1 Introduction

For \({n}\geq1\), \({R}_{+}^{{n}} = \{ {x}= ( {x}_{1},\ldots, {x}_{{n}} ): {x}_{{i}} > 0,i=1,\ldots,n \}\), \({a}_{{i}}, {b}_{{i}} > 0\ ({i}=1,\ldots,n)\), \(\omega(x) >0\ (x \in {R}_{+}^{{n}} )\), and \(\rho>0\), we set

$$\begin{aligned} &{u} ( {x} ) = \Biggl( \sum_{{i}=1}^{{n}} {a}_{{i}} {x}_{{i}}^{\rho} \Biggr)^{\frac{1}{\rho}},\qquad {v} ( {y} ) = \Biggl( \sum_{{i}=1}^{{n}} {b}_{{i}} {y}_{{i}}^{\rho} \Biggr)^{\frac{1}{\rho}}, \\ &{L}_{\omega}^{{p}} \bigl( {R}_{+}^{{n}} \bigr):= \biggl\{ {f} ( {x} ) \geq0: \Vert {f} \Vert _{{p},\omega} = \biggl( \int_{{R}_{+}^{{n}}} \omega ( {x} ) {f}^{{p}} ({x})\,{dx} \biggr)^{1/{p}} < + \infty \biggr\} . \end{aligned}$$

If \({p} >1\), \(\frac{1}{{p}} + \frac{1}{{q}} =1\), \({K}({u},{v})\geq0\ ({u},{v} >0)\), then the Hilbert-type multiple integral inequality is of the form

$$\begin{aligned} \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) {g} ( {y} ) {\,dx\,dy} \leq{M} \Vert {f} \Vert _{{p}, {u}^{\alpha}} \Vert {g} \Vert _{{q}, {v}^{\beta}}. \end{aligned}$$
(1)

Define a singular integral operator T:

$$\begin{aligned} {T} ( {f} ) ( {y} ):= \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) \,{dx},\quad{y}\in {R}_{+}^{{n}}, \end{aligned}$$
(2)

then (1) may be rewritten as follows:

$$\int_{{R}_{+}^{{n}}} {T} ( {f} ) ({y}){g} ( {y} ) {\,dy} \leq {M} \Vert {f} \Vert _{{p}, {u}^{\alpha}} \Vert {g} \Vert _{{q}, {v}^{\beta}}. $$

It is easy to prove that (1) is equivalent to the following inequality:

$$\begin{aligned} \bigl\Vert {T} ( {f} ) \bigr\Vert _{{p}, {v}^{\gamma}} \leq{M} \Vert {f} \Vert _{{p}, {u}^{\alpha}} \Vert {g} \Vert _{{q}, {v}^{\beta}}, \end{aligned}$$
(3)

where \(\gamma=\beta(1-p)\). When the operator T satisfies (3), T is called bounded operator from \({L}_{{u}^{\alpha}}^{{p}} ( {R}_{+}^{{n}} )\) to \({L}_{{v}^{\gamma}}^{{p}} ( {R}_{+}^{{n}} )\).

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

$${K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) ={G} \bigl( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) \bigr), $$

and calculate the best possible constant factor. Furthermore, its application in the operator theory is considered.

2 Some lemmas

Lemma 1

Suppose that \({p} > 1\), \(\frac{1}{{p}} + \frac{1}{{q}} =1\), \({n}\geq1\), \(\rho>0\), \(\lambda_{1} \lambda_{2} > 0\), \({a}_{{i}}, {b}_{{i}} > 0\) \(( {i}=1,\ldots,n )\), \({u} ( {x} ) = ( \sum_{{i}=1}^{{n}} {a}_{{i}} {x}_{{i}}^{\rho} )^{\frac{1}{\rho}}\), \({v} ( {y} ) = ( \sum_{{i}=1}^{{n}} {b}_{{i}} {y}_{{i}}^{\rho} )^{\frac{1}{\rho}}\).

If \({K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )\) is a non-negative measurable function, setting

$$\begin{aligned} &{W}_{1}:= \int_{{R}_{+}^{{n}}} \bigl( {v} ( {t} ) \bigr)^{- \frac{\beta+n}{{q}}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt},\\ & {W}_{2}:= \int_{{R}_{+}^{{n}}} \bigl( {u} ( {t} ) \bigr)^{- \frac{\alpha+n}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt}, \end{aligned}$$

we have the following:

$$\begin{aligned} &\omega_{1} ( {x} ):= \int_{{R}_{+}^{{n}}} \bigl( {v} ( {y} ) \bigr)^{- \frac{\beta+n}{{q}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {\,dy}= \bigl( {u} ( {x} ) \bigr)^{\frac{\lambda_{1}}{\lambda_{2}} ( \frac{\beta+n}{{q}} -{n})} {W}_{1}, \\ &\omega_{2} ( {x} ):= \int_{{R}_{+}^{{n}}} \bigl( {u} ( {x} ) \bigr)^{- \frac{\alpha+n}{{p}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {\,dx}= \bigl( {v} ( {y} ) \bigr)^{\frac{\lambda_{2}}{\lambda_{1}} ( \frac{\alpha+n}{{p}} -{n})} {W}_{2}. \end{aligned}$$

Proof

Since \(v(ay)=av(y)\ (a>0)\), in view of \({K} ( {tu},{v} ) ={K}({u}, {t}^{\frac{\lambda_{1}}{\lambda_{2}}} {v})\), setting \({t}= {u}^{\frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) {y}\), we find \({dy}= {u}^{\frac{-{n}\lambda_{1}}{\lambda_{2}}} ( {x} ) {\,dt}\) and

$$\begin{aligned} \omega_{1} ( {x} )& = \int_{{R}_{+}^{{n}}} \bigl( {v} ( {y} ) \bigr)^{- \frac{\beta+n}{{q}}} {K} \bigl( 1, {u}^{\frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) {v} ( {y} ) \bigr) {\,dy} \\ &= \int_{{R}_{+}^{{n}}} \bigl( {u}^{- \frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) {v} ( {t} ) \bigr)^{- \frac{\beta+n}{{q}}} {K} \bigl( 1,{v} ( {t} ) \bigr) {u}^{\frac{-{n}\lambda_{1}}{\lambda_{2}}} ( {x} ) {\,dt} \\ &= \bigl( {u} ( {x} ) \bigr)^{\frac{\lambda_{1}}{\lambda_{2}} ( \frac{\beta+n}{{q}} -{n})} {W}_{1}. \end{aligned}$$

In the same way, we have

$$\omega_{2} ( {x} ) = \bigl( {v} ( {y} ) \bigr)^{\frac{\lambda_{2}}{\lambda_{1}} ( \frac{\alpha+n}{{p}} -{n})} {W}_{2}. $$

The lemma is proved. □

Lemma 2

(cf. [15])

If \({p}_{{i}} > 0\), \({a}_{{i}} > 0\), \(\alpha_{{i}} > 0\ ({i}=1,\ldots,n)\) and \(\psi(t)\) is a measurable function, then we have the following:

$$\begin{aligned} & \int\cdots \int_{ \{ {x}_{{i}} > 0; \sum_{{i}=1}^{{n}} ( \frac{{x}_{{i}}}{{a}_{{i}}} )^{\alpha_{{i}}} \leq1 \}} \psi \Biggl( \sum_{{i}=1}^{{n}} \biggl( \frac{{x}_{{i}}}{{a}_{{i}}} \biggr)^{\alpha_{{i}}} \Biggr) {x}_{1}^{{p}_{1} -1} \cdots{x}_{{n}}^{{p}_{{n}} -1} {\,dx}_{1} \cdots{d} {x}_{{n}} \\ & \quad = \frac{{a}_{1}^{{p}_{1}} \cdots {a}_{{n}}^{{p}_{{n}}} \Gamma ( \frac{{p}_{1}}{\alpha_{1}} ) \cdots\Gamma ( \frac{{p}_{{n}}}{\alpha_{{n}}} )}{\alpha_{1} \cdots\alpha_{{n}} \Gamma ( \sum_{{i}=1}^{{n}} \frac{{p}_{{i}}}{\alpha_{{i}}} )} \int_{0}^{1} \psi ( {t} ) {t}^{\sum_{{i}=1}^{{n}} \frac{{p}_{{i}}}{\alpha_{{i}}} -1} {\,dt}, \end{aligned}$$

where \(\Gamma(t)\) is the gamma function. In particular, for \(\alpha_{{i}}=\rho\), \({p}_{{i}}=1\), \({b}_{{i}} = \frac{1}{{a}_{{i}}^{\rho}}\) \((i=1,\ldots,n)\), we have

$$\begin{aligned} & \int\cdots \int_{ \{ {x}_{{i}} > 0; \sum_{{i}=1}^{{n}} {b}_{{i}} {x}_{{i}}^{\rho} \leq1 \}} \psi \Biggl( \sum_{{i}=1}^{{n}} {b}_{{i}} {x}_{{i}}^{\rho} \Biggr) {\,dx}_{1} \cdots{d} {x}_{{n}} \\ & \quad = \frac{\prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \Gamma^{{n}} ( \frac{1}{\rho} )}{ \rho^{{n}} \Gamma ( {\frac{{n}}{\rho}} )} \int_{0}^{1} \psi ( {t} ) {t}^{\frac{{n}}{\rho} -1} {\,dt}. \end{aligned}$$

3 Main results

We set

$$\begin{aligned} &\Omega ( {a} < b ) =\bigl\{ {x}= ( {x}_{1},\ldots, {x}_{{n}} );{a} < u ( {x} ) < b\bigr\} , \\ &\Omega' ( {a} < {b} ) = \bigl\{ {x}= ( {x}_{1}, \ldots, {x}_{{n}} ); {a} < {v} ({y})< {b} \bigr\} . \end{aligned}$$

Theorem 1

Suppose that \({n}\geq1\), \(p > 1\), \(\frac{1}{ {p}} + \frac{1}{{q}} =1\), \(\rho>0\), \(\alpha,\beta\in R\), \(\lambda_{1} \lambda_{2} > 0\), \({a}_{{i}} > 0\), \({b}_{{i}} > 0\) (\({i}=1,\ldots,n\)), \({u} ( {x} ) = ( \sum_{ {i}=1}^{\infty} {a}_{{i}} {x}_{{i}}^{\rho} )^{1/\rho }\), \({v} ( {y} ) = ( \sum_{ {i}=1}^{\infty} {b}_{{i}} {y}_{{i}}^{\rho} )^{1/\rho }\), \({K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) ) \) is a non-negative measurable function,

$$\begin{aligned} &0 < {W}_{1} = \int_{{R}_{+}^{{n}}} \bigl( {v} ( {t} ) \bigr)^{- \frac{\beta+n}{{q}}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} < \infty, \\ &0 < {W}_{2} = \int_{{R}_{+}^{{n}}} \bigl( {u} ( {t} ) \bigr)^{- \frac{\alpha+n}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} < \infty, \end{aligned}$$

and for \(a=0\), \(b=1\) (or \(a=1\), \(b=+\infty\)),

$$\begin{aligned} &\int_{\Omega ( {a} < b )} \bigl( {v} ( {t} ) \bigr)^{- \frac{\beta+n}{{q}}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} >0,\\ &\int_{\Omega' ( {a} < b )} \bigl( {u} ( {t} ) \bigr)^{- \frac{\alpha+n}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} >0, \end{aligned}$$

then we have the following: There is a constant M such that, for \(f(x)\in {L}_{{u}^{\alpha} (x)}^{{p}} ( {R}_{+}^{{n}} )\) and \(g(y)\in {L}_{{v}^{\gamma} (y)}^{{p}} ( {R}_{+}^{{n}} )\), the following inequality

$$\begin{aligned} \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) {g} ( {y} ) {\,dx\,dy} \leq{M} \Vert {f} \Vert _{{p}, {u}^{\rho}} \Vert {g} \Vert _{{q}, {v}^{\rho}} \end{aligned}$$
(4)

holds true if and only if the equality \({\frac{{n}\lambda_{1}+\alpha\lambda_{2}}{{p}} = {\frac{{n}\lambda_{2}+\beta\lambda_{1}}{ {q}}}}\) is valid.

Proof

We assume that (4) is valid and set \({c}= {\frac{{n}\lambda_{2}+\beta\lambda_{1}}{{q}}} - \frac{{n}\lambda_{1}+\alpha\lambda_{2}}{ {p}}\).

(i) For \(\lambda_{1}\), \(\lambda_{2}>0\), if \(c>0\), putting \(\varepsilon>0\) small enough and

$$\begin{aligned} &{f}({x})= \textstyle\begin{cases} ({u} ( {x} ) )^{(-\alpha-n-\lambda_{1}\varepsilon)/p}, & {u} ( {x} ) > 1,\\ 0, & 0 < u(x) \leq1, \end{cases}\displaystyle \\ &{g}({y})= \textstyle\begin{cases} ({v} ( {y} ) )^{(-\beta-n+\lambda_{2}\varepsilon)/q}, & 0 < v(y)< 1,\\ 0, & {v}({y})\geq1, \end{cases}\displaystyle \end{aligned}$$

by Lemma 2, we have

$$\begin{aligned} & \Vert {f} \Vert _{{p}, {u}^{\rho}} \Vert {g} \Vert _{{q}, {v}^{\rho}} \\ &\quad = \biggl( \int_{\Omega ( 1 < +\infty )} \bigl( {u}({x}) \bigr)^{-{n}-\lambda_{1}\varepsilon} {\,dx} \biggr)^{1/{p}} \biggl( \int_{\Omega' ( 0 < 1 )} \bigl( {v}({y}) \bigr)^{-{n}+\lambda_{2}\varepsilon} {\,dy} \biggr)^{1/{q}} \\ &\quad = \biggl( \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{ {a}_{1}^{1/\rho} \cdots{a}_{{n}}^{1/\rho} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} )} \frac{1}{\lambda_{1} \varepsilon} \biggr)^{1/{p}} \\ &\qquad{}\times\biggl( \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{ {b}_{1}^{1/\rho} \cdots{b}_{{n}}^{1/\rho} \rho^{{n}-1} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} )} \frac{1}{\lambda_{2} \varepsilon} \biggr)^{1/{q}} \\ &\quad = \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{ \lambda_{1}^{1/{p}} \lambda_{2}^{1/{q}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{q}}, \end{aligned}$$
(5)
$$\begin{aligned} & \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) {g} ( {y} ) {\,dx\,dy} \\ &\quad = \int_{\Omega ( 1 < +\infty )} \bigl({u} ( {x} ) \bigr)^{(-\alpha-n-\lambda_{1}\varepsilon)/p} \biggl( \int_{\Omega' ( 0 < 1 )} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) \bigl({v} ( {y} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dy} \biggr) {\,dx} \\ &\quad = \int_{\Omega ( 1 < +\infty )} \bigl({u} ( {x} ) \bigr)^{(-\alpha-n-\lambda_{1}\varepsilon)/p} \biggl( \int_{\Omega' ( 0 < 1 )} {K}\bigl(1,{v}\bigl( {u}^{\frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) {y} \bigr)\bigr) \bigl({v} ( {y} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dy} \biggr) {\,dx} \\ & \quad= \int_{\Omega ( 1 < +\infty )} \bigl({u} ( {x} ) \bigr)^{(-\alpha-n-\lambda_{1}\varepsilon)/p} \biggl( \int_{\Omega' (0 < {u}^{\frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) )} {K} \bigl( 1,{v} ( {t} ) \bigr) \\ &\qquad {}\times \bigl( {u}^{- \frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {u}^{- \frac{{n} \lambda_{1}}{\lambda_{2}}} ( {x} ) {\,dt}\biggr){\,dx} \\ &\quad = \int_{\Omega ( 1 < +\infty )} \bigl({u} ( {x} ) \bigr)^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} \biggl( \int_{\Omega' (0 < {u}^{\frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) )} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl( {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt}\biggr){\,dx} \\ &\quad \geq \int_{\Omega ( 1 < +\infty )} \bigl({u} ( {x} ) \bigr)^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} {\,dx} \int_{\Omega' (0 < 1 )} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl( {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt}. \end{aligned}$$
(6)

Hence, by (4), (5) and (6), we have the following:

$$\begin{aligned} & \int_{\Omega ( 1 < +\infty )} \bigl({u} ( {x} ) \bigr)^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} {\,dx} \int_{\Omega' (0 < 1 )} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl( {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt} \\ & \quad \leq{M} \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{ \lambda_{1}^{1/{p}} \lambda_{2}^{1/{q}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{q}}. \end{aligned}$$
(7)

For \(\lambda_{2} > 0\), \(c>0\), \(\varepsilon>0\) small enough, \(-{n}+ {\frac{{c}}{\lambda_{2}}} - \lambda_{1} \varepsilon>-n\), it follows that \(\int_{\Omega ( 1 <+\infty )} ({u} ( {x} ) )^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} {\,dx}=+\infty\), which contradicts inequality (7) in view of \(\int_{\Omega' (0 <1 )} {K} ( 1,{v} ( {t} ) ) ( {v} ( {t} ) )^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt} >0\). Hence it is not valid for \({c} >0\).

If \(c<0\), putting \(\varepsilon>0\) small enough and

$$\begin{aligned} &{f}({x})= \textstyle\begin{cases} ({u} ( {x} ) )^{(-\alpha-n+\lambda_{1}\varepsilon)/p}, & 0 < {u} ( {x} ) < 1,\\ 0, & {u}({x})\geq1, \end{cases}\displaystyle \\ &{g}({y})= \textstyle\begin{cases} ({v} ( {y} ) )^{(-\beta-n+\lambda_{2}\varepsilon)/q}, & {v} ( {y} ) > 1,\\ 0, & 0 < {v}({y})\leq1, \end{cases}\displaystyle \end{aligned}$$

in the same way, we have the following:

$$\begin{aligned} & \int_{\Omega' ( 1 < +\infty )} \bigl({v}({y})\bigr)^{{-{n}- {\frac{{c}}{\lambda _{1}}} - \lambda_{2} \varepsilon}} {\,dy} \int_{\Omega(0 < 1 )} {K} \bigl( {u} ( {t} ),1 \bigr) \bigl( {u} ( {t} ) \bigr)^{(-\alpha-n+\lambda_{1}\varepsilon)/p} {\,dt} \\ &\quad \leq{M} \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{ \lambda_{1}^{\frac{1}{{p}}} \lambda_{2}^{\frac{1}{{q}}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{- \frac{1}{\rho}} \Biggr)^{\frac{1}{{p}}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \Biggr)^{\frac{1}{{q}}}. \end{aligned}$$
(8)

For \(\lambda_{2} > 0\), \(c<0\), \(\varepsilon>0\) small enough, hence \(-{n}- {\frac{{c}}{\lambda_{1}}} - \lambda_{2} \varepsilon>-n\), it follows that \(\int_{\Omega' ( 1 <+\infty )} ({v}({y}))^{{-{n}- {\frac{{c}}{\lambda _{1}}} - \lambda_{2} \varepsilon}} {\,dy}=+\infty\), which contradicts inequality (8) in view of \(\int_{\Omega(0 <1 )} {K} ( {u} ( {t} ),1 ) ( {u} ( {t} ) )^{(-\alpha-n+\lambda_{1}\varepsilon)/p} {\,dt} >0\). Hence, it is not valid for \({c} <0\).

Therefore, we prove that \(c=0\), namely \({\frac{{n}\lambda_{1}+\alpha\lambda_{2}}{{p}} = {\frac{{n}\lambda_{2}+\beta\lambda_{1}}{ {q}}}}\) is valid.

(ii) For \(\lambda_{1}\), \(\lambda_{2}<0\), we prove that \(\frac{{n}\lambda_{1}+\alpha\lambda_{2}}{{p}} = {\frac{{n}\lambda_{2}+\beta\lambda_{1}}{ {q}}}\) is valid as follows.

If \(c>0\), putting \(\varepsilon>0\) small enough and

$$\begin{aligned} &{f}({x})= \textstyle\begin{cases} ({u} ( {x} ) )^{(-\alpha-n-\lambda_{1}\varepsilon)/p}, & 0 < {u} ( {x} ) < 1,\\ 0, & {u}({x})\geq1, \end{cases}\displaystyle \\ &{g}({y})= \textstyle\begin{cases} ({v} ( {y} ) )^{(-\beta-n+\lambda_{2}\varepsilon)/q}, & {v} ( {y} ) > 1,\\ 0, & 0 < {v}({y})\leq1, \end{cases}\displaystyle \end{aligned}$$

we have

$$\begin{aligned} & \Vert {f} \Vert _{{p}, {u}^{\rho}} \Vert {g} \Vert _{{q}, {v}^{\rho}} \\ & \quad= \biggl( \int_{\Omega ( 0 < 1 )} \bigl( {u}({x}) \bigr)^{-{n}-\lambda_{1}\varepsilon} {\,dx} \biggr)^{1/{p}} \biggl( \int_{\Omega' ( 1 < +\infty )} \bigl( {v}({y}) \bigr)^{-{n}+\lambda_{2}\varepsilon} {\,dy} \biggr)^{1/{q}} \\ &\quad = \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{(- \lambda_{1} )^{1/{p}} (- \lambda_{2} )^{1/{q}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{q}}, \end{aligned}$$
(9)
$$\begin{aligned} & \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) {g} ( {y} ) {\,dx\,dy} \\ & \quad= \int_{\Omega ( 0 < 1 )} \bigl({u} ( {x} ) \bigr)^{(-\alpha-n-\lambda_{1}\varepsilon)/p} \biggl( \int_{\Omega' ( 1 < +\infty )} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) \bigl({v} ( {y} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dy} \biggr) {\,dx} \\ &\quad = \int_{\Omega ( 0 < 1 )} \bigl({u} ( {x} ) \bigr)^{(-\alpha-n-\lambda_{1}\varepsilon)/p} \biggl( \int_{\Omega' ( {u}^{\frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) < +\infty)} {K} \bigl( 1,{v} ( {t} ) \bigr) \\ & \qquad{}\times \bigl( {u}^{- \frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {u}^{- \frac{{n} \lambda_{1}}{\lambda_{2}}} ( {x} ) {\,dt}\biggr){\,dx} \\ & \quad= \int_{\Omega ( 0 < 1 )} \bigl({u} ( {x} ) \bigr)^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} \biggl( \int_{\Omega' ( {u}^{\frac{\lambda_{1}}{\lambda_{2}}} ( {x} ) < +\infty)} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl( {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt}\biggr){\,dx} \\ & \quad\geq \int_{\Omega ( 0 < 1 )} \bigl({u} ( {x} ) \bigr)^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} {\,dx} \int_{\Omega' ( 1< +\infty)} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl( {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt}. \end{aligned}$$
(10)

Hence, by (4), (9) and (10), we have the following:

$$\begin{aligned} & \int_{\Omega ( 0 < 1 )} \bigl({u} ( {x} ) \bigr)^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} {\,dx} \int_{\Omega' ( 1< +\infty)} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl( {v} ( {t} ) \bigr)^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt} \\ &\quad \leq{M} \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{ (- \lambda_{1} )^{1/{p}} (- \lambda_{2} )^{1/{q}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{q}}. \end{aligned}$$
(11)

It is obvious that \(\int_{\Omega ( 0 < 1 )} ({u} ( {x} ) )^{{-{n}+ {\frac{{c}}{\lambda_{2}}} -\lambda_{1}\varepsilon}} {\,dx}=+\infty\), which contradicts inequality (11) in view of \(\int_{\Omega' ( 1<+\infty)} {K} ( 1,{v} ( {t} ) ) ( {v} ( {t} ) )^{(-\beta-n+\lambda_{2}\varepsilon)/q} {\,dt} >0\). Hence it is not valid for \(c>0\).

If \(c<0\), putting \(\varepsilon>0\) small enough and

$$\begin{aligned} &{f}({x})= \textstyle\begin{cases} ({u} ( {x} ) )^{(-\alpha-n+\lambda_{1}\varepsilon)/p}, & {u} ( {x} ) > 1,\\ 0, & 0 < u(x) \leq1, \end{cases}\displaystyle \\ &{g}({y})= \textstyle\begin{cases} ({v} ( {y} ) )^{(-\beta-n-\lambda_{2}\varepsilon)/q}, & 0 < v(y)< 1,\\ 0, & {v}({y})\geq1, \end{cases}\displaystyle \end{aligned}$$

in the same way, we have

$$\begin{aligned} & \int_{\Omega' (0 < 1)} \bigl({v}({y})\bigr)^{{-{n}- {\frac{{c}}{\lambda_{1}}} - \lambda_{2} \varepsilon}} {\,dy} \int_{\Omega( 1< +\infty)} {K} \bigl( {u} ( {t} ),1 \bigr) \bigl( {u} ( {t} ) \bigr)^{(-\alpha-n+\lambda_{1}\varepsilon)/p} {\,dt} \\ & \leq{M} \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{ (- \lambda_{1} )^{1/{p}} (- \lambda_{2} )^{1/{q}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{q}}. \end{aligned}$$
(12)

In virtue of \(\int_{\Omega' (0 <1)} ({v}({y}))^{{-{n}- {\frac{{c}}{\lambda_{1}}} - \lambda_{2} \varepsilon}} {\,dy}=+ \infty\), (12) is a contradiction in view of \(\int_{\Omega( 1<+\infty)} {K} ( {u} ( {t} ),1 ) ( {u} ( {t} ) )^{(-\alpha-n+\lambda_{1}\varepsilon)/p} {\,dt} >0\). Hence, \(c<0\) is not valid.

Therefore, we prove that \(c=0\) is valid.

On the other hand, we assume that \(\frac{{n}\lambda_{1}+\alpha\lambda_{2}}{{p}} = {\frac{{n}\lambda_{2}+\beta\lambda_{1}}{ {q}}}\) is valid.

Setting \({a}= \frac{\alpha}{{pq}} + {\frac{{n}}{ {pq}}}\), \({b}= \frac{\beta}{{pq}} + {\frac{{n}}{ {pq}}}\), by Holder’s inequality with weight and Lemma 1, we find

$$\begin{aligned} & \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) {g} ( {y} ) {\,dx\,dy} \\ & \quad= \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} \biggl( {f}({x}) \frac{{u}^{{a}} ( {x} )}{{v}^{{b}} (y)} \biggr) \biggl({g} ( {y} ) \frac{{v}^{{b}} (y)}{ {u}^{{a}} ( {x} )} \biggr){K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {\,dx\,dy} \\ &\quad\leq \biggl( \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {f}^{{p}} ({x}) \frac{{u}^{{ap}} ( {x} )}{ {v}^{{bp}} (y)} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {\,dx\,dy} \biggr)^{1/{p}} \\ &\qquad {}\times \biggl( \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {g}^{{q}} ({y}) \frac{{v}^{{bq}} (y)}{{u}^{{aq}} ( {x} )} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {\,dx\,dy} \biggr)^{1/{q}} \\ & \quad= \biggl( \int_{{R}_{+}^{{n}}} \bigl( {u} ( {x} ) \bigr)^{{\frac{\alpha+n}{{q}}}} {f}^{{p}} ( {x} ) \omega_{1} ({x}){\,dx} \biggr)^{1/{p}} \biggl( \int_{{R}_{+}^{{n}}} \bigl( {v} ( {y} ) \bigr)^{{\frac{\beta+n}{{p}}}} {g}^{{q}} ( {y} ) \omega_{2} ({y}){\,dy} \biggr)^{1/{q}} \\ &\quad = {W}_{1}^{1/{p}} {W}_{2}^{1/{q}} \biggl( \int_{{R}_{+}^{{n}}} \bigl( {u} ( {x} ) \bigr)^{{\frac{\alpha+n}{{q}} + \frac{\lambda_{1}}{\lambda_{2}} ( \frac{\beta+n}{{q}} -{n})}} {f}^{{p}} ( {x} ) {\,dx} \biggr)^{1/{p}} \\ & \qquad{}\times \biggl( \int_{{R}_{+}^{{n}}} \bigl( {v} ( {y} ) \bigr)^{{\frac{\beta+n}{{p}} + \frac{\lambda_{2}}{\lambda_{1}} ( \frac{\alpha+n}{{p}} -{n})}} {g}^{{q}} ( {y} ) {\,dy} \biggr)^{1/{q}} \\ & \quad= {W}_{1}^{1/{p}} {W}_{2}^{1/{q}} \biggl( \int_{{R}_{+}^{{n}}} \bigl( {u} ( {x} ) \bigr)^{{\alpha}} {f}^{{p}} ( {x} ) {\,dx} \biggr)^{1/{p}} \biggl( \int_{{R}_{+}^{{n}}} \bigl( {v} ( {y} ) \bigr)^{{\beta}} {g}^{{q}} ( {y} ) {\,dy} \biggr)^{1/{q}} \\ &\quad= {W}_{1}^{1/{p}} {W}_{2}^{1/{q}} \Vert {f} \Vert _{{p}, {u}^{\alpha}} \Vert {g} \Vert _{{q}, {v}^{\beta}}. \end{aligned}$$

Taking \(M\geq{W}_{1}^{1/{p}} {W}_{2}^{1/{q}}\), we prove that (4) is valid. □

Theorem 2

With regards to the assumption of Theorem  1, the best possible constant factor of (4) is \(\operatorname{inf} M= {W}_{1}^{1/{p}} {W}_{2}^{1/{q}}\) when (4) holds true.

Proof

We assume that (4) is valid. If there exists a positive number \(M_{0}< {W}_{1}^{1/{p}} {W}_{2}^{1/{q}}\) such that (4) is still valid when replacing M by \(M_{0}\), then, \(\forall f(x)\in{L}_{{u}^{\alpha} (x)}^{{p}} ( {R}_{+}^{{n}} )\) and \(g(y)\in {L}_{{v}^{\beta} (y)}^{{p}} ( {R}_{+}^{{n}} )\), we have

$$\begin{aligned} \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) {g} ( {y} ) {\,dx\,dy} \leq{M}_{0} \Vert {f} \Vert _{{p}, {u}^{\alpha}} \Vert {g} \Vert _{{q}, {v}^{\beta}}. \end{aligned}$$
(13)

Taking \(\varepsilon>0\) and \(\delta>0\) small enough and setting

$$\begin{aligned} &{f}({x})= \textstyle\begin{cases} ({u} ( {x} ) )^{(-\alpha-n- \vert \lambda_{1} \vert \varepsilon)/p}, & {u} ( {x} ) > \delta,\\ 0, & 0 < u(x) \leq\delta, \end{cases}\displaystyle \\ &{g}({y})= \textstyle\begin{cases} ({v} ( {y} ) )^{(-\beta-n+ \vert \lambda_{2} \vert \varepsilon)/q}, & 0 < v(y)< 1,\\ 0, & {v}({y})\geq1, \end{cases}\displaystyle \end{aligned}$$

we have

$$\begin{aligned} & \Vert {f} \Vert _{{p}, {u}^{\alpha}} \Vert {g} \Vert _{{q}, {v}^{\beta}} \\ &\quad = \biggl( \int_{\Omega ( \delta< +\infty )} \bigl( {u}({x}) \bigr)^{-{n}- \vert \lambda_{1} \vert \varepsilon} {\,dx} \biggr)^{1/{p}} \biggl( \int_{\Omega' ( 0 < 1 )} \bigl( {v}({y}) \bigr)^{-{n}+ \vert \lambda_{2} \vert \varepsilon} {\,dy} \biggr)^{1/{q}} \\ & \quad= \frac{\Gamma^{{n}} ( \frac{1}{\rho} ) ( \frac{1}{\delta^{ \vert \lambda_{1} \vert \varepsilon/\rho}} )^{1/{p}}}{ \vert \lambda_{1} \vert ^{1/{p}} \vert \lambda_{2} \vert ^{1/{q}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{q}}. \end{aligned}$$
(14)

And we have the following by using \(\frac{{n}\lambda_{1}+\alpha\lambda_{2}}{{p}} = {\frac{{n}\lambda_{2}+\beta\lambda_{1}}{ {q}}}\):

$$\begin{aligned} & \int_{{R}_{+}^{{n}}} \int_{{R}_{+}^{{n}}} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {f} ( {x} ) {g} ( {y} ) {\,dx\,dy} \\ & \quad= \int_{\Omega' ( 0 < 1 )} \bigl({v} ( {y} ) \bigr)^{(-\beta-n+ \vert \lambda_{2} \vert \varepsilon)/q} \biggl( \int_{\Omega ( \delta< +\infty )} \bigl({u} ( {x} ) \bigr)^{(-\alpha-n- \vert \lambda_{1} \vert \varepsilon)/p} {K} \bigl( {u} ( {x} ),{v} ( {y} ) \bigr) {\,dx} \biggr) {\,dy} \\ &\quad = \int_{\Omega' ( 0 < 1 )} \bigl({v} ( {y} ) \bigr)^{(-\beta-n+ \vert \lambda_{2} \vert \varepsilon)/q} \\ &\qquad {}\times \biggl( \int_{\Omega ( \delta< +\infty )} \bigl( {u} ( {x} ) \bigr)^{\frac{-\alpha-n- \vert \lambda_{1} \vert \varepsilon}{{p}}} {K} \bigl({u}\bigl( {v}^{\frac{\lambda_{2}}{\lambda_{1}}} ({y}){x},1\bigr) \bigr){\,dx} \biggr) {\,dy} \\ &\quad = \int_{\Omega' ( 0 < 1 )} \bigl({v} ( {y} ) \bigr)^{(-\beta-n+ \vert \lambda_{2} \vert \varepsilon)/q} \biggl( \int_{\Omega(\delta{v}^{\frac{\lambda_{2}}{\lambda_{1}}} ({y}) < +\infty)} \bigl( {v}^{- \frac{\lambda_{2}}{\lambda_{1}}} ( {y} ) {u} ( {t} ) \bigr)^{\frac{-\alpha-n- \vert \lambda_{1} \vert \varepsilon}{{p}}} \\ & \qquad{}\times{K}\bigl({u}({t}),1\bigr) {v}^{- \frac{{n} \lambda_{2}}{\lambda_{1}}} ({y}){\,dt}\biggr){\,dy} \\ &\quad = \int_{\Omega' ( 0 < 1 )} \bigl({v} ( {y} ) \bigr)^{-{n}+ \vert \lambda_{2} \vert \varepsilon} \biggl( \int_{\Omega(\delta{v}^{\frac{\lambda_{2}}{\lambda_{1}}} ({y}) < +\infty)} \bigl({u} ( {t} ) \bigr)^{\frac{-\alpha-n- \vert \lambda_{1} \vert \varepsilon}{{p}}} {K} \bigl({u}({t}),1\bigr){\,dt}\biggr){\,dy} \\ & \quad\geq \int_{\Omega' ( 0 < 1 )} \bigl({v} ( {y} ) \bigr)^{-{n}+ \vert \lambda_{2} \vert \varepsilon} {\,dy} \int_{\Omega(\delta< +\infty)} \bigl({u} ( {t} ) \bigr)^{\frac{-\alpha-n- \vert \lambda_{1} \vert \varepsilon}{{p}}} {K} \bigl({u}({t}),1\bigr){\,dt} \\ &\quad = \frac{\Gamma^{{n}} ( \frac{1}{\rho} ) \prod_{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho}}{ \vert \lambda_{2} \vert \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \varepsilon} \int_{\Omega ( \delta< +\infty )} \bigl( {u} ( {t} ) \bigr)^{\frac{-\alpha-n- \vert \lambda_{1} \vert \varepsilon}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt}. \end{aligned}$$
(15)

Combining (13), (14) and (15), we have

$$\begin{aligned} & \int_{\Omega ( \delta< +\infty )} {K} \bigl( {u} ( {t} ),1 \bigr) \bigl( {u} ( {t} ) \bigr)^{\frac{-\alpha-n- \vert \lambda_{1} \vert \varepsilon}{{p}}} {\,dt} \\ & \quad\leq{M}_{0} \Biggl( \frac{1}{ \vert \lambda_{1} \vert } \prod _{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \Biggl( \frac{1}{ \vert \lambda_{2} \vert } \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{p}} \biggl( \frac{1}{\delta^{ \vert \lambda_{1} \vert \varepsilon/\rho}} \biggr)^{1/{p}}. \end{aligned}$$
(16)

If we set

$$\begin{aligned} &{f}({x})= \textstyle\begin{cases} ({u} ( {x} ) )^{(-\alpha-n- \vert \lambda_{1} \vert \varepsilon)/p}, & 0 < {u} ( {x} ) < 1,\\ 0, & u(x)\geq1, \end{cases}\displaystyle \\ &{g}({y})= \textstyle\begin{cases} ({v} ( {y} ) )^{(-\beta-n+ \vert \lambda_{2} \vert \varepsilon)/q}, & v(y) > \delta,\\ 0, & 0 < {v}({y})\leq\delta, \end{cases}\displaystyle \end{aligned}$$

then, in the same way, we have

$$\begin{aligned} & \int_{\Omega' ( \delta< +\infty )} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl({v} ( {y} ) \bigr)^{(-\beta-n+ \vert \lambda_{2} \vert \varepsilon)/q} {\,dt} \\ & \quad\leq{M}_{0} \Biggl( \frac{1}{ \vert \lambda_{1} \vert } \prod _{{i}=1}^{{n}} {a}_{{i}}^{-1/\rho} \Biggr)^{1/{q}} \Biggl( \frac{1}{ \vert \lambda_{2} \vert } \prod _{{i}=1}^{{n}} {b}_{{i}}^{-1/\rho} \Biggr)^{1/{q}} \biggl( \frac{1}{\delta^{ \vert \lambda_{2} \vert \varepsilon/\rho}} \biggr)^{1/{q}}. \end{aligned}$$
(17)

Hence, by (16) and (17), we have

$$\begin{aligned} & \biggl( \int_{\Omega' ( \delta< +\infty )} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl({v} ( {y} ) \bigr)^{(-\beta-n+ \vert \lambda_{2} \vert \varepsilon)/q} {\,dt} \biggr)^{1/{p}} \\ &\qquad{}\times \biggl( \int_{\Omega ( \delta< +\infty )} {K} \bigl( {u} ( {t} ),1 \bigr) \bigl( {u} ( {t} ) \bigr)^{\frac{-\alpha-n- \vert \lambda_{1} \vert \varepsilon}{{p}}} {\,dt} \biggr)^{1/{q}} \\ &\quad\leq{M}_{0} \biggl( \frac{1}{\delta^{ \vert \lambda_{2} \vert \varepsilon/\rho}} \biggr)^{1/({pq})} \biggl( \frac{1}{ \delta^{ \vert \lambda_{1} \vert \varepsilon/\rho}} \biggr)^{1/({pq})}. \end{aligned}$$

For \(\varepsilon\rightarrow0^{+}\), using Fatou’s lemma, we obtain

$$\biggl( \int_{\Omega' ( \delta< +\infty )} {K} \bigl( 1,{v} ( {t} ) \bigr) \bigl({v} ( {y} ) \bigr)^{{- \frac{\beta+n}{{q}}}} {\,dt} \biggr)^{{\frac{1}{{p}}}} \biggl( \int_{\Omega ( \delta < +\infty )} {K} \bigl( {u} ( {t} ),1 \bigr) \bigl( {u} ( {t} ) \bigr)^{- \frac{\alpha+n}{{p}}} {\,dt} \biggr)^{\frac{1}{{q}}} \leq{M}_{0}, $$

and then it follows that, for \(\delta\rightarrow0^{+}\),

$${W}_{1}^{\frac{1}{{p}}} {W}_{2}^{\frac{1}{{q}}} = \biggl( \int_{{R}_{+}^{{n}}} \bigl({v} ( {y} ) \bigr)^{{- \frac{\beta+n}{{q}}}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} \biggr)^{{\frac{1}{{p}}}} \biggl( \int_{{R}_{+}^{{n}}} \bigl( {u} ( {t} ) \bigr)^{- \frac{\alpha+n}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} \biggr)^{\frac{1}{{q}}} \leq{M}_{0}. $$

This is a contradiction, which leads to the fact that \({W}_{1}^{1/{p}} {W}_{2}^{1/{q}}\) is the best possible constant factor of (4). □

4 Application in the operator theory

For \(\gamma=\beta(1-p)\), there is \({- \frac{\beta+n}{{q}} =} \frac{\gamma+n}{{p}} -{n}\), and it follows that \(\frac{{n}\lambda_{1}+\alpha\lambda_{2}}{{p}} = {\frac{{n}\lambda_{2}+\beta\lambda_{1}}{ {q}}}\) is equivalent to \(\lambda_{1}(n+\gamma)+\lambda_{2}(n+\alpha)=\lambda_{2}np\). In view of the fact that (1) is equivalent to (3), by Theorems 1-2, we have the following.

Theorem 3

Suppose that \({n}\geq1\), \(p > 1\), \(\rho >0\), \(\alpha,\gamma\in R\), \(\lambda_{1} \lambda_{2} > 0\), \({a}_{{i}} > 0\), \({b}_{{i}} > 0\), \({u} ( {x} ) = ( \sum_{{i}=1}^{\infty} {a}_{{i}} {x}_{{i}}^{\rho} )^{1/\rho}\), \({v} ( {y} ) = ( \sum_{{i}=1}^{\infty} {b}_{{i}} {y}_{{i}}^{\rho} )^{1/\rho}\), \({K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )\) is a non-negative measurable function, the operator T is defined by (2),

$$\begin{aligned} &0 < \tilde{{W}}_{1} = \int_{{R}_{+}^{{n}}} \bigl( {v} ( {t} ) \bigr)^{\frac{\gamma+n}{{p}} -{n}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} < \infty, \\ &0 < \tilde{{W}}_{2} = \int_{{R}_{+}^{{n}}} \bigl( {u} ( {t} ) \bigr)^{- \frac{\alpha+ {n}}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} < \infty, \end{aligned}$$

and for \(a=0\), \(b=1\) (or \(a=1\), \(b=+\infty\)),

$$\int_{\Omega' ( a< b )} \bigl( {v} ( {t} ) \bigr)^{\frac{\gamma+n}{{p}} -{n}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} >0,\qquad \int_{\Omega' ( a< b )} \bigl( {u} ( {t} ) \bigr)^{- \frac{\alpha+ {n}}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} >0, $$

then we have the following:

(i) T is a bounded operator from \({L}_{{u}^{\alpha}}^{{p}} ( {R}_{+}^{{n}} )\) to \({L}_{{v}^{\gamma}}^{{p}} ( {R}_{+}^{{n}} )\) if and only if the equality \(\lambda_{1}(n+\gamma)+\lambda_{2}(n+\alpha)=\lambda_{2}np\) is valid.

(ii) If the operator T is a bounded operator from \({L}_{{u}^{\alpha}}^{{p}} ( {R}_{+}^{{n}} )\) to \({L}_{{v}^{\gamma}}^{{p}} ( {R}_{+}^{{n}} )\), then we obtain the norm of the operator T as follows:

$$\Vert T \Vert := \sup_{{f}\in {L}_{{u}^{\alpha}}^{{p}} ( {R}_{+}^{{n}} )} \frac{ \Vert {T} ( {f} ) \Vert _{{p}, {v}^{\gamma}}}{ \Vert {f} \Vert _{{p}, {u}^{\rho}}} = \tilde{{W}}_{1}^{\frac{1}{{p}}} \tilde{{W}}_{2}^{\frac{1}{{q}}}. $$

Taking \(\alpha=\gamma=0\) in Theorem 3, we have the result as follows.

Corollary 1

Suppose that \({n}\geq1\), \(p > 1\), \(\rho>0\), \(\lambda_{1} \lambda_{2} > 0\), \({a}_{{i}} > 0\), \({b}_{{i}} > 0\) \(( {i}=1,\ldots,n )\), \({u} ( {x} ) = ( \sum_{{i}=1}^{\infty} {a}_{{i}} {x}_{{i}}^{\rho} )^{1/\rho}\), \({v} ( {y} ) = ( \sum_{{i}=1}^{\infty} {b}_{{i}} {y}_{{i}}^{\rho} )^{1/\rho}\), \({K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )\) is a non-negative measurable function, the operator T is defined by (2),

$$\begin{aligned} &0 < \tilde{{W}}_{1} = \int_{{R}_{+}^{{n}}} \bigl( {v} ( {t} ) \bigr)^{\frac{{n}}{{p}} -{n}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} < \infty, \\ &0 < \tilde{{W}}_{2} = \int_{{R}_{+}^{{n}}} \bigl( {u} ( {t} ) \bigr)^{- \frac{{n}}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} < \infty, \end{aligned}$$

and for \(a=0\), \(b=1\) (or \(a=1\), \(b=+\infty\)),

$$\int_{\Omega' ( a< b )} \bigl( {v} ( {t} ) \bigr)^{\frac{{n}}{{p}} -{n}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} >0,\qquad \int_{\Omega' ( a< b )} \bigl( {u} ( {t} ) \bigr)^{- \frac{{n}}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} >0, $$

then we have the following:

(i) T is a bounded operator in \({L}^{{p}} ( {R}_{+}^{{n}} )\) if and only if \(\lambda_{1}=(p-1)\lambda_{2}\).

(ii) If the operator T is a bounded operator in \({L}^{{p}} ( {R}_{+}^{{n}} )\), then the norm of the operator T is

$$\Vert T\Vert=\tilde{{W}}_{1}^{\frac{1}{{p}}} \tilde{{W}}_{2}^{\frac{1}{{q}}}. $$

Theorem 4

Suppose that \({n}\geq1\), \(p > 1\), \(\frac{1}{ {p}} + \frac{1}{{q}} =1\), \(\rho>0\), \(\lambda_{1}, \lambda_{2} > 0\), \({a}_{{i}} > 0\), \({b}_{{i}} > 0\ ({i}=1,\ldots,n )\), \({b} > \frac{n}{\lambda_{2} {p}}\), \({a} >b- \frac{{n}}{\lambda_{2} {p}}\), \({u} ( {x} ) = ( \sum_{{i}=1}^{\infty} {a}_{{i}} {x}_{{i}}^{\rho} )^{\frac{1}{\rho}}\), \({v} ( {y} ) = ( \sum_{{i}=1}^{\infty} {b}_{{i}} {y}_{{i}}^{\rho} )^{\frac{1}{\rho}}\), the operator T is defined by

$${T} ( {f} ) ( {y} ) = \int_{{R}_{+}^{{n}}} \frac{( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )^{{b}}}{(1+ {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )^{{a}}} {f}({x}) {\,dx},\quad {y}\in {R}_{+}^{{n}}, $$

then we have the following:

(i) T is a bounded operator in \({L}^{{p}} ( {R}_{+}^{{n}} )\) if and only if \(\frac{{\lambda}_{1}}{{p}} = {\frac{\lambda_{2}}{ {q}}}\).

(ii) If the operator T is a bounded operator in \({L}^{{p}} ( {R}_{+}^{{n}} )\), then the norm of the operator T is as follows:

$$\Vert T \Vert = \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{1}^{\frac{1}{{q}}} \lambda_{2}^{\frac{1}{{p}}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \Gamma ( {{a}} )} \Biggl( \prod _{{i}=1}^{{n}} {a}_{{i}}^{- \frac{1}{\rho}} \Biggr)^{\frac{1}{{q}}} \Biggl( \prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \Biggr)^{\frac{1}{{p}}} \Gamma \biggl( {b}- \frac{{n}}{\lambda_{2} {p}} \biggr) \Gamma \biggl( {a}-{b}+ \frac{{n}}{\lambda_{2} {p}} \biggr). $$

Proof

(ii) In view of \(\frac{{\lambda}_{1}}{{p}} = {\frac{\lambda_{2}}{{q}}}\), we have the following by using Lemma 2:

$$\begin{aligned} \tilde{{W}}_{1} ={}& \int_{{R}_{+}^{{n}}} \bigl( {v} ( {t} ) \bigr)^{- \frac{{n}}{{q}}} {K} \bigl( 1,{v} ( {t} ) \bigr) {\,dt} \\ ={}& \int_{{R}_{+}^{{n}}} \Biggl( \sum_{{i}=1}^{{n}} {b}_{{i}} {t}_{{i}}^{\rho} \Biggr)^{\frac{\lambda_{2} {b}}{\rho} - \frac{{n}}{{q}\rho}} \frac{1}{ [ 1+ ( \sum_{{i}=1}^{{n}} {b}_{{i}} {t}_{{i}}^{\rho} )^{\lambda_{2} /\rho} ]^{{a}}} {\,dt} \\ = {}&\prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \int_{{R}_{+}^{{n}}} \Biggl( \sum_{{i}=1}^{{n}} {x}_{{i}}^{\rho} \Biggr)^{\frac{\lambda_{2} {b}}{\rho} - \frac{{n}}{{q}\rho}} \frac{1}{ [ 1+ ( \sum_{{i}=1}^{{n}} {x}_{{i}}^{\rho} )^{\lambda_{2} /\rho} ]^{{a}}} {\,dt} \\ = {}&\prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \lim_{{r}\rightarrow\infty} \int\cdots \int_{{x}_{{i}} > 0; {x}_{1}^{{p}} +\cdots+ {x}_{{n}}^{{p}} \leq{r}^{{p}}} \frac{{r}^{\lambda_{2} {b}-{n}/{q}}}{ [ 1+ {r}^{\lambda_{2}} ( \sum_{{i}=1}^{{n}} ( {\frac{{x}_{{i}}}{{r}}} )^{\rho} )^{\lambda_{2} /\rho} ]^{{a}}} \\ &{} \times \Biggl( \sum_{{i}=1}^{{n}} \biggl( { \frac{{x}_{{i}}}{{r}}} \biggr)^{\rho} \Biggr)^{\frac{\lambda_{2} {b}}{\rho} - \frac{{n}}{{q}\rho}} {x}_{1}^{1-1} \cdots{x}_{{n}}^{1-1} {\,dx}_{1} \cdots{d} {x}_{{n}} \\ ={}& \prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \lim_{{r}\rightarrow\infty} {r}^{\lambda_{2} {b}- \frac{{n}}{{q}}} \frac{{r}^{{n}} \Gamma^{{n}} ( \frac{1}{\rho} )}{\rho^{{n}} \Gamma ( {\frac{{n}}{\rho}} )} \int_{0}^{1} \frac{{u}^{\frac{\lambda_{2} {b}}{\rho} - \frac{{n}}{{q}\rho} + \frac{{n}}{\rho} -1}}{(1+ {r}^{\lambda_{2}} {u}^{\lambda_{2} /\rho} )^{{a}}} {\,du} \\ ={}& \prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \lim_{{r}\rightarrow\infty} \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{2} \rho^{{n} - 1} \Gamma ( {\frac{{n}}{\rho}} )} \int_{0}^{{r}^{\lambda_{2}}} \frac{1}{(1+{t})^{{a}}} {t}^{{b}- \frac{{n}}{\lambda_{2} {p}} -1} {\,dt} \\ = {}&\frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{2} \rho^{{n} - 1} \Gamma ( {\frac{{n}}{\rho}} )} \prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \int_{0}^{+\infty} \frac{1}{(1+{t})^{{a}}} {t}^{{b}- \frac{{n}}{\lambda_{2} {p}} -1} {\,dt} \\ ={}& \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{2} \rho^{{n} - 1} \Gamma ( {\frac{{n}}{\rho}} )} \prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} {B}\biggl({b}- \frac{{n}}{\lambda_{2} {p}},{a}- \biggl({b}- \frac{{n}}{\lambda_{2} {p}} \biggr)\biggr) \\ ={}& \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{2} \rho^{{n} - 1} \Gamma ( {\frac{{n}}{\rho}} ) \Gamma ( {a} )} \prod_{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \Gamma \biggl( {b}- \frac{{n}}{\lambda_{2} {p}} \biggr) \Gamma \biggl( {a}-{b}+ \frac{{n}}{\lambda_{2} {p}} \biggr). \end{aligned}$$

In the same way, we still have the following:

$$\begin{aligned} \tilde{{W}}_{2}& = \int_{{R}_{+}^{{n}}} \bigl[{u} ( {t} ) \bigr]^{- \frac{{n}}{{p}}} {K} \bigl( {u} ( {t} ),1 \bigr) {\,dt} \\ &= \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{1} \rho^{{n} - 1} \Gamma ( {\frac{{n}}{\rho}} ) \Gamma ( {a} )} \prod_{{i}=1}^{{n}} {a}_{{i}}^{- \frac{1}{\rho}} \Gamma \biggl( {b}- \frac{{n}}{\lambda_{1} {q}} \biggr) \Gamma \biggl( {a}-{b}+ \frac{{n}}{\lambda_{1} {q}} \biggr) \\ &= \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{1} \rho^{{n} - 1} \Gamma ( {\frac{{n}}{\rho}} ) \Gamma ( {a} )} \prod_{{i}=1}^{{n}} {a}_{{i}}^{- \frac{1}{\rho}} \Gamma \biggl( {b}- \frac{{n}}{\lambda_{2} {p}} \biggr) \Gamma \biggl( {a}-{b}+ \frac{{n}}{\lambda_{2} {p}} \biggr). \end{aligned}$$

It follows that

$$\tilde{{W}}_{1}^{\frac{1}{{p}}} \tilde{{W}}_{2}^{\frac{1}{{q}}} = \frac{\Gamma^{{n}} ( \frac{1}{\rho} )}{\lambda_{1}^{\frac{1}{{q}}} \lambda_{2}^{\frac{1}{{p}}} \rho^{{n}-1} \Gamma ( {\frac{{n}}{\rho}} ) \Gamma ( {{a}} )} \Biggl( \prod_{{i}=1}^{{n}} {a}_{{i}}^{- \frac{1}{\rho}} \Biggr)^{\frac{1}{{q}}} \Biggl( \prod _{{i}=1}^{{n}} {b}_{{i}}^{- \frac{1}{\rho}} \Biggr)^{\frac{1}{{p}}} \Gamma \biggl( {b}- \frac{{n}}{\lambda_{2} {p}} \biggr) \Gamma \biggl( {a}-{b}+ \frac{{n}}{\lambda_{2} {p}} \biggr). $$

Hence, we prove that (ii) is valid by Corollary 1. □

5 Conclusions

In this paper, 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 \({K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )\) and the best possible constant factor are discussed in Theorems 1-2. Furthermore, its application in the operator theory is considered in Theorems 3-4. The method of real analysis is very important as itis the key to prove the equivalent inequalities with the best possible constant factor. The lemmas and theorems provide an extensive account of this type of inequalities.