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Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

Abstract

For the Hilbert type multiple integral inequality

$$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$

with a nonhomogeneous kernel \(K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })\) (\(\lambda _{1}\lambda _{2}> 0\)), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, \(\lambda _{1}\), \(\lambda _{2}\), α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

Introduction and preparatory knowledge

Let \(k \in \mathbb{N}=\{1, 2, 3, \ldots \}\), \(\rho >0\), \(x=(x_{1}, x_{2}, \ldots , x_{k})\), \(\mathbb{R}_{+}^{k}=\{x= (x_{1}, x_{2}, \ldots , x_{k} ) : x_{i}>0, i=1,2, \ldots , k\}\), \(\|x\|_{k,\rho }= (x_{1}^{\rho }+x_{2}^{\rho }+\cdots +x_{k}^{\rho } )^{1 / \rho }\). Define

$$ L_{p}^{\alpha } \bigl(\mathbb{R}_{+}^{k} \bigr)= \biggl\{ f(x) \geq 0 : \Vert f \Vert _{p, \alpha }= \biggl( \int _{\mathbb{R}_{+}^{k}} \Vert x \Vert ^{\alpha }_{k, \rho } f^{p}(x) \,\mathrm{d} x \biggr)^{1 / p}< +\infty \biggr\} . $$

In this paper, for a class of nonhomogeneous kernels \(K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })\) (\(\lambda _{1}\lambda _{2}>0\)), we discuss the equivalent parameter conditions for the validity of Hilbert type multiple integral inequality

$$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta }. $$
(1)

That is, what conditions do parameters \(\lambda _{1}\), \(\lambda _{2}\), p, q, α, β satisfy to make (1) hold? On the contrary, what conditions do the parameters satisfy when (1) holds? Meanwhile, the best constant factor and its application in operator theory are also considered.

In [1], we studied the necessary and sufficient conditions for the validity of Hilbert type multiple integral inequalities with kernel \(K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho } \|y\|^{\lambda _{2}}_{n, \rho })\) (\(\lambda _{1}\lambda _{2}>0\)). The present paper is a supplement and improvement of [1], more relevant research can be referred to [220].

Lemma 1.1

([21])

Let \(p_{i}>0\), \(a_{i}>0\), \(\alpha _{i}>0(i=1,2, \ldots , n), \psi (u)\) be measurable. Then

$$\begin{aligned}& \int _{ (\frac{x_{1}}{a_{1}} )^{\alpha _{1}}+\cdots + (\frac{x_{n}}{a_{n}} )^{\alpha _{n}} \leq 1; x_{i}>0} { \psi } \biggl( \biggl(\frac{x_{1}}{a_{1}} \biggr)^{\alpha _{1}}+\cdots + \biggl(\frac{x_{n}}{a_{n}} \biggr)^{\alpha _{n}} \biggr) x_{1}^{p_{1}-1} \cdots x_{n}^{p_{n}-1} \,\mathrm{d} x_{1} \cdots \, \mathrm{d} x_{n} \\& \quad = \frac{a_{1}^{p_{1}} \cdots a_{n}^{p_{n}} \Gamma (\frac{p_{1}}{\alpha _{1}} ) \cdots \Gamma (\frac{p_{n}}{\alpha _{n}} )}{\alpha _{1} \cdots \alpha _{n} \Gamma (\frac{p_{1}}{\alpha _{1}}+\cdots +\frac{p_{n}}{\alpha _{n}} )} \int _{0}^{1} \psi (u) u^{\frac{p_{1}}{\alpha _{1}}+\cdots + \frac{p_{n}}{\alpha _{n}}-1} \, \mathrm{d} u, \end{aligned}$$

where \(\Gamma (t)\) represents the gamma function.

By using Lemma 1.1, under the same conditions, it is not difficult to obtain: Let \(\varphi (u)\) be measurable, \(\rho >0\), \(n\geq 1\), \(x=(x_{1}, x_{2}, \ldots , x_{n}) \in \mathbb{R}_{+}^{n}\). Then

$$\begin{aligned}& \begin{gathered} \int _{ \Vert x \Vert _{n,\rho } \leq r} \varphi \bigl( \Vert x \Vert _{n,\rho } \bigr) \,\mathrm{d} x= \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{0}^{r} \varphi (u) u^{n-1} \,\mathrm{d} u, \\ \int _{ \Vert x \Vert _{n,\rho } \geq r} \varphi \bigl( \Vert x \Vert _{n,\rho } \bigr) \,\mathrm{d} x= \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{r}^{+ \infty } \varphi (u) u^{n-1} \, \mathrm{d} u. \end{gathered} \end{aligned}$$
(2)

Suppose that \(K(u, v)=G(u^{\lambda _{1}}/{v^{\lambda _{2}}})\), then obviously \(K(u, v)\) satisfies the following property:

$$ K(u, v)=K\bigl(1,u^{{-\lambda _{1}}/{\lambda _{2}}}v\bigr)=K\bigl(v^{{-\lambda _{2}}/{ \lambda _{1}}}u, 1\bigr). $$

Lemma 1.2

Let \(\frac{1}{p}+\frac{1}{q}=1(p>1)\), \(\rho >0\), \(m,n\in \mathbb{N}\), \(K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/\|y\|^{ \lambda _{2}}_{n, \rho })\), \(\alpha , \beta \in \mathbb{R}\). Then

$$\begin{aligned} \omega _{1}(m, \alpha , p, y) =& \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \Vert x \Vert ^{-\frac{\alpha +m}{p}}_{m, \rho } \,\mathrm{d} x \\ =& \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \Vert y \Vert _{n,\rho }^{\frac{\lambda _{2}}{\lambda _{1}} (- \frac{\alpha +m}{p}+m )} \int _{0}^{+\infty } K(t, 1) t^{- \frac{\alpha +m}{p}+m-1} \, \mathrm{d} t \\ :=& \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \Vert y \Vert _{n,\rho }^{\frac{\lambda _{2}}{\lambda _{1}} (- \frac{\alpha +m}{p}+m )}W_{1}, \end{aligned}$$
(3)
$$\begin{aligned} \omega _{2}(n, \beta , q, x) =& \int _{\mathbb{R}_{+}^{n}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) { \Vert y \Vert _{n, \rho }}^{-\frac{\beta +n}{q}} \,\mathrm{d} y \\ =& \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \Vert x \Vert _{m,\rho }^{\frac{\lambda _{1}}{\lambda _{2}} (- \frac{\beta +n}{q}+n )} \int _{0}^{+\infty } K(1, t) t^{- \frac{\beta +n}{q}+n-1} \, \mathrm{d} t \\ :=& \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \Vert x \Vert _{m,\rho }^{\frac{\lambda _{1}}{\lambda _{2}} (- \frac{\beta +n}{q}+n )}W_{2}. \end{aligned}$$
(4)

Moreover, if \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\), then \(\lambda _{1}W_{1}=\lambda _{2}W_{2}\).

Proof

It follows from (2) that

$$\begin{aligned} \omega _{1}(m, \alpha , p, y) =& \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{0}^{+ \infty } K\bigl(u, \Vert y \Vert _{n,\rho }\bigr) u^{-\frac{\alpha +m}{p}+m-1} \,\mathrm{d} u \\ =& \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{0}^{+\infty } K\bigl( \Vert y \Vert _{n,\rho }^{-{\lambda _{2}}/{\lambda _{1}}}u, 1\bigr) u^{-\frac{\alpha +m}{p}+m-1} \,\mathrm{d} u \\ =& \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \Vert y \Vert _{n,\rho }^{\frac{\lambda _{2}}{\lambda _{1}} (- \frac{\alpha +m}{p}+m-1 )+\frac{\lambda _{2}}{\lambda _{1}}} \\ &{}\times \int _{0}^{+\infty } K(t, 1) t^{-\frac{\alpha +m}{p}+m-1} \, \mathrm{d} t \\ =& \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \Vert y \Vert _{n,\rho }^{\frac{\lambda _{2}}{\lambda _{1}} (- \frac{\alpha +m}{p}+m )}W_{1}. \end{aligned}$$

(4) can be proved at the same time.

When \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\), notice that \(\lambda _{1}\lambda _{2}>0\), we have

$$\begin{aligned} W_{1} =& \int _{0}^{+\infty } K(t, 1) t^{-\frac{\alpha +m}{p}+m-1} \, \mathrm{d} t \\ =& \int _{0}^{+\infty } K\bigl(1, t^{-\lambda _{1}/\lambda _{2}} \bigr) t^{- \frac{\alpha +m}{p}+m-1} \,\mathrm{d} t \\ =& \frac{\lambda _{2}}{\lambda _{1}} \int _{0}^{+\infty } K(1, u) u^{- \frac{\lambda _{2}}{\lambda _{1}}{ (-\frac{\alpha +m}{p}+m-1 )}-\frac{\lambda _{2}}{\lambda _{1}}-1} \, \mathrm{d} u \\ =& \frac{\lambda _{2}}{\lambda _{1}} \int _{0}^{+\infty } K(1, u) u^{{- \frac{\beta +n}{q}+n-1}} \, \mathrm{d} u \\ =& \frac{\lambda _{2}}{\lambda _{1}}W_{2}. \end{aligned}$$

Thus \(\lambda _{1}W_{1}=\lambda _{2}W_{2}\). □

Main results

Theorem 2.1

Let \(\frac{1}{p}+\frac{1}{q}=1(p>1)\), \(\rho >0\), \(m,n\in \mathbb{N}\), \(\lambda _{1} \lambda _{2}>0\), \(\alpha , \beta \in \mathbb{R}\), \(K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho }) (\lambda _{1}\lambda _{2}>0)\) be nonnegative measurable and

$$ W_{0}= \vert \lambda _{1} \vert \int _{0}^{+\infty } K(t, 1) t^{- \frac{\alpha +m}{p}+m-1} \, \mathrm{d} t $$

be convergent. Then

(i) If and only if \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\), there exists a constant \(M>0\) such that

$$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta }, $$
(5)

where \(f(x) \in L_{p}^{\alpha }(\mathbb{R}_{+}^{m})\), \(g(y) \in L_{q}^{\beta } (\mathbb{R}_{+}^{n} )\).

(ii) When (5) holds, the best constant factor is

$$ \inf M=\frac{W_{0}}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} . $$

Proof

Let \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=c\).

(i) Suppose that (5) holds. We prove that \(c=0\). Consider the case of \(\lambda _{1}>0\), \(\lambda _{2}>0\). If \(c>0\), take \(0<\varepsilon <\frac{c}{\lambda _{1}\lambda _{2}}\) and

$$\begin{aligned}& f(x)= \textstyle\begin{cases} { \Vert x \Vert _{m,\rho }^{(-\alpha -m-\lambda _{1}\varepsilon ) / p},} & { \Vert x \Vert _{m,\rho } \geq 1}, \\ {0,} & {0< \Vert x \Vert _{m,\rho }< 1},\end{cases}\displaystyle \\& g(y)= \textstyle\begin{cases} { \Vert y \Vert _{n,\rho }^{(-\beta -n-\lambda _{2}\varepsilon ) / q},} & { \Vert y \Vert _{n, \rho } \geq 1}, \\ {0,} & {0< \Vert y \Vert _{n, \rho }< 1}.\end{cases}\displaystyle \end{aligned}$$

Then

$$\begin{aligned}& \begin{aligned} M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } ={}&M \biggl( \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{-m-\lambda _{1}\varepsilon } \,\mathrm{d} x \biggr)^{1 / p} \biggl( \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{-n- \lambda _{2}\varepsilon } \,\mathrm{d} y \biggr)^{1 / q} \\ ={}& M \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q} \\ &{}\times \biggl( \int _{1}^{+\infty } u^{-1-\lambda _{1}\varepsilon } \,\mathrm{d} u \biggr)^{1 / p} \biggl( \int _{1}^{+\infty } u^{-1-\lambda _{2} \varepsilon } \,\mathrm{d} u \biggr)^{1 / q} \\ ={} & \frac{M}{\varepsilon \lambda ^{1/p}_{1}\lambda ^{1/q}_{2}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}, \end{aligned} \\& \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \\& \quad = \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{(-\alpha -m-\lambda _{1} \varepsilon ) / p} \biggl( \int _{ \Vert y \Vert _{m,\rho } \geq 1} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \Vert y \Vert _{n,\rho }^{(-\beta -n-\lambda _{2} \varepsilon ) / q} \,\mathrm{d} y \biggr) \,\mathrm{d} x \\& \quad = \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{- \frac{\alpha +m+\lambda _{1}\varepsilon }{p}} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{1}^{+ \infty } K \bigl({ \Vert x \Vert _{m,\rho }}, u \bigr) {u}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \,\mathrm{d} u \biggr) \, \mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{- \frac{\alpha +m+\lambda _{1}\varepsilon }{p}} \biggl( \int _{1}^{+\infty } K \bigl({1, u \Vert x \Vert ^{-\lambda _{1}/\lambda _{2}}_{m,\rho }} \bigr) {u}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \,\mathrm{d} u \biggr) \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{- \frac{\alpha +m+\lambda _{1}\varepsilon }{p}+ \frac{\lambda _{1}}{\lambda _{2}} (- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1 )+ \frac{\lambda _{1}}{\lambda _{2}}} \\& \qquad {}\times \biggl( \int _{ \Vert x \Vert ^{-\lambda _{1}/\lambda _{2}}_{m,\rho }}^{+ \infty } K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \biggr) \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{{-m+ \frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \biggl( \int _{ \Vert x \Vert ^{- \lambda _{1}/\lambda _{2}}_{m,\rho }}^{+\infty } K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \biggr) \,\mathrm{d} x \\& \quad \geq \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{{-m+ \frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \biggl( \int _{1}^{+ \infty } K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \biggr) \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{1}^{+\infty } K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{ \Vert x \Vert _{m,\rho } \geq 1} \Vert x \Vert _{m,\rho }^{{-m+\frac{c}{\lambda _{2}}- \lambda _{1}\varepsilon }} \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{m+n}(1 / \rho )}{\rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{1}^{+\infty } K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{1}^{+ \infty } u^{{-1+\frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \,\mathrm{d} u. \end{aligned}$$

It follows that

$$\begin{aligned}& \frac{\Gamma ^{m+n}(1 / \rho )}{\rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{1}^{+\infty } K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{1}^{+ \infty } u^{{-1+\frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \,\mathrm{d} u \\& \quad \leq \frac{M}{\varepsilon \lambda ^{1/p}_{1}\lambda ^{1/q}_{2}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}< + \infty . \end{aligned}$$

But since \(0<\varepsilon <\frac{c}{\lambda _{1}\lambda _{2}}\), we have \(\frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon >0\) and \(\int _{1}^{+\infty } u^{-1+\frac{c}{\lambda _{2}}-\lambda _{1} \varepsilon } \,\mathrm{d} u=+\infty \), which is contradictory, hence \(c > 0\) is not valid.

If \(c<0\), take \(0<\varepsilon <\frac{-c}{\lambda _{1}\lambda _{2}}\) and

$$\begin{aligned}& f(x)= \textstyle\begin{cases} { \Vert x \Vert _{m,\rho }^{(-\alpha -m+\lambda _{1}\varepsilon ) / p},} & {0< \Vert x \Vert _{m,\rho } \leq 1}, \\ {0,} & { \Vert x \Vert _{m,\rho }>1}.\end{cases}\displaystyle \\& g(y)= \textstyle\begin{cases} { \Vert y \Vert _{n,\rho }^{(-\beta -n+\lambda _{2}\varepsilon ) / q},} & {0< \Vert y \Vert _{n,\rho } \leq 1}, \\ {0,} & { \Vert y \Vert _{n, \rho }>1}.\end{cases}\displaystyle \end{aligned}$$

Similarly, we can get

$$\begin{aligned}& \frac{\Gamma ^{m+n}(1 / \rho )}{\rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{0}^{1} K(1, t) {t}^{- \frac{\beta +n-\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{0}^{1} u^{{-1+\frac{c}{\lambda _{2}}+\lambda _{1}\varepsilon }} \,\mathrm{d} u \\& \quad \leq \frac{M}{\varepsilon \lambda ^{1/p}_{1}\lambda ^{1/q}_{2}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}< + \infty . \end{aligned}$$

Since \(0<\varepsilon <\frac{-c}{\lambda _{1}\lambda _{2}}\), we obtain \(\frac{c}{\lambda _{2}}+\lambda _{1}\varepsilon <0\) and \(\int _{0}^{1} u^{-1+\frac{c}{\lambda _{2}}+\lambda _{1}\varepsilon } \,\mathrm{d} u=+\infty \), this is still a contradiction, hence \(c< 0\) cannot hold.

To sum up, when \(\lambda _{1}>0\), \(\lambda _{2}>0\), we have \(c=0\), that is, \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\).

Moreover, consider the case of \(\lambda _{1}<0\), \(\lambda _{2}<0\). If \(c>0\), take \(0<\varepsilon <\frac{c}{\lambda _{1}\lambda _{2}}\) and

$$\begin{aligned}& f(x)= \textstyle\begin{cases} { \Vert x \Vert _{m,\rho }^{(-\alpha -m-\lambda _{1}\varepsilon ) / p},} & {0< \Vert x \Vert _{m,\rho } \leq 1}, \\ {0,} & { \Vert x \Vert _{m,\rho }>1},\end{cases}\displaystyle \\& g(y)= \textstyle\begin{cases} { \Vert y \Vert _{n,\rho }^{(-\beta -n-\lambda _{2}\varepsilon ) / q},} & {0< \Vert y \Vert _{n,\rho } \leq 1}, \\ {0,} & { \Vert y \Vert _{n, \rho }>1}.\end{cases}\displaystyle \end{aligned}$$

Then, by calculation,

$$\begin{aligned}& M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } = \frac{M}{\varepsilon (-\lambda _{1})^{1/p}(-\lambda _{2})^{1/q}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}, \\& { \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y} \\& \quad = \int _{0< \Vert x \Vert _{m,\rho } \leq 1} \Vert x \Vert _{m,\rho }^{(-\alpha -m- \lambda _{1}\varepsilon ) / p} \biggl( \int _{0< \Vert y \Vert _{n,\rho } \leq 1} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \Vert y \Vert _{n,\rho }^{(-\beta -n-\lambda _{2} \varepsilon ) / q} \,\mathrm{d} y \biggr) \,\mathrm{d} x \\& \quad = \int _{0< \Vert x \Vert _{m,\rho } \leq 1} \Vert x \Vert _{m,\rho }^{- \frac{\alpha +m+\lambda _{1}\varepsilon }{p}} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{0}^{1} K \bigl({ \Vert x \Vert _{m,\rho }}, u \bigr) {u}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \,\mathrm{d} u \biggr) \, \mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{0< \Vert x \Vert _{m,\rho } \leq 1} \Vert x \Vert _{m,\rho }^{- \frac{\alpha +m+\lambda _{1}\varepsilon }{p}} \biggl( \int _{0}^{1} K \bigl({1, u \Vert x \Vert ^{-\lambda _{1}/\lambda _{2}}_{m,\rho }} \bigr) {u}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \,\mathrm{d} u \biggr) \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{0< \Vert x \Vert _{m,\rho } \leq 1} \Vert x \Vert _{m,\rho }^{- \frac{\alpha +m+\lambda _{1}\varepsilon }{p}+ \frac{\lambda _{1}}{\lambda _{2}} (- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1 )+ \frac{\lambda _{1}}{\lambda _{2}}} \\& \qquad {}\times \biggl( \int _{0}^{ \Vert x \Vert ^{-\lambda _{1}/\lambda _{2}}_{m,\rho }} K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \biggr) \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{0< \Vert x \Vert _{m,\rho } \leq 1} \Vert x \Vert _{m,\rho }^{{-m+ \frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \biggl( \int _{0}^{ \Vert x \Vert ^{-\lambda _{1}/\lambda _{2}}_{m,\rho }} K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \biggr) \,\mathrm{d} x \\& \quad \geq \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{0< \Vert x \Vert _{m,\rho } \leq 1} \Vert x \Vert _{m,\rho }^{{-m+ \frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \biggl( \int _{0}^{1} K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \biggr) \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \int _{0}^{1} K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{0< \Vert x \Vert _{m,\rho } \leq 1} \Vert x \Vert _{m,\rho }^{{-m+\frac{c}{\lambda _{2}}- \lambda _{1}\varepsilon }} \,\mathrm{d} x \\& \quad = \frac{\Gamma ^{m+n}(1 / \rho )}{\rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{0}^{1} K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{0}^{1} u^{{-1+\frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \,\mathrm{d} u. \end{aligned}$$

It follows that

$$\begin{aligned}& \frac{\Gamma ^{m+n}(1 / \rho )}{\rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{0}^{1} K(1, t) {t}^{- \frac{\beta +n+\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{0}^{1} u^{{-1+\frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon }} \,\mathrm{d} u \\& \quad \leq \frac{M}{\varepsilon (-\lambda _{1})^{1/p}(-\lambda _{2})^{1/q}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}< + \infty . \end{aligned}$$

Since \(0<\varepsilon <\frac{c}{\lambda _{1}\lambda _{2}}\) and \(\lambda _{1}<0\), then \(\frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon <0\) and \(\int _{0}^{1} u^{-1+\frac{c}{\lambda _{2}}-\lambda _{1}\varepsilon } \,\mathrm{d} u=+\infty \). This is a contradiction, therefore \(c > 0\) cannot hold.

If \(c<0\), take \(0<\varepsilon <\frac{-c}{\lambda _{1}\lambda _{2}}\) and

$$\begin{aligned}& f(x)= \textstyle\begin{cases} { \Vert x \Vert _{m,\rho }^{(-\alpha -m+\lambda _{1}\varepsilon ) / p},} & { \Vert x \Vert _{m,\rho }\geq 1}, \\ {0,} & {0< \Vert x \Vert _{m,\rho }< 1}.\end{cases}\displaystyle \\& g(y)= \textstyle\begin{cases} { \Vert y \Vert _{n,\rho }^{(-\beta -n+\lambda _{2}\varepsilon ) / q},} & { \Vert y \Vert _{n, \rho } \geq 1}, \\ {0,} & {0< \Vert y \Vert _{n, \rho }< 1}.\end{cases}\displaystyle \end{aligned}$$

Similarly,

$$\begin{aligned}& \frac{\Gamma ^{m+n}(1 / \rho )}{\rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{1}^{+\infty } K(1, t) {t}^{- \frac{\beta +n-\lambda _{2}\varepsilon }{q}+n-1} \, \mathrm{d} t \int _{1}^{+ \infty } u^{{-1+\frac{c}{\lambda _{2}}+\lambda _{1}\varepsilon }} \,\mathrm{d} u \\& \quad \leq \frac{M}{\varepsilon (-\lambda _{1})^{1/p}(-\lambda _{2})^{1/q}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}< + \infty . \end{aligned}$$

Since \(0<\varepsilon <\frac{-c}{\lambda _{1}\lambda _{2}}\) and \(\lambda _{1}<0\), we have \(\frac{c}{\lambda _{2}}+\lambda _{1}\varepsilon >0\) and \(\int _{1}^{+\infty } u^{-1+\frac{c}{\lambda _{2}}+\lambda _{1} \varepsilon } \,\mathrm{d} u=+\infty \). That is still a contradiction, so \(c< 0\) does not hold either.

To sum up, when \(\lambda _{1}<0\), \(\lambda _{2}<0\), we still have \(c=0\), that is, \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\).

Conversely, if \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\), set \(a=\frac{\alpha +m}{p q}\), \(b=\frac{\beta +n}{p q}\), it follows from Hölder’s inequality and Lemma 1.2 that

$$ \begin{aligned} & { \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y} \\ &\quad = \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \biggl( \frac{ \Vert x \Vert _{m,\rho }^{a}}{ \Vert y \Vert _{n,\rho }^{b}} f(x) \biggr) \biggl( \frac{ \Vert y \Vert _{n,\rho }^{b}}{ \Vert x \Vert _{m,\rho }^{a}} g(y) \biggr) \,\mathrm{d} x \,\mathrm{d} y \\ &\quad \leq \biggl( \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \frac{ \Vert x \Vert _{m,\rho }^{a p}}{ \Vert y \Vert _{n,\rho }^{b p}} f^{p}(x) \,\mathrm{d} x \,\mathrm{d} y \biggr)^{1 / p} \\ &\qquad {} \times \biggl( \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \frac{ \Vert y \Vert _{n,\rho }^{b q}}{ \Vert x \Vert _{m,\rho }^{a q}} g^{q}(y) \,\mathrm{d} x \,\mathrm{d} y \biggr)^{1 / q} \\ &\quad = \biggl[ \int _{\mathbb{R}_{+}^{m}} \Vert x \Vert _{m,\rho }^{ \frac{\alpha +m}{q}} f^{p}(x) \biggl( \int _{\mathbb{R}_{+}^{n}} \Vert y \Vert _{n, \rho }^{-\frac{\beta +n}{q}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \,\mathrm{d} y \biggr)\,\mathrm{d} x \biggr]^{1 / p} \\ &\qquad {} \times \biggl[ \int _{\mathbb{R}_{+}^{n}} \Vert y \Vert _{n,\rho }^{ \frac{\beta +n}{p}} g^{q}(y) \biggl( \int _{\mathbb{R}_{+}^{m}} \Vert x \Vert _{m, \rho }^{-\frac{\alpha +m}{p}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \,\mathrm{d} x \biggr) \,\mathrm{d} y \biggr]^{1 / q} \\ &\quad = \biggl( \int _{\mathbb{R}_{+}^{m}} \Vert x \Vert _{m,\rho }^{ \frac{\alpha +m}{q}} f^{p}(x) \omega _{2}(n, \beta , q,x)\,\mathrm{d} x \biggr)^{1 / p} \biggl( \int _{\mathbb{R}_{+}^{n}} \Vert y \Vert _{n,\rho }^{ \frac{\beta +n}{p}} g^{q}(y) \omega _{1}(m,\alpha , p, y) \,\mathrm{d} y \biggr)^{1 / q} \\ &\quad = \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \\ &\qquad {}\times W_{1}^{1 / q} W_{2}^{1 / p} \biggl( \int _{\mathbb{R}_{+}^{m}} \Vert x \Vert _{m,\rho }^{\frac{\alpha +m}{q}+\frac{\lambda _{1}}{\lambda _{2}} (-\frac{\beta +n}{q}+n )} f^{p}(x)\,\mathrm{d} x \biggr)^{1 / p} \\ &\qquad {} \times \biggl( \int _{\mathbb{R}_{+}^{n}} \Vert y \Vert _{n,\rho }^{ \frac{\beta +n}{p}+\frac{\lambda _{2}}{\lambda _{1}} (- \frac{\alpha +m}{p}+m )} g^{q}(y) \,\mathrm{d} y \biggr)^{1 / q} \\ &\quad = \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} W_{1}^{1 / q} W_{2}^{1 / p} \\ &\qquad {} \times \biggl( \int _{\mathbb{R}_{+}^{m}} \Vert x \Vert _{m,\rho }^{\alpha } f^{p}(x) \,\mathrm{d} x \biggr)^{1 / p} \biggl( \int _{\mathbb{R}_{+}^{n}} \Vert y \Vert _{n, \rho }^{\beta } g^{q}(y) \,\mathrm{d} y \biggr)^{1 / q} \\ &\quad = \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} W_{1}^{1 / q} W_{2}^{1 / p} \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta }. \end{aligned} $$

Arbitrarily take a constant M satisfying

$$ M \geq \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} W_{1}^{1 / q} W_{2}^{1 / p}, $$

then

$$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta }. $$

Thus (5) holds.

(ii) Assume that there is a constant \(M_{0}\) satisfying

$$\begin{aligned} M_{0}< \frac{W_{0}}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \end{aligned}$$
(6)

such that, for any \(f(x) \in L_{p}^{\alpha }(\mathbb{R}_{+}^{m})\), \(g(y) \in L_{q}^{\beta } (\mathbb{R}_{+}^{n} )\), we have

$$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M_{0} \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta }. $$

Take sufficiently small \(\varepsilon >0\), \(\delta >0\), and set

$$\begin{aligned}& f(x)= \textstyle\begin{cases} { \Vert x \Vert _{m,\rho }^{(-\alpha -m- \vert \lambda _{1} \vert \varepsilon ) / p},} & { \Vert x \Vert _{m,\rho }\geq \delta }, \\ {0,} & {0< \Vert x \Vert _{m,\rho }< \delta }.\end{cases}\displaystyle \\& g(y)= \textstyle\begin{cases} { \Vert y \Vert _{n,\rho }^{(-\beta -n- \vert \lambda _{2} \vert \varepsilon ) / q},} & { \Vert y \Vert _{n,\rho } \geq 1}, \\ {0,} & {0< \Vert y \Vert _{n, \rho }< 1}.\end{cases}\displaystyle \end{aligned}$$

It can be obtained by calculation that

$$\begin{aligned} M_{0} \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } =& M_{0} \biggl( \int _{ \Vert x \Vert _{m, \rho }\geq \delta } \Vert x \Vert _{m,\rho }^{-m- \vert \lambda _{1} \vert \varepsilon } \,\mathrm{d} x \biggr)^{1 / p} \biggl( \int _{ \Vert y \Vert _{m,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{-n- \vert \lambda _{2} \vert \varepsilon } \,\mathrm{d} y \biggr)^{1 / q} \\ =& \frac{M_{0}\cdot \delta ^{-{ \vert \lambda _{1} \vert \varepsilon /p}}}{\varepsilon \vert \lambda _{1} \vert ^{1/p} \vert \lambda _{2} \vert ^{1/q}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}. \end{aligned}$$

Since \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\),

$$\begin{aligned} & \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \\ &\quad = \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{(-\beta -n- \vert \lambda _{2} \vert \varepsilon ) / q} \biggl( \int _{ \Vert x \Vert _{m,\rho } \geq \delta } \Vert x \Vert _{m, \rho }^{(-\alpha -m- \vert \lambda _{1} \vert \varepsilon ) / p} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \,\mathrm{d} x \biggr)\,\mathrm{d} y \\ &\quad = \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{(-\beta -n- \vert \lambda _{2} \vert \varepsilon ) / q} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{ \delta }^{+\infty } u^{-\frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K\bigl(u, \Vert y \Vert _{n, \rho }\bigr) \,\mathrm{d} u \biggr)\,\mathrm{d} y \\ &\quad = \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{(-\beta -n- \vert \lambda _{2} \vert \varepsilon ) / q} \\ &\qquad {}\times \biggl( \int _{\delta }^{+\infty } u^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K\bigl(u \Vert y \Vert ^{- \lambda _{2}/\lambda _{1}}_{n, \rho }, 1\bigr) \,\mathrm{d} u \biggr) \,\mathrm{d} y \\ &\quad = \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{- \frac{\beta +n+ \vert \lambda _{2} \vert \varepsilon }{q}+ \frac{\lambda _{2}}{\lambda _{1}} (- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p} +m-1 )+ \frac{\lambda _{2}}{\lambda _{1}}} \\ &\qquad {}\times \biggl( \int _{\delta \Vert y \Vert ^{-\lambda _{2}/\lambda _{1}}_{n, \rho }}^{+\infty } {t}^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K(t, 1) \, \mathrm{d} t \biggr) \,\mathrm{d} y \\ &\quad = \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{\frac{-1}{\lambda _{1}} (n\lambda _{1}+\frac{\lambda _{1} \vert \lambda _{2} \vert \varepsilon }{q}+ \frac{ \vert \lambda _{1} \vert \lambda _{2} \varepsilon }{p} )} \\ &\qquad {}\times \biggl( \int _{\delta \Vert y \Vert _{n,p}^{-\lambda _{2}/\lambda _{1}}}^{+\infty } t^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K(t, 1) \, \mathrm{d} t \biggr)\,\mathrm{d} y \\ &\quad \geq \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{-n- \vert \lambda _{2} \vert \varepsilon } \biggl( \int _{\delta }^{+\infty } t^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K(t, 1) \, \mathrm{d} t \biggr)\,\mathrm{d} y \\ &\quad = \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \int _{\delta }^{+\infty } t^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K(t, 1) \, \mathrm{d} t \int _{ \Vert y \Vert _{n,\rho } \geq 1} \Vert y \Vert _{n,\rho }^{-n- \vert \lambda _{2} \vert \varepsilon } \,\mathrm{d} y \\ &\quad = \frac{\Gamma ^{m+n}(1 / \rho )}{\rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{\delta }^{+\infty } t^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K(t, 1) \, \mathrm{d} t \int _{1}^{+\infty } u^{-1- \vert \lambda _{2} \vert \varepsilon } \,\mathrm{d} u \\ &\quad = \frac{\Gamma ^{m+n}(1 / \rho )}{\varepsilon \vert \lambda _{2} \vert \rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{\delta }^{+\infty } t^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K(t, 1) \, \mathrm{d} t. \end{aligned}$$

Consequently,

$$ \begin{aligned} & \frac{\Gamma ^{m+n}(1 / \rho )}{ \vert \lambda _{2} \vert \rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{\delta }^{+\infty } t^{- \frac{\alpha +m+ \vert \lambda _{1} \vert \varepsilon }{p}+m-1} K(t, 1) \, \mathrm{d} t \\ &\quad \leq \frac{M_{0} \delta ^{- \vert \lambda _{1} \vert \varepsilon /p }}{ \vert \lambda _{1} \vert ^{1/p} \vert \lambda _{2} \vert ^{1/q}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}. \end{aligned} $$

Let \(\varepsilon \rightarrow 0^{+}\), and by using the famous Fatou lemma, we obtain

$$ \begin{aligned} & \frac{\Gamma ^{m+n}(1 / \rho )}{ \vert \lambda _{2} \vert \rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \int _{\delta }^{+\infty } t^{-\frac{\alpha +m}{p}+m-1} K(t, 1) \, \mathrm{d} t \\ &\quad \leq \frac{M_{0}}{ \vert \lambda _{1} \vert ^{1/p} \vert \lambda _{2} \vert ^{1/q}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}. \end{aligned} $$

Let again \(\delta \rightarrow 0^{+}\), then

$$ \frac{\Gamma ^{m+n}(1 / \rho )W_{1}}{ \vert \lambda _{2} \vert \rho ^{m+n-2} \Gamma (n / \rho )\Gamma (m / \rho )} \leq \frac{M_{0}}{ \vert \lambda _{1} \vert ^{1/p} \vert \lambda _{2} \vert ^{1/q}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / p} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / q}. $$

It follows that

$$ \frac{W_{0}}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p}\leq M_{0}. $$

This contradicts (6). Thus

$$ \frac{W_{0}}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} $$

is the best constant factor of (5). □

Applications in operator theory

Let \(p>1\), \(\rho >0\), \(m,n\in \mathbb{N}\), \(\alpha ,\beta \in \mathbb{R}\), \(K(u,v)\) be nonnegative measurable. Define

$$ T(f) (y)= \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) f(x) \,\mathrm{d} x,\quad f(x) \in L_{p}^{\alpha }\bigl(\mathbb{R}_{+}^{m} \bigr). $$
(7)

Then T is a singular integral operator defined on \(L_{p}^{\alpha }(\mathbb{R}_{+}^{m})\). Using this operator and according to Hilbert type integral operator theory, (5) is equivalent to

$$ \bigl\Vert T(f) \bigr\Vert _{p,\beta (1-p)}\leq M \Vert f \Vert _{p, \alpha }, $$

so we get the following.

Theorem 3.1

Under the same conditions as in Theorem 2.1, let the singular integral operator T be defined as in (7). Then

(i) T is a bounded operator from \(L_{p}^{\alpha }(\mathbb{R}_{+}^{m})\) to \(L_{p}^{\beta (1-p)} (\mathbb{R}_{+}^{n} )\) if and only if \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\).

(ii) When T is a bounded operator from \(L_{p}^{\alpha }(\mathbb{R}_{+}^{m})\) to \(L_{p}^{\beta (1-p)} (\mathbb{R}_{+}^{n} )\), the operator norm of T is

$$ \Vert T \Vert = \frac{W_{0}}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p}. $$

Corollary 3.1

Let \(\frac{1}{p}+\frac{1}{q}=1(p>1)\), \(\rho >0\), \(\lambda >0\), \(\lambda _{1} \lambda _{2}>0\), \(m,n\in \mathbb{N}\), \(\alpha ,\beta \in \mathbb{R}\), \(0< \frac{1}{\rho \lambda _{1}} ( \frac{m}{q}-\frac{\alpha }{p} )<\lambda \). Define a singular integral operator T by

$$ T(f) (y)= \int _{\mathbb{R}_{+}^{m}} \frac{f(x)}{ [1+(\sum_{k=1}^{m} x_{k}^{\rho })^{\lambda _{1}}/(\sum_{k=1}^{n} y_{k}^{\rho })^{\lambda _{2}} ]^{\lambda }} \,\mathrm{d} x. $$

Then \(T: L_{p}^{\alpha }(\mathbb{R}_{+}^{m}) \rightarrow L_{p}^{\beta (1-p)} (\mathbb{R}_{+}^{n} )\) is a bounded operator if and only if \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\). And when T is bounded, its operator norm is

$$\begin{aligned} \Vert T \Vert =& \frac{1}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} B \biggl( \frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}-\frac{\alpha }{p} \biggr), \lambda - \frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}- \frac{\alpha }{p} \biggr) \biggr) \\ &{}\times \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n} \Gamma (n / \rho )} \biggr)^{1 / p}, \end{aligned}$$

where \(B(u,v)\) represents the beta function.

Proof

First, notice that

$$\begin{aligned} \frac{1}{ [1+(\sum_{k=1}^{m} x_{k}^{\rho })^{\lambda _{1}}/(\sum_{k=1}^{n} y_{k}^{\rho })^{\lambda _{2}} ]^{\lambda }} &= \frac{1}{ (1+ \Vert x \Vert _{m, \rho }^{\rho \lambda _{1}}/ \Vert y \Vert _{n, \rho }^{\rho \lambda _{2}} )^{\lambda }} \\ &=G\bigl( \Vert x \Vert ^{\rho \lambda _{1}}_{m, \rho }/ \Vert y \Vert ^{\rho \lambda _{2}}_{n, \rho }\bigr) =K\bigl( \Vert x \Vert _{m, \rho }, \Vert y \Vert _{n, \rho }\bigr) \end{aligned}$$

and

$$ \frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0 $$

is equivalent to

$$ \frac{n(\rho \lambda _{1})-\alpha (\rho \lambda _{2})}{p}+ \frac{m(\rho \lambda _{2})-\beta (\rho \lambda _{1})}{q}=0. $$

Since

$$\begin{aligned} W_{0} =& \vert \rho \lambda _{1} \vert W_{1} = \vert \rho \lambda _{1} \vert \int _{0}^{+ \infty } K(t, 1) t^{-\frac{\alpha +m}{p}+m-1} \, \mathrm{d} t \\ =& \vert \rho \lambda _{1} \vert \int _{0}^{+\infty } \frac{1}{ (1+t^{\rho \lambda _{1}} )^{\lambda }} t^{- \frac{\alpha +m}{p}+m-1} \,\mathrm{d} t = \int _{0}^{+\infty } \frac{1}{(1+u)^{\lambda }} u^{ \frac{1}{\rho \lambda _{1}} (\frac{m}{q}-\frac{\alpha }{p} )-1} \,\mathrm{d} u \\ =& B \biggl(\frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}- \frac{\alpha }{p} \biggr), \lambda -\frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}-\frac{\alpha }{p} \biggr) \biggr), \end{aligned}$$

we have

$$\begin{aligned}& \frac{W_{0}}{ \vert \rho \lambda _{1} \vert ^{1/q} \vert \rho \lambda _{2} \vert ^{1/p}} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \\& \quad = \frac{1}{\rho \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} B \biggl( \frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}-\frac{\alpha }{p} \biggr), \lambda - \frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}- \frac{\alpha }{p} \biggr) \biggr) \\& \qquad {}\times \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \\& \quad = \frac{1}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} B \biggl( \frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}-\frac{\alpha }{p} \biggr), \lambda - \frac{1}{\rho \lambda _{1}} \biggl( \frac{m}{q}- \frac{\alpha }{p} \biggr) \biggr) \\& \qquad {}\times \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n} \Gamma (n / \rho )} \biggr)^{1 / p}. \end{aligned}$$

According to Theorem 3.1, Corollary 3.1 holds. □

Corollary 3.2

Let \(\frac{1}{p}+\frac{1}{q}=1(p>1)\), \(\rho >0\), \(\lambda _{1}>0\), \(\lambda _{2}>0\), \(m,n\in \mathbb{N}\), \(\alpha ,\beta \in \mathbb{R}\), \(-\lambda _{1}< \frac{m}{q}-\frac{\alpha }{p} <\lambda _{1}\). Define a singular integral operator T by

$$ T(f) (y)= \int _{\mathbb{R}_{+}^{m}} \frac{\min \{1, \Vert x \Vert ^{\lambda _{1}}_{m, \rho }/ \Vert y \Vert ^{\lambda _{2}}_{n, \rho }\}}{\max \{1, \Vert x \Vert ^{\lambda _{1}}_{m, \rho }/ \Vert y \Vert ^{\lambda _{2}}_{n, \rho }\}} f(x) \,\mathrm{d} x, \quad f(x) \in L_{p}^{\alpha }\bigl(\mathbb{R}_{+}^{m} \bigr). $$

Then \(T: L_{p}^{\alpha }(\mathbb{R}_{+}^{m}) \rightarrow L_{p}^{\beta (1-p)} (\mathbb{R}_{+}^{n} )\) is a bounded operator if and only if \(\frac{n\lambda _{1}-\alpha \lambda _{2}}{p}+ \frac{m\lambda _{2}-\beta \lambda _{1}}{q}=0\), and when T is bounded, its operator norm is

$$ \Vert T \Vert = \frac{2\lambda ^{2}_{1}}{\lambda _{1}^{1/q} \lambda _{2}^{1/p} (\lambda _{1}+\frac{m}{q}-\frac{\alpha }{p} ) (\lambda _{1}-\frac{m}{q}+\frac{\alpha }{p} )} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p}. $$

Proof

Since \(-\lambda _{1}< \frac{m}{q}-\frac{\alpha }{p} <\lambda _{1}\), then \(\frac{m}{q}-\frac{\alpha }{p} +\lambda _{1}>0\) and \(\frac{m}{q}-\frac{\alpha }{p} -\lambda _{1}<0\), therefore

$$\begin{aligned} W_{0} =&\lambda _{1} \int _{0}^{+\infty } K(t, 1) t^{- \frac{\alpha +m}{p}+m-1} \, \mathrm{d} t \\ =& \lambda _{1} \int _{0}^{+\infty } \frac{\min \{1,t^{\lambda _{1}}\}}{\max \{1,t^{\lambda _{1}}\}} t^{ \frac{m}{q}-\frac{\alpha }{p}-1} \,\mathrm{d} t\\ =& \lambda _{1} \int _{0}^{1} t^{\frac{m}{q}-\frac{\alpha }{p}+\lambda _{1}-1} \,\mathrm{d} t + \lambda _{1} \int _{1}^{+\infty } t^{\frac{m}{q}-\frac{\alpha }{p}- \lambda _{1}-1} \,\mathrm{d} t \\ =& \frac{\lambda _{1}}{\frac{m}{q}-\frac{\alpha }{p}+\lambda _{1}} - \frac{\lambda _{1}}{\frac{m}{q}-\frac{\alpha }{p}-\lambda _{1}} = \frac{-2\lambda ^{2}_{1}}{ (\frac{m}{q}-\frac{\alpha }{p}+\lambda _{1} ) (\frac{m}{q}-\frac{\alpha }{p}-\lambda _{1} )}. \end{aligned}$$

It follows that

$$\begin{aligned}& \frac{W_{0}}{ \vert \lambda _{1} \vert ^{1/q} \vert \lambda _{2} \vert ^{1/p}} \biggl( \frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p} \\& \quad = \frac{2\lambda ^{2}_{1}}{\lambda _{1}^{1/q} \lambda _{2}^{1/p} (\lambda _{1}+\frac{m}{q}-\frac{\alpha }{p} ) (\lambda _{1}-\frac{m}{q}+\frac{\alpha }{p} )} \biggl(\frac{\Gamma ^{m}(1 / \rho )}{\rho ^{m-1} \Gamma (m / \rho )} \biggr)^{1 / q} \biggl( \frac{\Gamma ^{n}(1 / \rho )}{\rho ^{n-1} \Gamma (n / \rho )} \biggr)^{1 / p}. \end{aligned}$$

According to Theorem 3.1, Corollary 3.2 holds. □

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Acknowledgements

The authors thank the referee for his useful suggestions to reform the paper.

Funding

This work is supported by the Innovation Team Construction Project of Guangdong Province (2018KCXTD020).

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BH carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. ZL and YH participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.

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Correspondence to Zhen Li.

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He, B., Hong, Y. & Li, Z. Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels. J Inequal Appl 2021, 64 (2021). https://doi.org/10.1186/s13660-021-02593-z

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MSC

  • 26D15
  • 47A07

Keywords

  • Nonhomogeneous kernel
  • Hilbert type multiple integral inequality
  • Parameter condition
  • Boundedness of operator
  • Operator norm
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