follows that
and \(l_{\gamma }(\sigma ) \in \mathbf{R}_{+}\). We find
For γ ≥ 0, we obtain
Setting t = −lnv, we find
In view of Theorem 1 and (39), we have
Example 3.
For \(h(v) = \frac{\vert \ln v\vert ^{\gamma }} {(\max \{1,v\})^{\lambda }}(\gamma \geq 0,\mu,\sigma > 0,\mu +\sigma =\lambda )\), we have
Setting t = −lnv, we find
In view of Theorem 1 and (39), we have
Example 4.
For \(h(v) = \frac{\vert \ln v\vert ^{\gamma }} {\vert 1-v\vert ^{\lambda }}(\gamma \geq 0,\mu,\sigma > 0,\mu +\sigma =\lambda < 1)\), we have
We find
For γ ≥ 0, we obtain
Setting \(0 <\delta <\min \{\mu,\sigma \}\), since \((-\ln v)^{\gamma }v^{\delta } \rightarrow 0(v \rightarrow 0^{+})\), there exists a constant L > 0, such that \(0 < (-\ln v)^{\gamma }v^{\delta } \leq L(v \in (0,1])\), and then it follows
Hence \(\tilde{k}_{\gamma }(\sigma ) \in \mathbf{R}_{+}\), and
Setting t = −lnv, we find
In view of Theorem 1 and (39), we have
Example 5.
For \(h(v) = \frac{\vert \ln v\vert ^{\gamma }} {\vert v^{\lambda }-1\vert }(\gamma > 0,\mu,\sigma > 0,\mu +\sigma =\lambda )\), we have
We find
For γ > 0, we obtain
Setting t = −lnv, we find
In view of Theorem 1 and (39), we have
Lemma 6.
If C is the set of complex numbers and \(\mathbf{C}_{\infty } = \mathbf{C} \cup \{\infty \}\),\(z_{k} \in \mathbf{C}\setminus \{z\vert Rez \geq 0\) , Imz = 0}(k = 1,2,…,n) are different points, the function f(z) is analytic in C ∞ except for z i (i = 1,2,…,n), and z = ∞ is a zero point of f(z) whose order is not less than 1, then for α ∈ R , we have
where \(0 < Im\ln z =\arg z < 2\pi\) . In particular, if z k (k = 1,…,n) are all poles of order 1, setting \(\varphi _{k}(z) = (z - z_{k})f(z)(\varphi _{k}(z_{k})\neq 0)\) , then
Proof.
By Pan et al. [22, p. 118], we have (86). We find
In particular, since \(f(z)z^{\alpha -1} = \frac{1} {z-z_{k}}(\varphi _{k}(z)z^{\alpha -1})\), it is obvious that
Then by (86), we obtain (87). The lemma is proved.
Example 6.
For s ∈ N, 0 < a 1 < ⋯ < a s , we set
By (87), setting u = v λ∕s, we find
In view of Theorem 1 and (39), we have
Example 7.
For c > 0, 0 < γ < π, We set
Putting \(z_{1} = -\frac{\sqrt{c}} {2} e^{i\gamma }\), \(z_{2} = -\frac{\sqrt{c}} {2} e^{-i\gamma }\), by (87), it follows
In view of Theorem 1 and (39), we have
Example 8.
We set
Then we find
In view of Theorem 1 and (39), we have
Example 9.
We set
We find
For a > 0, by (87), we have
By using the simple way, we still can obtain (94) for a = 0.
In view of Theorem 1 and (39), we have
Example 10.
We set
Setting u = ρ v γ, we find
In view of Theorem 1 and (39), we have
Example 11.
We set
We find
In view of Theorem 1 and (39), we have
Example 12.
We set
where \(\csc h(u) = \frac{2} {e^{u}-e^{-u}}\) is hyperbolic cosecant function [23]. We find
Setting u = (2k + 1)ρ v γ, we obtain
where, \(\zeta \left (\frac{\sigma }{\gamma }\right ) =\sum _{ k=1}^{\infty } \frac{1} {k^{\sigma /\gamma }}\left (\frac{\sigma }{\gamma } > 1\right )\) (ζ(⋅ ) is the Riemann’s zeta function [24]).
In view of Theorem 1 and (39), we have
Example 13.
We set
where \(\sec h(u) = \frac{2} {e^{u}+e^{-u}}\) is hyperbolic secant function. We find
Setting u = (2k + 1)ρ v γ, we obtain
where
In view of Theorem 1 and (39), we have
Example 14.
We set
where \(\coth h(u) = \frac{e^{u}+e^{-u}} {e^{u}-e^{-u}}\) is hyperbolic cotangent function. We find
Setting u = 2k ρ v γ, we obtain
In view of Theorem 1 and (39), we have
Example 15.
We set
where \(\tan h(u) = \frac{e^{u}+e^{-u}} {e^{u}-e^{-u}}\) is hyperbolic tangent function. We find
Setting u = 2k ρ v γ, we obtain
where, \(\xi (\frac{\sigma }{\gamma }):=\sum _{ k=1}^{\infty }\frac{(-1)^{k-1}} {k^{\sigma /\gamma }}\).
In view of Theorem 1 and (39), we have
Note.
The following references [24–31] provide an extensive theory and applications of Analytic Number Theory relating to Riemann’s zeta function that will provide a source study for further research on Hilbert-type inequalities.
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Acknowledgements
This work is supported by The National Natural Science Foundation of China (No. 61370186) and 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).
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Yang, B. (2014). Multidimensional Hilbert-Type Integral Inequalities and Their Operators Expressions. In: Rassias, T., Tóth, L. (eds) Topics in Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-06554-0_34
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