Skip to main content

A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications

Abstract

Let \(x=(x_{1},x_{2},\ldots,x_{n})\), and let \(K(u(x),v(y))\) satisfy \(u(rx)=ru(x)\), \(v(ry)=rv(y)\), \(K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)\), and \(K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)\). In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel \(K(u(x),v(y))\) and discuss its applications in the theory of operators.

Preliminary

Let \(n\ge1\), \(x=(x_{1},x_{2},\ldots, x_{n})\), \(\|x\|_{\rho}=(x_{1}^{\rho}+\cdots +x_{n}^{\rho})^{1/\rho}\), and \(\mathbf {R}_{+}^{n}=\{x=(x_{1},\ldots, x_{n}): x_{1}>0, \ldots, x_{n}>0\}\).

Define the function space

$$L^{p}_{\omega(x)} \bigl(\mathbf {R}^{n}_{+} \bigr)= \biggl\{ f(x)\ge0: \Vert f \Vert _{p,\omega(x)}= \biggl( \int _{\mathbf {R}^{n}_{+}}f^{p}(x)\omega(x)\,dx \biggr)^{\frac{1}{p}}< +\infty \biggr\} . $$

Definition 1

Let λ, \(\lambda_{1}\), and \(\lambda_{2}\) be constants, and let \(u(x)\), \(v(y)\) and \(K(u,v)\) satisfy: for all \(r>0\), \(u(rx)=ru(x)\), \(v(ry)=rv(y)\), and

$$K(ru,v)=r^{\lambda\lambda_{1}}K \bigl(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v \bigr),\qquad K(u,rv)=r^{\lambda\lambda_{2}}K \bigl(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v \bigr). $$

Then we call \(K(u(x), v(y))\) a generalized homogeneous function with parameters \((\lambda,\lambda_{1},\lambda_{2})\). Obviously, \(K(u(x), v(y))\) is a homogeneous function of order \(\lambda\lambda_{1}\) when \(\lambda _{1}=\lambda_{2}\).

If \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), then we call the inequality

$$ \int_{\mathbf {R}^{n}_{+}} \int_{\mathbf {R}^{n}_{+}}K \bigl(u(x),v(y) \bigr)f(x)g(y)\,dx \,dy \le M \Vert f \Vert _{p,u^{\alpha}(x)} \Vert g \Vert _{q,v^{\beta}(y)} $$
(1.1)

the Hilbert-type multiple integral inequality with \(f\in L^{p}_{u^{\alpha}(x)}(\mathbf {R}^{n}_{+})\) and \(g\in L^{q}_{v^{\beta}(y)}(\mathbf {R}^{n}_{+})\).

Define the integral operator T with kernel \(K(u(x),v(y))\) as follows:

$$ T(f) (y)= \int_{\mathbf {R}^{n}_{+}}K \bigl(u(x),v(y) \bigr)f(x)\,dx,\quad y\in \mathbf {R}^{n}_{+}. $$
(1.2)

If there exists a constant M such that

$$\bigl\Vert T(f) \bigr\Vert _{p,\omega_{2}(y)}\le M \Vert f \Vert _{p,\omega_{1}(x)},\quad f\in L^{p}_{\omega _{1}(x)} \bigl(\mathbf {R}^{n}_{+} \bigr), $$

then T is called a bounded operator from \(L^{p}_{\omega_{1}}(\mathbf {R}^{n}_{+})\) to \(L^{p}_{\omega_{2}}(\mathbf {R}^{n}_{+})\). If T is a bounded operator from \(L^{p}_{\omega_{1}}(\mathbf {R}^{n}_{+})\) to itself, then we call T a bounded operator in \(L^{p}_{\omega_{1}}(\mathbf {R}^{n}_{+})\). The operator norm of T is defined as

$$\Vert T \Vert = \inf M=\sup_{f\in L^{p}_{\omega_{1}}(\mathbf {R}^{n}_{+}) } \frac{ \Vert T(f) \Vert _{p,\omega_{2}}}{ \Vert f \Vert _{p,\omega_{1}}}. $$

By (1.2) inequality (1.1) can be rewritten as

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

It is not hard to prove that this inequality is equivalent to

$$ \bigl\Vert T(f) \bigr\Vert _{p,v^{\beta(1-p)}(y)}\le M \Vert f \Vert _{p,u^{\alpha}(x)}. $$
(1.3)

In this paper, we discuss a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with the integral kernel of the generalized homogeneous function \(K(u(x),v(y))\). Our research is of some theoretical and application value for the research of Hilbert-type inequalities. Further, these results are used to study the boundedness and norm of the operator. Related studies can be found in [116].

Lemma 1

Let\(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(n \ge1\), \(\lambda>0\), \(\lambda _{1}\lambda_{2}>0\), and let a nonnegative measurable function\(K(u(x), v(y))\)be a generalized homogeneous function with parameters\((\lambda, \lambda_{1}, \lambda_{2})\). Denote

$$\begin{gathered} W_{1}= \int_{\mathbf {R}^{n}_{+}} \bigl[v(t) \bigr]^{-\frac{\beta+n}{q}}K \bigl(1,v(t) \bigr)\,dt,\\ W_{2}= \int_{\mathbf {R}^{n}_{+}} \bigl[u(t) \bigr]^{-\frac{\alpha+n}{p}}K \bigl(u(t),1 \bigr)\,dt.\end{gathered} $$

Then

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

Proof

Since \(K(u(x), v(y))\) is a generalized homogeneous function with parameters \((\lambda, \lambda_{1}, \lambda_{2})\), we have

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

By the same method we can obtain \(\omega_{2}(y)=[v(y)]^{\lambda\lambda _{2}-\frac{\lambda_{2}}{\lambda_{1}}(\frac{\alpha+n}{p}-n)}W_{2}\). □

Lemma 2

([17])

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

$$\begin{aligned} & \int\cdots \int_{x_{1}>0,\ldots,x_{n}>0;\sum_{i=1}^{n}(\frac {x_{i}}{a_{i}})^{\alpha_{i}}\le1} \psi \Biggl(\sum_{i=1}^{n} \biggl(\frac{x_{i}}{a_{i}} \biggr)^{\alpha_{i}} \Biggr) x_{1}^{p_{1}-1} \cdots x_{n}^{p_{n}-1} \,dx_{1}\cdots dx_{n} \\ &\quad=\frac{a_{1}^{p_{1}}\cdots a_{n}^{p_{n}}\varGamma(\frac{p_{1}}{\alpha_{1}})\cdots \varGamma(\frac{p_{n}}{\alpha_{n}})}{\alpha_{1}\cdots\alpha_{n}\varGamma(\frac {p_{1}}{\alpha_{1}}+\cdots+\frac{p_{n}}{\alpha_{n}})} \int_{0}^{1} \psi(t)t^{\frac{p_{1}}{\alpha_{1}}+\cdots+\frac{p_{n}}{\alpha _{n}}-1}\,dt, \end{aligned}$$

whereΓis the gamma function.

Main results

Theorem 1

Let\(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(n\ge1\), \(\rho>0\), \(\lambda>0\), \(\lambda _{1}\lambda_{2}>0\), let there exist positive constants\(C_{1}\)and\(C_{2}\)such that\(C_{1}\|x\|_{\rho}\le u(x)\le C_{2}\|x\|_{\rho}\), \(C_{1}\|y\|_{\rho}\le v(y)\le C_{2}\|y\|_{\rho}\), let a nonnegative measurable function\(K(u(x),v(y))\)be a generalized homogeneous function with parameters\((\lambda, \lambda_{1}, \lambda_{2})\), and let the convergent integrals\(W_{1}\)and\(W_{2}\)be defined as in Lemma 1. Then we have:

  1. (i)

    There exists a constantMsuch that the Hilbert-type multiple integral inequality in (1.1) holds if and only if\(\frac{\lambda_{2}\alpha-n\lambda_{1}}{p} +\frac{\lambda_{1}\beta-n\lambda_{2}}{q}=\lambda\lambda_{1}\lambda_{2}\).

  2. (ii)

    The best constant factor in (1.1) is\(\inf M=W_{1}^{\frac{1}{p}}W_{2}^{\frac{1}{q}}\).

Proof

Let \(\varOmega(a< b)=\{x=(x_{1},\ldots, x_{n}):a< \|x\|_{\rho}< b \}\).

(i) Suppose there exists a constant M such that (1.1) holds. Denote \(l=\frac{\lambda_{2}\alpha-n\lambda_{1}}{p}+\frac{\lambda_{1}\beta -n\lambda_{2}}{q}-\lambda\lambda_{1}\lambda_{2}\). First, we let \(\lambda_{1}>0\), \(\lambda_{2}>0\). For \(l>0\) and \(\varepsilon>0\) sufficiently small, we set

$$\begin{aligned}& f(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} [u(x)]^{(-\alpha-n+\lambda_{1}\varepsilon)/p},& 0< \Vert x \Vert _{\rho}< 1, \\ 0,& \Vert x \Vert _{\rho}\geq1. \end{array}\displaystyle \right . \\& g(y)= \left \{ \textstyle\begin{array}{l@{\quad}l} [v(y)]^{(-\beta-n+\lambda_{2}\varepsilon)/q}, &0< \Vert y \Vert _{\rho}< 1, \\ 0, &\|y\|_{\rho}\geq1. \end{array}\displaystyle \right . \end{aligned}$$

Thus we have

$$ \Vert f \Vert _{p,u^{\alpha}(x)} \Vert g \Vert _{q,v^{\beta}(y)}= \biggl( \int _{\varOmega(0< 1)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon}\,dx \biggr)^{\frac {1}{p}} \biggl( \int_{\varOmega(0< 1)} \bigl[v(y) \bigr]^{-n+\lambda_{2}\varepsilon}\, dy \biggr)^{\frac{1}{q}}. $$
(2.1)

In view of \(\lambda_{1}>0\), \(\lambda_{2}>0\), \(C_{1}\|x\|_{\rho}\le u(x)\le C_{2}\|x\| _{\rho}\), \(C_{1}\|y\|_{\rho}\le v(y)\le C_{2}\|y\|_{\rho}\), the two integrals in (2.1) are all convergent.

Also, since \(-\frac{\lambda_{1}}{\lambda_{2}}<0\) and \((C_{2}\|x\|_{\rho})^{-\frac{\lambda_{1}}{\lambda_{2}}}\le u^{-\frac{\lambda_{1}}{\lambda _{2}}}(x)\le(C_{1}\|x\|_{\rho})^{-\frac{\lambda_{1}}{\lambda_{2}}}\), we have

$$\begin{aligned} & \int_{\mathbf {R}^{n}_{+}} \int_{\mathbf {R}^{n}_{+}}K \bigl(u(x),v(y) \bigr)f(x)g(y)\,dx\, dy \\ &\quad= \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{(-\alpha-n+\lambda_{1}\varepsilon)/p} \biggl( \int_{\varOmega(0< 1)}K \bigl(u(x),v(y) \bigr) \bigl[v(y) \bigr]^{(-\beta-n+\lambda_{2}\varepsilon )/q}\,dy \biggr)\,dx \\ &\quad= \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{\lambda\lambda_{1}+(-\alpha-n+\lambda _{1}\varepsilon)/p} \biggl( \int_{\varOmega(0< 1)}K \bigl(1,v \bigl(u^{-\frac{\lambda _{1}}{\lambda_{2}}}(x)y \bigr) \bigr) \bigl[v(y) \bigr]^{(-\beta-n+\lambda_{2}\varepsilon)/q}\,dy \biggr)\,dx \\ &\quad= \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{\lambda\lambda_{1}+(-\alpha-n+\lambda _{1}\varepsilon)/p} \\ &\qquad{}\times \biggl( \int_{\varOmega(0< u^{-\frac {\lambda_{1}}{\lambda_{2}}}(x))}K \bigl(1,v(t) \bigr) \bigl[u^{\frac{\lambda_{1}}{\lambda_{2}}} (x)v(t) \bigr]^{(-\beta-n+\lambda_{2}\varepsilon)/q}u^{\frac{n\lambda _{1}}{\lambda_{2}}}(x)\,dt \biggr)\,dx \\ &\quad= \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac {l}{\lambda_{2}}} \biggl( \int_{\varOmega(0< u^{-\frac{\lambda_{1}}{\lambda _{2}}}(x))} K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n-\lambda_{2}\varepsilon}{q}} \,dt \biggr)\,dx \\ &\quad\ge \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac {l}{\lambda_{2}}} \biggl( \int_{\varOmega(0< (C_{2}\|x\|_{\rho})^{-\frac{\lambda _{1}}{\lambda_{2}}})} K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n-\lambda_{2}\varepsilon }{q}} \,dt \biggr)\,dx \\ &\quad\ge \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac{l}{\lambda_{2}}}\,dx \int_{\varOmega (0< C_{2}^{-\frac{\lambda_{1}}{\lambda_{2}}})} K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta +n-\lambda_{2}\varepsilon}{q}} \,dt. \end{aligned}$$

Combining this with (1.1) and (2.1), we get

$$\begin{aligned} & \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac{l}{\lambda _{2}}}\,dx \int_{\varOmega(0< C_{2}^{-\frac{\lambda_{1}}{\lambda _{2}}})}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n-\lambda_{2}\varepsilon}{q}} \, dt \\ &\quad\le M \biggl( \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon}\, dx \biggr)^{\frac{1}{p}} \biggl( \int_{\varOmega(0< 1)} \bigl[v(y) \bigr]^{-n+\lambda _{2}\varepsilon}\,dy \biggr)^{\frac{1}{q}}. \end{aligned}$$
(2.2)

Since \(l>0\) and ε is sufficiently small, \(-n+\lambda _{1}\varepsilon-\frac{l}{\lambda_{2}}<-n\), and additionally \(C_{1}\|x\|_{\rho}\le u(x)\le C_{2}\|x\|_{\rho}\), then \(\int_{\varOmega(0<1)}[u(x)]^{-n+\lambda_{1}\varepsilon-\frac{l}{\lambda _{2}}}\,dx=+\infty\). So (2.2) is a contradiction to \(l>0\).

If \(l<0\), let \(\varepsilon>0\) be sufficient small. Then we set

$$\begin{aligned}& f(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} [u(x)]^{(-\alpha-n-\lambda_{1}\varepsilon)/p}, & \Vert x \Vert _{\rho}>1,\\ 0 &\mbox{otherwise}, \end{array}\displaystyle \right . \\& g(y)=\left \{ \textstyle\begin{array}{l@{\quad}l} [v(y)]^{(-\beta-n-\lambda_{2}\varepsilon)/q}, &\Vert y \Vert _{\rho}>1,\\ 0 &\mbox{otherwise}. \end{array}\displaystyle \right . \end{aligned}$$

Similarly, we can get

$$\begin{aligned} & \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n-\lambda_{2}\varepsilon-\frac {l}{\lambda_{1}}}\,dy \int_{\varOmega(C_{1}^{-\frac{\lambda_{2}}{\lambda _{1}}}< +\infty)}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+\beta+\lambda_{1}\varepsilon }{p}} \,dt \\ &\quad\le M \biggl( \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{-n-\lambda_{1}\varepsilon }\,dx \biggr)^{\frac{1}{p}} \biggl( \int_{\varOmega(1< +\infty )} \bigl[v(y) \bigr]^{-n-\lambda_{2}\varepsilon}\,dy \biggr)^{\frac{1}{q}}. \end{aligned}$$
(2.3)

Since \(C_{1}\|x\|_{\rho}\le u(x)\le C_{2}\|x\|_{\rho}\), \(C_{1}\|y\|_{\rho}\le v(y)\le C_{2}\|y\|_{\rho}\), \(l<0\), \(\lambda_{1}>0\), \(\lambda_{2}>0\), and \(\varepsilon>0\) is sufficient small, the right-hand side of (2.3) converges; also, \(\int_{\varOmega(1<+\infty)}[v(y)]^{-n-\lambda_{2}\varepsilon-\frac {l}{\lambda_{1}}}\,dy\) diverges, and thus (2.3) is a contradiction to \(l<0\).

In conclusion, when \(\lambda_{1}>0\), \(\lambda_{2}>0\), then we have \(l=0\), that is, \(\frac{\lambda_{2}\alpha-n\lambda_{1}}{p}+\frac{\lambda_{1}\beta -n\lambda_{2}}{q}=\lambda\lambda_{1}\lambda_{2}\).

Again, suppose \(\lambda_{1}<0\), \(\lambda_{2}<0\). If \(l>0\), then taking \(\varepsilon>0\) sufficiently small, we set

$$\begin{aligned}& f(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} [u(x)]^{(-\alpha-n+\lambda_{1}\varepsilon)/p}, &\Vert x \Vert _{\rho}>1,\\ 0 &\mbox{otherwise}, \end{array}\displaystyle \right . \\& g(y)=\left \{ \textstyle\begin{array}{l@{\quad}l} [v(y)]^{(-\beta-n+\lambda_{2}\varepsilon)/q}, & \Vert y \Vert _{\rho}>1,\\ 0 &\mbox{otherwise}. \end{array}\displaystyle \right . \end{aligned}$$

We thus have

$$\begin{aligned} \Vert f \Vert _{p,u^{\alpha}(x)} \Vert g \Vert _{q,v^{\beta}(y)}= \biggl( \int_{\varOmega (1< +\infty)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon}\,dx \biggr)^{\frac{1}{p}} \biggl( \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n+\lambda_{2}\varepsilon}\,dy \biggr)^{\frac{1}{q}} . \end{aligned}$$
(2.4)

Meanwhile, using \(C_{1}\|x\|_{\rho}\le u(x)\le C_{2}\|x\|_{\rho}\), \((C_{2}\|x\| _{\rho})^{-\frac{\lambda_{1}}{\lambda_{2}}}\le u^{-\frac{\lambda_{1}}{\lambda _{2}}}\le(C_{1}\|x\|_{\rho})^{-\frac{\lambda_{1}}{\lambda_{2}}}\), we have

$$\begin{aligned} & \int_{R_{+}^{n}} \int_{R_{+}^{n}}K \bigl(u(x),v(y) \bigr)f(x)g(y)\,dx\,dy \\ &\quad= \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{(-\alpha-n+\lambda_{1}\varepsilon )/p} \biggl( \int_{\varOmega(1< +\infty)}K \bigl(u(x),v(y) \bigr) \bigl[v(y) \bigr]^{(-\beta -n+\lambda_{2}\varepsilon)/q}\,dy \biggr)\,dx \\ &\quad= \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{\lambda\lambda_{1}+(-\alpha -n+\lambda_{1}\varepsilon)/p} \\ &\qquad{}\times \biggl( \int_{\varOmega(1< +\infty )}K \bigl(1,v \bigl(u^{-\frac{\lambda_{1}}{\lambda_{2}}}(x)y \bigr) \bigr) \bigl[v(y) \bigr]^{(-\beta-n+\lambda _{2}\varepsilon)/q}\,dy \biggr)\,dx \\ &\quad= \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{\lambda\lambda_{1}-\frac{\alpha +n-\lambda_{1}\varepsilon}{p}} \\ &\qquad{}\times\biggl( \int_{\varOmega(u^{-\frac{\lambda _{1}}{\lambda_{2}}}(x)< +\infty)}K \bigl(1,v(t) \bigr) \bigl[u^{\frac{\lambda_{1}}{\lambda_{2}}}(x) v(t) \bigr]^{-\frac{\beta+n-\lambda_{2}\varepsilon}{q}}u^{\frac{n\lambda _{1}}{\lambda_{2}}}(x)\,dt \biggr)\,dx \\ &\quad\ge \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac {l}{\lambda_{2}}} \biggl( \int_{\varOmega((C_{1}\|x\|_{\rho})^{-\frac{\lambda _{1}}{\lambda_{2}}}< +\infty)}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n-\lambda _{2}\varepsilon}{q}} \,dt \biggr)\,dx \\ &\quad\ge \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac {l}{\lambda_{2}}}\,dx \int_{\varOmega(C_{1}^{-\frac{\lambda_{1}}{\lambda _{2}}}< +\infty)}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n-\lambda_{2}\varepsilon }{q}} \,dt. \end{aligned}$$

Combining this with (1.1) and (2.4), we obtain

$$\begin{aligned} & \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac {l}{\lambda_{2}}}\,dx \int_{\varOmega(C_{1}^{-\frac{\lambda_{1}}{\lambda _{2}}}< +\infty)}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n-\lambda_{2}\varepsilon }{q}} \,dt \\ &\quad\le M \biggl( \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon} \,dx \biggr)^{\frac{1}{p}} \biggl( \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n+\lambda_{2}\varepsilon}\,dy \biggr)^{\frac{1}{q}}. \end{aligned}$$
(2.5)

Since the two integrals of the right-hand side of (2.5) converge, but the integral

$$\int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{-n+\lambda_{1}\varepsilon-\frac {l}{\lambda_{2}}}\,dx $$

diverges, (2.5) is a contradiction to \(l>0\).

If \(l<0\) and \(\varepsilon>0\) is sufficiently small, then we set

$$\begin{aligned}& f(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} [u(x)]^{(-\alpha-n-\lambda_{1}\varepsilon)/p},& 0< \Vert x \Vert _{\rho}< 1,\\ 0 &\mbox{otherwise}, \end{array}\displaystyle \right . \\& g(y)=\left \{ \textstyle\begin{array}{l@{\quad}l} [v(y)]^{(-\beta-n-\lambda_{2}\varepsilon)/q}, &0< \Vert y \Vert _{\rho}< 1,\\ 0 &\mbox{otherwise}. \end{array}\displaystyle \right . \end{aligned}$$

Similarly, we can get

$$\begin{aligned} & \int_{\varOmega(0< 1)} \bigl[v(y) \bigr]^{-n-\lambda_{2}\varepsilon-\frac{l}{\lambda _{1}}}\,dy \int_{\varOmega(0< C_{2}^{-\frac{\lambda_{2}}{\lambda _{1}}})}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+\beta+\lambda_{1}\varepsilon }{p}} \,dt \\ &\quad\le M \biggl( \int_{\varOmega(0< 1)} \bigl[u(x) \bigr]^{-n-\lambda_{1}\varepsilon}\, dx \biggr)^{\frac{1}{p}} \biggl( \int_{\varOmega(0< 1)} \bigl[v(y) \bigr]^{-n-\lambda _{2}\varepsilon}\,dy \biggr)^{\frac{1}{q}}. \end{aligned}$$
(2.6)

We now easily get that both integrals on the right-hand side of (2.6) converge, but

$$\int_{\varOmega(0< 1)} \bigl[v(y) \bigr]^{-n-\lambda_{2}\varepsilon-\frac{l}{\lambda _{1}}}\,dy $$

diverges, and thus (2.6) is a contradiction to \(l<0\).

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

On the contrary, if \(\frac{\lambda_{2}\alpha-n\lambda_{1}}{p}+\frac{\lambda _{1}\beta-n\lambda_{2}}{q}=\lambda\lambda_{1}\lambda_{2}\), then let \(a=\frac {\alpha}{pq}+\frac{n}{pq}\), \(b=\frac{\beta}{pq}+\frac{n}{pq}\). By the Hölder inequality and Lemma 1 we have

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

Taking arbitrary \(M\ge W_{1}^{\frac{1}{p}}W_{2}^{\frac{1}{q}}\), inequality (1.1) holds.

(ii) Suppose inequality (1.1) holds. If \(\inf M \neq W_{1}^{\frac {1}{p}}W_{2}^{\frac{1}{q}}\), then there exists a constant \(M_{0}< W_{1}^{\frac {1}{p}}W_{2}^{\frac{1}{q}}\) such that

$$\begin{aligned} \int_{R_{+}^{n}} \int_{R_{+}^{n}}K \bigl(u(x),v(y) \bigr)f(x)g(y)\,dx\,dy\le M_{0} \Vert f \Vert _{p,u^{\alpha}(x)} \Vert g \Vert _{q,v^{\beta}(y)} \end{aligned}$$
(2.7)

for all \(f\in L^{p}_{u^{\alpha}(x)}(R_{+}^{n})\) and \(g\in L^{q}_{v^{\beta}(y)}(R_{+}^{n})\).

Let \(\varepsilon>0\) and \(\delta>0\) be sufficient small. We take

$$\begin{aligned}& f(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} [u(x)]^{(-\alpha-n- \vert \lambda_{1} \vert \varepsilon)/p}, &\Vert x \Vert _{\rho}>\delta,\\ 0& \mbox{otherwise}, \end{array}\displaystyle \right . \\& g(y)=\left \{ \textstyle\begin{array}{l@{\quad}l} [v(y)]^{(-\beta-n- \vert \lambda_{2} \vert \varepsilon)/q}, &\Vert y \Vert _{\rho}>1,\\ 0& \mbox{otherwise}. \end{array}\displaystyle \right . \end{aligned}$$

Then we have

$$\begin{aligned} \Vert f \Vert _{p,u^{\alpha}(x)} \Vert g \Vert _{q,v^{\beta}(y)}= \biggl( \int_{\varOmega(\delta < +\infty)} \bigl[u(x) \bigr]^{-n- \vert \lambda_{1} \vert \varepsilon}\,dx \biggr)^{\frac{1}{p}} \biggl( \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n- \vert \lambda_{2} \vert \varepsilon}\, dy \biggr)^{\frac{1}{q}}. \end{aligned}$$
(2.8)

Since \(\frac{\lambda_{2}\alpha-n\lambda_{1}}{p}+\frac{\lambda_{1}\beta -n\lambda_{2}}{q}=\lambda\lambda_{1}\lambda_{2}\) and \(v^{-\frac{\lambda _{2}}{\lambda_{1}}}(y)\le(C_{1}\|y\|_{\rho})^{-\frac{\lambda_{2}}{\lambda_{1}}}\), we have

$$\begin{aligned} & \int_{R_{+}^{n}} \int_{R_{+}^{n}}K \bigl(u(x),v(y) \bigr)f(x)g(y)\,dx\,dy \\ &\quad= \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{(-\beta-n-|\lambda_{2}|\varepsilon )/q} \biggl( \int_{\varOmega(\delta< +\infty)} K \bigl(u(x),v(y) \bigr) \bigl[u(x) \bigr]^{(-\alpha-n-|\lambda_{1}|\varepsilon)/p}\,dx \biggr)\,dy \\ &\quad= \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{\lambda\lambda_{2}-\frac{\beta +n+|\lambda_{2}|\varepsilon}{q}} \biggl( \int_{\varOmega(\delta< +\infty)} K \bigl(u \bigl(v^{-\frac{\lambda_{2}}{\lambda_{2}}}(y)x \bigr),1 \bigr) \bigl[u(x) \bigr]^{-\frac{\alpha +n+|\lambda_{1}|\varepsilon}{p}}\,dx \biggr)\,dy \\ &\quad= \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{\lambda\lambda_{2}-\frac{\beta +n+|\lambda_{2}|\varepsilon}{q}} \\ &\qquad{}\times \biggl( \int_{\varOmega(\delta v^{-\frac{\lambda_{2}}{\lambda _{1}}}(y)< +\infty)} K \bigl(u(t),1 \bigr) \bigl[v^{\frac{\lambda_{2}}{\lambda_{1}}}(y)u(t) \bigr]^{-\frac{\alpha +n+|\lambda_{1}|\varepsilon}{p}}v^{\frac{n\lambda_{2}}{\lambda_{1}}}(y)\, dt \biggr)\,dy \\ &\quad= \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n-|\lambda_{2}|\varepsilon} \biggl( \int_{\varOmega(\delta v^{-\frac{\lambda_{2}}{\lambda_{1}}}(y)< +\infty )}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n+|\lambda_{1}|\varepsilon}{p}} \,dt \biggr)\,dy \\ &\quad\ge \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n-|\lambda_{2}|\varepsilon} \biggl( \int_{\varOmega(\delta(C_{1}\|y\|_{\rho})^{-\frac{\lambda_{2}}{\lambda _{1}}}< +\infty)}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n+|\lambda_{1}|\varepsilon }{p}} \,dt \biggr)\,dy \\ &\quad\ge \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n-|\lambda_{2}|\varepsilon} \int _{\varOmega(\delta C_{1}^{-\frac{\lambda_{2}}{\lambda_{1}}}< +\infty )}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n+|\lambda_{1}|\varepsilon}{p}} \, dt . \end{aligned}$$

Combining this with (2.7) and (2.8), we obtain

$$\begin{aligned} & \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n-|\lambda_{2}|\varepsilon}\,dy \int _{\varOmega(\delta C_{1}^{-\frac{\lambda_{2}}{\lambda_{1}}}< +\infty )}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n+|\lambda_{1}|\varepsilon}{p}} \, dt \\ &\quad\le M_{0} \biggl( \int_{\varOmega(\delta< +\infty)} \bigl[u(x) \bigr]^{-n-|\lambda _{1}|\varepsilon}\,dx \biggr)^{\frac{1}{p}} \biggl( \int_{\varOmega(1< +\infty )} \bigl[v(y) \bigr]^{-n-|\lambda_{2}|\varepsilon}\,dy \biggr)^{\frac{1}{q}}. \end{aligned}$$

Thus

$$\begin{aligned} & \biggl( \int_{\varOmega(1< +\infty)} \bigl[v(y) \bigr]^{-n-|\lambda_{2}|\varepsilon}\, dy \biggr)^{\frac{1}{p}} \int_{\varOmega(\delta C_{1}^{-\frac{\lambda _{2}}{\lambda_{1}}}< +\infty)}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n+|\lambda _{1}|\varepsilon}{p}} \,dt \\ &\quad\le M_{0} \biggl( \int_{\varOmega(\delta< +\infty)} \bigl[u(x) \bigr]^{-n-|\lambda _{1}|\varepsilon}\,dx \biggr)^{\frac{1}{p}}. \end{aligned}$$
(2.9)

We also take

$$\begin{aligned}& f(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} [u(x)]^{(-\alpha-n- \vert \lambda_{1} \vert \varepsilon)/p}, & \Vert x \Vert _{\rho}>1,\\ 0 &\mbox{otherwise}, \end{array}\displaystyle \right . \\& g(y)=\left \{ \textstyle\begin{array}{l@{\quad}l} [v(y)]^{(-\beta-n- \vert \lambda_{2} \vert \varepsilon)/q},& \Vert y \Vert _{\rho}>\delta,\\ 0 &\mbox{otherwise}. \end{array}\displaystyle \right . \end{aligned}$$

Similarly, we can get

$$\begin{aligned} & \biggl( \int_{\varOmega(1< +\infty)} \bigl[u(x) \bigr]^{-n-|\lambda_{1}|\varepsilon}\, dx \biggr)^{\frac{1}{q}} \int_{\varOmega(\delta C_{1}^{-\frac{\lambda _{2}}{\lambda_{1}}}< +\infty)}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n+|\lambda _{2}|\varepsilon}{q}} \,dt \\ &\quad\le M_{0} \biggl( \int_{\varOmega(\delta< +\infty)} \bigl[v(y) \bigr]^{-n-|\lambda _{2}|\varepsilon}\,dy \biggr)^{\frac{1}{q}}. \end{aligned}$$
(2.10)

By (2.9) and (2.10) we have

$$\begin{aligned} & \biggl( \int_{\varOmega(\delta C_{1}^{-\frac{\lambda_{1}}{\lambda_{2}}}< +\infty )}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n+|\lambda_{2}|\varepsilon}{q}} \,dt \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int_{\varOmega(\delta C_{1}^{-\frac{\lambda_{2}}{\lambda _{1}}}< +\infty)}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n+|\lambda_{1}|\varepsilon }{p}} \,dt \biggr)^{\frac{1}{q}} \\ &\quad\le M_{0} \biggl(\frac{ \int_{\varOmega(\delta< +\infty)}[u(x)]^{-n-|\lambda _{1}|\varepsilon}\,dx}{ \int_{\varOmega(1< +\infty)}[u(x)]^{-n-|\lambda _{1}|\varepsilon}\,dx} \biggr)^{\frac{1}{pq}} \biggl(\frac{ \int_{\varOmega(\delta< +\infty)}[v(y)]^{-n-|\lambda _{2}|\varepsilon}\,dy}{ \int_{\varOmega(1< +\infty)}[v(y)]^{-n-|\lambda _{2}|\varepsilon}\,dy} \biggr)^{\frac{1}{pq}} \\ &\quad=M_{0} \biggl(1+\frac{ \int_{\varOmega(\delta < 1)}[u(x)]^{-n-|\lambda_{1}|\varepsilon}\,dx}{ \int_{\varOmega(1< +\infty )}[u(x)]^{-n-|\lambda_{1}|\varepsilon}\,dx} \biggr)^{\frac{1}{pq}} \biggl(1+\frac{ \int_{\varOmega(\delta< 1)}[v(y)]^{-n-|\lambda _{2}|\varepsilon}\,dy}{ \int_{\varOmega(1< +\infty)}[v(y)]^{-n-|\lambda _{2}|\varepsilon}\,dy} \biggr)^{\frac{1}{pq}}. \end{aligned}$$
(2.11)

Since \(C_{1}\|x\|_{\rho}\le u(x)\le C_{2}\|x\|_{\rho}\), \(\int_{\varOmega(\delta <1)}[u(x)]^{-n}\,dx\) is a usual integral, but \(\int_{\varOmega(1<+\infty )}[u(x)]^{-n}\,dx\) diverges, and thus

$$\lim_{\varepsilon\to0^{+}}\frac{ \int_{\varOmega(\delta < 1)}[u(x)]^{-n-|\lambda_{1}|\varepsilon}\,dx}{ \int_{\varOmega(1< +\infty )}[u(x)]^{-n-|\lambda_{1}|\varepsilon}\,dx}=0. $$

In the same way, we have

$$\lim_{\varepsilon\to0^{+}}\frac{ \int_{\varOmega(\delta < 1)}[v(y)]^{-n-|\lambda_{2}|\varepsilon}\,dy}{ \int_{\varOmega(1< +\infty )}[v(y)]^{-n-|\lambda_{2}|\varepsilon}\,dy}=0. $$

Letting \(\varepsilon\to0^{+}\) in (2.11), we get

$$\begin{gathered} \biggl( \int_{\varOmega(\delta C_{1}^{-\frac{\lambda_{1}}{\lambda_{2}}}< +\infty )}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n}{q}} \,dt \biggr)^{\frac{1}{p}}\\\quad{}\times \biggl( \int_{\varOmega(\delta C_{1}^{-\frac{\lambda_{2}}{\lambda_{1}}}< +\infty )}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n}{p}} \,dt \biggr)^{\frac{1}{q}} \le M_{0}.\end{gathered} $$

Letting \(\delta\to0^{+}\), we obtain

$$\begin{gathered} W_{1}^{\frac{1}{p}}W_{2}^{\frac{1}{q}}= \biggl( \int_{\varOmega(0< +\infty )}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n}{q}} \,dt \biggr)^{\frac{1}{p}}\\\quad{}\times \biggl( \int_{\varOmega(0< +\infty)}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n}{p}} \, dt \biggr)^{\frac{1}{q}}\le M_{0}.\end{gathered} $$

This is a contradiction, and hence \(\inf M=W_{1}^{\frac{1}{p}}W_{2}^{\frac{1}{q}}\). □

Theorem 2

Let\(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(n\ge1\), \(\lambda>0\), \(\lambda _{1}\lambda_{2}>0\), \(\gamma=(1-p)\beta\), and let there exist constants\(C_{1}\)and\(C_{2}\)such that\(C_{1}\|x\|_{\rho}\le u(x)\le C_{2}\|x\|_{\rho}\)and\(C_{1}\|y\|_{\rho}\le v(y)\le C_{2}\|y\|_{\rho}\). Let a nonnegative measurable function\(K(u(x), v(y))\)be a generalized homogeneous function for parameters\((\lambda, \lambda_{1}, \lambda_{2})\). Let the operatorTbe defined by (1.2), and let\(W_{1}\)and\(W_{2}\)defined by Lemma 1be also convergent. Then

  1. (i)

    Tis a bounded operator from\(L^{p}_{u^{\alpha}(x)}(R^{n}_{+})\)to\(L^{p}_{v^{\gamma}(y)}(R^{n}_{+})\)if and only if\(\frac{1}{p}[\lambda_{2}(\alpha +n)-\lambda_{1}(\gamma+n)]=n\lambda_{2}+\lambda\lambda_{1}\lambda_{2}\).

  2. (ii)

    IfTis a bounded operator from\(L^{p}_{u^{\alpha}(x)}(R^{n}_{+})\)to\(L^{p}_{v^{\gamma}(y)}(R^{n}_{+})\), then the operator norm ofTis\(\|T\|=W_{1}^{\frac{1}{p}}W_{2}^{\frac{1}{q}}\).

Proof

Since \(\frac{1}{p}+\frac{1}{q}=1\), \(\beta=\frac{\gamma}{1-p}\), \(\frac {\lambda_{2}\alpha-n\lambda_{1}}{p}+\frac{\lambda_{1}\beta-n\lambda _{2}}{q}=\lambda\lambda_{1}\lambda_{2}\) leads to \(\frac{1}{p} [\lambda_{2}(\alpha+n)-\lambda_{1}(\gamma+n)]=n\lambda_{2}+\lambda\lambda _{1}\lambda_{2}\), and since equality (1.1) is equivalent to (1.3), Theorem 2 holds by Theorem 1. □

Applications

Theorem 3

Let\(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(n\ge1\), \(\rho>0\), \(\lambda >0\), \(\lambda_{1}>0\), \(\lambda_{2}>0\), \(a_{i}>0\), \(b_{i}>0\), \(\alpha< n(p-1)\), \(\beta< n(q-1)\), \(u(x)=(\sum_{i=1}^{n} a_{i}x_{i}^{\rho})^{1/\rho}\), and\(v(y)=(\sum_{i=1}^{n} b_{i}y_{i}^{\rho})^{1/\rho}\). Then:

  1. (i)

    There exists a constantMsuch that

    $$\begin{aligned} \int_{R_{+}^{n}} \int_{R_{+}^{n}}\frac{1}{(u^{\lambda _{1}}(x)+v^{\lambda_{2}}(y))^{\lambda}}f(x)g(y)\,dx\,dy\le M \Vert f \Vert _{p,u^{\alpha}(x)} \Vert g \Vert _{q,v^{\beta}(y)} \end{aligned}$$
    (3.1)

    if and only if\(\frac{n\lambda_{1}-\lambda_{2}\alpha}{p}+\frac{n\lambda _{2}-\lambda_{1}\beta}{q}=\lambda\lambda_{1}\lambda_{2}\), where\(f\in L^{p}_{u^{\alpha}(x)}(R_{+}^{n})\)and\(g\in L^{q}_{v^{\beta}(y)}(R_{+}^{n})\).

  2. (ii)

    If inequality (3.1) holds, then its best constant factor is

    $$\inf M= \Biggl(\prod_{i=1}^{n} a_{i}^{-\frac{1}{\rho}} \Biggr)^{\frac{1}{q}} \Biggl(\prod _{i=1}^{n} b_{i}^{-\frac{1}{\rho}} \Biggr)^{\frac{1}{p}}\frac{\varGamma^{n}(\frac {1}{\rho})}{\rho^{n-1}\varGamma(\lambda)\varGamma(\frac{n}{\rho})} \varGamma \biggl( \frac{1}{\lambda_{1}} \biggl(\frac{n}{q}-\frac{\alpha}{p} \biggr) \biggr)\varGamma \biggl(\frac{1}{\lambda_{2}} \biggl(\frac{n}{p}- \frac{\beta}{q} \biggr) \biggr). $$

Proof

Set \(K(u(x), v(y))=\frac{1}{(u^{\lambda_{1}}(x)+v^{\lambda_{2}}(y))^{\lambda}}\). Then \(K(u(x),v(y))\) is a generalized homogeneous function for parameters \((\lambda, -\lambda_{1}, -\lambda_{2})\), and \(\frac{n\lambda_{1}-\lambda _{2}\alpha}{p}+\frac{n\lambda_{2}-\lambda_{1}\beta}{q}=\lambda\lambda_{1}\lambda _{2}\) is equivalent to \(\frac{(-\lambda_{2})\alpha-n(-\lambda_{1})}{ p}+\frac{(-\lambda_{1})\beta-n(-\lambda_{2})}{q}=\lambda(-\lambda _{1})(-\lambda_{2})\). Further, we have \(\lambda-\frac{1}{\lambda_{2}}(\frac {n}{p}-\frac{\beta}{q})=\frac{1}{\lambda_{1}}(\frac{n}{q}-\frac{\alpha}{p})\), and \(\frac{n}{p}-\frac{\beta}{q}>0\) and \(\frac{n}{q}-\frac{\alpha }{p}>0\) when \(\alpha< n(p-1)\) and \(\beta< n(q-1)\). By Lemma 1 we have

$$\begin{aligned} W_{1} =& \int_{R_{+}^{n}} \bigl[v(t) \bigr]^{-\frac{\beta+n}{q}}K \bigl(1,v(t) \bigr)\,dt\\ =& \int _{R_{+}^{n}}\frac{1}{[1+(\sum_{i=1}^{n} b_{i}t_{i}^{\rho})^{\lambda_{2}/\rho }]^{\lambda}} \Biggl(\sum _{i=1}^{n} b_{i}t_{i}^{\rho}\Biggr)^{-\frac{\beta+n}{q\rho}}\, dt \\ =&\prod_{i=1}^{n} b_{i}^{-\frac{1}{\rho}} \int_{R_{+}^{n}}\frac{1}{[1+(\sum_{i=1}^{n}x_{i}^{\rho})^{\lambda_{2}/\rho}]^{\lambda}} \Biggl(\sum _{i=1}^{n}x_{i}^{\rho}\Biggr)^{-\frac{\beta+n}{q\rho}}\,dx \\ =&\prod_{i=1}^{n} b_{i}^{-\frac{1}{\rho}}\lim_{r\to+\infty} \int\cdots \int _{x_{i}>0,x_{1}^{\rho}+\cdots+x_{n}^{\rho}\le r^{\rho}}\frac{1}{[1+r^{\lambda _{2}}(\sum_{i=1}^{n}(\frac{x_{i}}{r})^{\rho})^{\lambda_{2}/\rho}]^{\lambda}} \\ & {}\times r^{-\frac{\beta+n}{q}} \Biggl(\prod_{i=1}^{n} \biggl(\frac{x_{i}}{r} \biggr)^{\rho}\Biggr)^{-\frac{\beta+n}{q\rho }}x_{1}^{1-1} \cdots x_{n}^{1-1}\,dx_{1}\cdots dx_{2} \\ =&\prod_{i=1}^{n} b_{i}^{-\frac{1}{\rho}}\lim_{r\to+\infty}r^{-\frac {\beta+n}{q}} \frac{r^{n}\varGamma^{n}(\frac{1}{\rho})}{\rho^{n}\varGamma(\frac {n}{\rho})} \int_{0}^{1}\frac{1}{(1+r^{\lambda_{2}}u^{\lambda_{2}/\rho})^{\lambda}}u^{-\frac {\beta+n}{q\rho}}u^{\frac{n}{\rho}-1}du \\ =&\prod_{i=1}^{n} b_{i}^{-\frac{1}{\rho}}\frac{\varGamma^{n}(\frac{1}{\rho })}{\rho^{n-1}\varGamma(\frac{n}{\rho})\lambda_{2}} \int_{0}^{\infty}\frac {1}{(1+t)^{\lambda}}t^{\frac{1}{\lambda_{2}}(\frac{n}{p}-\frac{\beta }{q})-1} \,dt \\ =&\prod_{i=1}^{n} b_{i}^{-\frac{1}{\rho}}\frac{\varGamma^{n}(\frac{1}{\rho })}{\lambda_{2}\rho^{n-1}\varGamma(\frac{n}{\rho})}B \biggl( \frac{1}{\lambda _{2}} \biggl(\frac{n}{p}-\frac{\beta}{q} \biggr), \lambda-\frac{1}{\lambda_{2}} \biggl(\frac {n}{p}-\frac{\beta}{q} \biggr) \biggr) \\ =&\prod_{i=1}^{n} b_{i}^{-\frac{1}{\rho}}\frac{\varGamma^{n}(\frac{1}{\rho })}{\lambda_{2}\rho^{n-1}\varGamma(\frac{n}{\rho})\varGamma(\lambda )}\varGamma \biggl( \frac{1}{\lambda_{2}} \biggl(\frac{n}{p}-\frac{\beta}{q} \biggr) \biggr)\varGamma \biggl(\frac{1}{\lambda_{1}} \biggl(\frac{n}{q}- \frac{\alpha}{p} \biggr) \biggr). \end{aligned}$$

In the same way, we get

$$\begin{aligned} W_{2}&= \int_{R_{+}^{n}} \bigl[u(x) \bigr]^{-\frac{\alpha+n}{p}}K \bigl(u(t),1 \bigr)\,dt\\&=\prod_{i=1}^{n} a_{i}^{-\frac{1}{\rho}}\frac{\varGamma^{n}(\frac{1}{\rho})}{\lambda _{1}\rho^{n-1}\varGamma(\frac{n}{\rho})\varGamma(\lambda)}\varGamma \biggl( \frac {1}{\lambda_{1}} \biggl(\frac{n}{q}-\frac{\alpha}{p} \biggr) \biggr)\varGamma \biggl(\frac{1}{\lambda _{2}} \biggl(\frac{n}{p}- \frac{\beta}{q} \biggr) \biggr).\end{aligned} $$

Thus

$$\begin{aligned} W_{1}^{\frac{1}{p}}W_{2}^{\frac{1}{q}}&= \Biggl( \prod_{i=1}^{n}a_{i}^{-\frac{1}{\rho }} \Biggr)^{\frac{1}{q}} \Biggl(\prod_{i=1}^{n}b_{i}^{-\frac{1}{\rho}} \Biggr)^{\frac{1}{p}} \frac{\varGamma^{n}(\frac{1}{\rho})}{\lambda_{1}^{\frac{1}{q}}\lambda _{2}^{\frac{1}{p}}\rho^{n-1}\varGamma(\lambda)\varGamma(\frac{n}{\rho })}\\&\quad\times\varGamma \biggl( \frac{1}{\lambda_{1}} \biggl(\frac{n}{q}-\frac{\alpha}{p} \biggr) \biggr)\varGamma \biggl(\frac{1}{\lambda_{2}} \biggl(\frac{n}{p}- \frac{\beta}{q} \biggr) \biggr).\end{aligned} $$

Hence Theorem 3 holds by Theorem 1. □

Corollary 1

Let\(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(n\ge1\), \(\rho>0\), \(\lambda>0\), \(\lambda_{1}>0\), \(\lambda_{2}>0\), \(u(x)=(\sum_{i=1}^{n} x_{i}^{\rho})^{\frac{1}{\rho }}\), and\(v(y)=(\sum_{i=1}^{n} y_{i}^{\rho})^{\frac{1}{\rho}}\). Then:

  1. (i)

    The operatorTdefined by

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

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

  2. (ii)

    WhenTis a bounded operator in\(L^{p}(R_{+}^{n})\), the operator norm ofTis

    $$\Vert T \Vert =\frac{\varGamma^{n}(\frac{1}{\rho})}{\rho^{n-1}\lambda_{1}^{\frac {1}{q}}\lambda_{2}^{\frac{1}{p}}\varGamma(\lambda)\varGamma(\frac{n}{\rho })}\varGamma \biggl(\frac{n}{\lambda_{1}q} \biggr)\varGamma \biggl(\frac{n}{\lambda_{2}p} \biggr). $$

Proof

The corollary follows from Theorems 2 and 3. □

Theorem 4

Let\(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(n\ge1\), \(\rho>0\), \(\lambda >0\), \(\lambda_{1}>0\), \(\lambda_{2}>0\), \(\alpha< n(p-1)\), \(\beta< n(q-1)\), \(u(x)=(\sum_{i=1}^{n} x_{i}^{\rho})^{1/\rho}\), and\(v(y)=(\sum_{i=1}^{n} y_{i}^{\rho})^{1/\rho}\). Then

  1. (i)

    There existsMsuch that

    $$\begin{aligned} \int_{R_{+}^{n}} \int_{R_{+}^{n}}\frac{1}{(\max\{u^{\lambda _{1}}(x),v^{\lambda_{2}}(y)\})^{\lambda}}f(x)g(y)\,dx\,dy\le M \Vert f \Vert _{p,u^{\alpha}(x)} \Vert g \Vert _{q,v^{\beta}(y)} \end{aligned}$$
    (3.2)

    if and only if\(\frac{n\lambda_{1}-\lambda_{2}\alpha}{p}+\frac{n\lambda _{2}-\lambda_{1}\beta}{q}=\lambda\lambda_{1}\lambda_{2}\), where\(f\in L^{p}_{u^{\alpha}(x)}(R_{+}^{n})\)and\(g\in L^{q}_{v^{\beta}(y)}(R_{+}^{n})\).

  2. (ii)

    If inequality (3.2) holds, then its best constant factor is

    $$\inf M=\frac{\varGamma^{n}(\frac{1}{\rho})}{\lambda_{1}^{\frac{1}{q}}\lambda _{2}^{\frac{1}{p}}\rho^{n-1}\varGamma(\frac{n}{\rho})} \biggl[ \biggl(\frac{1}{\lambda _{1}} \biggl( \frac{n}{q}-\frac{\alpha}{p} \biggr) \biggr)^{-1} + \biggl(\frac{1}{\lambda_{2}} \biggl(\frac{n}{p}-\frac{\beta}{q} \biggr) \biggr)^{-1} \biggr]. $$

Proof

Set \(K(u(x), v(y))=\frac{1}{(\max\{u^{\lambda_{1}}(x),v^{\lambda_{2}}(y)\} )^{\lambda}}\). Then \(K(u(x),v(y))\) is a generalized homogeneous function for parameters \((\lambda, -\lambda_{1}, -\lambda_{2})\). By Lemma 2 we get

$$\begin{aligned} W_{1} =& \int_{R_{+}^{n}}K \bigl(1,v(t) \bigr) \bigl[v(t) \bigr]^{-\frac{\beta+n}{q}} \,dt \\ =& \int_{v(t)\le1} \bigl[v(t) \bigr]^{-\frac{\beta+n}{q}}\,dt+ \int _{v(t)>1} \bigl[v(t) \bigr]^{-\lambda\lambda_{2}-\frac{\beta+n}{q}}\,dt \\ =&\frac{\varGamma^{n}(\frac{1}{\rho})}{\lambda_{2}\rho^{n-1}\varGamma(\frac {n}{\rho})} \biggl(\frac{1}{\lambda_{2}} \biggl( \frac{n}{p}- \frac{\beta}{q} \biggr) \biggr)^{-1}+ \frac{\varGamma^{n}(\frac{1}{\rho})}{\lambda_{2}\rho^{n-1}\varGamma(\frac {n}{\rho})} \biggl(\frac{1}{\lambda_{1}} \biggl(\frac{n}{q}- \frac{\alpha}{p} \biggr) \biggr)^{-1} \\ =&\frac{\varGamma^{n}(\frac{1}{\rho})}{\lambda_{2}\rho^{n-1}\varGamma(\frac {n}{\rho})} \biggl[ \biggl(\frac{1}{\lambda_{2}} \biggl( \frac{n}{p}-\frac{\beta }{q} \biggr) \biggr)^{-1}+ \biggl(\frac{1}{\lambda_{1}} \biggl(\frac{n}{q}-\frac{\alpha}{p} \biggr) \biggr)^{-1} \biggr]. \end{aligned}$$

Similarly, we obtain

$$\begin{aligned} W_{2} =& \int_{R_{+}^{n}}K \bigl(u(t),1 \bigr) \bigl[u(t) \bigr]^{-\frac{\alpha+n}{p}} \,dt \\ =&\frac{\varGamma^{n}(\frac{1}{\rho})}{\lambda_{1}\rho^{n-1}\varGamma(\frac {n}{\rho})} \biggl[ \biggl(\frac{1}{\lambda_{1}} \biggl( \frac{n}{q}-\frac{\alpha }{p} \biggr) \biggr)^{-1}+ \biggl(\frac{1}{\lambda_{2}} \biggl(\frac{n}{p}-\frac{\beta}{q} \biggr) \biggr)^{-1} \biggr]. \end{aligned}$$

Then we have

$$W_{1}^{\frac{1}{p}}W_{2}^{\frac{1}{q}}= \frac{\varGamma^{n}(\frac{1}{\rho })}{\lambda_{1}^{\frac{1}{q}}\lambda_{2}^{\frac{1}{p}}\rho^{n-1}\varGamma (\frac{n}{\rho})} \biggl[ \biggl(\frac{1}{\lambda_{1}} \biggl( \frac{n}{q}-\frac{\alpha}{p} \biggr) \biggr)^{-1}+ \biggl(\frac{1}{\lambda_{2}} \biggl(\frac {n}{p}-\frac{\beta}{q} \biggr) \biggr)^{-1} \biggr]. $$

In summary, Theorem 4 holds by Theorem 1. □

Corollary 2

Let\(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(n\ge1\), \(\rho>0\), \(\lambda>0\), \(\lambda _{1}>0\), \(\lambda_{2}>0\), \(u(x)=(\sum_{i=1}^{n} x_{i}^{\rho})^{\frac{1}{\rho}}\), and\(v(y)=(\sum_{i=1}^{n} y_{i}^{\rho})^{\frac{1}{\rho}}\). Then

  1. (i)

    The operatorTdefined by

    $$T(f) (y)= \int_{R_{+}^{n}}\frac{1}{\max\{u^{\lambda_{1}}(x),v^{\lambda_{2}}(y)\} )^{\lambda}}f(x)\,dx, y\in R_{+}^{n}, $$

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

  2. (ii)

    WhenTis a bounded operator in\(L^{p}(R_{+}^{n})\), the operator norm ofTis

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

Proof

The corollary follows from Theorems 2 and 4. □

References

  1. 1.

    Hong, Y.: A Hilbert-type integral inequality with quasi-homogeneous kernel and several functions. Acta Math. Sin. Chin. Ser., 57, 833–840 (2014)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    He, B., Cao, J.F., Yang, B.C.: A brand new multiple Hilbert-type integral inequality. Acta Math. Sin. Chin. Ser. 58, 661–672 (2015)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Perić, I., Vuković, P.: Hardy–Hilbert’s inequalities with a general homogeneous kernel. Math. Inequal. Appl. 12, 525–536 (2009)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Yang, B.C.: On an extension of Hilbert’s integral inequality with some parameters. Aust. J. Math. Anal. Appl. 1(1), 1–8 (2004)

    MathSciNet  Google Scholar 

  5. 5.

    Hong, Y., Wen, Y.M.: A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel and the best constant factor. Chin. Ann. Math. 37A(3), 329–336 (2017)

    MATH  Google Scholar 

  6. 6.

    Rassias, M.Th., Yang, B.C.: On a Hardy–Hilbert-type inequality with a general homogeneous kernel. Int. J. Nonlinear Anal. Appl. 7(1), 249–269 (2016)

    MATH  Google Scholar 

  7. 7.

    Chen, Q., Shi, Y.P., Yang, B.C.: A relation between two simple Hardy–Mulholland-type inequalities with parameters. J. Inequal. Appl. 2016, Article ID 75 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Yang, B.C., Qiang, Q.: On a more accurate Hardy–Mulholland-type inequality. J. Inequal. Appl. 2016, Article ID 82 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Yang, B.C.: On a more accurate multidimensional Hilbert-type inequality with parameters. Math. Inequal. Appl. 18, 429–441 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Zhong, W.Y., Yang, B.C.: On multiple Hardy–Hilbert’s integral inequality with kernel. J. Inequal. Appl. 2007, Article ID 27962 (2007)

    MATH  Article  Google Scholar 

  11. 11.

    Xin, D.M., Yang, B.C., Chen, Q.: A discrete Hilbert-type inequality in the whole plane. J. Inequal. Appl. 2016, Article ID 133 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Kuang, J.C.: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004)

    Google Scholar 

  13. 13.

    Yang, B.C., Chen, Q.: On a Hardy–Hilbert-type inequality with parameters. J. Inequal. Appl. 2015, Article ID 339 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Yang, B.C., Chen, Q.: A new extension of Hardy–Hilbert’s inequality in the whole plane. J. Funct. Spaces 2016, Article ID 9197476 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Hong, Y.: On multiple Hardy–Hilbert integral inequalities with some parameters. J. Inequal. Appl. 2006, Article ID 94960 (2006)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Huang, Q.L., Yang, B.C.: On a multiple Hilbert-type integral operator and applications. J. Inequal. Appl. 2009, Article ID 192197 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Fichtigoloz, G.H.: A Course in Differential and Integral Calculus. Renmin Jiaoyu Publishers, Beijing (1957) (in Chinese)

    Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous reviewers for their insightful and detailed comments on the paper.

Availability of data and materials

All data generated or analyzed during this study are included in this paper.

Funding

The first author was supported by the National Natural Science Foundation of China (No. 61300204). The second author was supported by the National Natural Science Foundation of China (No. 11401113) and the Characteristic Innovation Project (Natural Science) of Guangdong Province (No. 2017KTSCX133).

Author information

Affiliations

Authors

Contributions

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

Corresponding author

Correspondence to Jianquan Liao.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hong, Y., Liao, J., Yang, B. et al. A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications. J Inequal Appl 2020, 140 (2020). https://doi.org/10.1186/s13660-020-02401-0

Download citation

MSC

  • 26D15

Keywords

  • Generalized homogeneous kernel
  • Hilbert-type multiple integral inequality
  • Necessary and sufficient condition
  • The best constant factor
  • Bounded operator
  • Operator norm