构建一类非齐次核的Hilbert型积分不等式的等价参数条件

洪勇; 陈强

doi:10.6054/j.jscnun.2020085

构建一类非齐次核的Hilbert型积分不等式的等价参数条件

洪勇^1, ,,
陈强²

1.
广东白云学院数学教研室，广州 510450
2.
广东第二师范学院计算机科学系，广州 510303

基金项目:

国家自然科学基金项目 61640222

详细信息

通讯作者:
洪勇，教授，Email:hongyonggdcc@yeah.net

中图分类号: O178
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出版历程
- 收稿日期: 2020-03-25
- 网络出版日期: 2020-11-02
- 刊出日期: 2020-10-24

Equivalent Parameter Conditions for the Construction of Hilbert-type Integral Inequalities with a Class of Non-homogeneous Kernels

HONG Yong^1, ,,
CHEN Qiang²

1.
Department of Mathematics, Guangdong Baiyun University, Guangzhou 510450, China
2.
Department of Computer Science, Guangdong University of Education, Guangzhou 510303, China

摘要

摘要: 利用权函数方法，讨论了非齐次核K(x, y)=φ^λ(x^λ₁y^λ₂)φ′(x^λ₁y^λ₂)的Hilbert型积分不等式成立的等价参数条件及最佳常数因子，得到了构建此类Hilbert型不等式的充分必要条件及最佳常数因子的表达公式；对一些具体的非齐次核，得到了若干具有最佳常数因子的新的Hilbert型不等式; 最后，讨论了相应奇异积分算子的有界性及其范数.
- 非齐次核 /
- Hilbert型积分不等式 /
- 等价参数条件 /
- 有界算子 /
- 算子范数
Abstract: Using the weight function methods, the equivalent parameter conditions for the validity of Hilbert-type integral inequalities with non-homogeneous kernel K(x, y)=φ^λ(x^λ₁y^λ₂)φ′(x^λ₁y^λ₂) and the best constant factor are discussed. The necessary and sufficient conditions for constructing such Hilbert-type inequalities and the formula for expressing the best constant factor are obtained. Many new Hilbert-type integral inequalities with some specific non-homogeneous kernels and the best constant factors are also obtained. Finally, the norm and boundedness of corresponding singular integral operators are discussed.
- non-homogeneous kernel /
- Hilbert-type integral inequality /
- equivalent parameter condition /
- bounded operator /
- operator norm

HTML全文

设r>0，α∈ $\mathbb{R}$ ，定义函数空间：

$L_r^\alpha (0, + \infty ) = \{ f(x) \ge 0:{\left\| f \right\|_{r,\alpha }} = {\left( {\int_0^{ + \infty } {{x^\alpha }} {f^r}(x){\rm{d}}x} \right)^{1/r}} < + \infty \} .$

若 $\frac{1}{p} + \frac{1}{q} = 1\; (p > 1)$ ，K(x, y)≥0，f(x)∈L_p^α(0, +∞)，g(y)∈L_q^β(0, +∞)，称

$\int_0^{ + \infty } {\int_0^{ + \infty } K } (x,y)f(x)g(y){\rm{d}}x{\rm{d}}y \le M{\left\| f \right\|_{p,\alpha }}{\left\| g \right\|_{q,\beta }}$

(1)

为以K(x, y)为核的Hilbert型积分不等式.

对于齐次核的情况，选取适当的搭配参数，已获得了许多具有最佳常数因子的Hilbert型不等式^[1-10]，研究结果表明：对于不同类型的核，最佳的搭配参数有不同的规律，找出其规律具有重要意义.但在已有研究中，搭配参数的选取大多数是凭研究者的经验得到的^[11-15].

设λ>0，λ₁>0，λ₂>0，在K(x, y)=φ^λ(x^λ₁y^λ₂)×φ′(x^λ₁y^λ₂)的非齐次核情况下，本文讨论不等式(1)成立的等价参数条件，并讨论式(1)成立时的最佳常数因子.

1. 主要结论及证明

首先给出本文主要定理证明时需用的关于权函数的引理:

引理1 设 $\frac{1}{p} + \frac{1}{q} = 1\; (p > 1)$ ，λ₁>0，λ₂>0，φ(t)非负可导，φ′(t)保持定号，φ^λ+1(+∞)及φ^λ+1(0)存在，则

$\begin{array}{*{20}{l}} {{W_1} = \int_0^{ + \infty } {{\varphi ^\lambda }} ({t^{{\lambda _2}}})|{\varphi ^\prime }({t^{{\lambda _2}}})|{t^{{\lambda _2} - 1}}{\rm{d}}t = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{{\lambda _2}(\lambda + 1)}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)|,} \end{array}$

$\begin{array}{*{20}{l}} {{W_2} = \int_0^{ + \infty } {{\varphi ^\lambda }} ({t^{{\lambda _1}}})|{\varphi ^\prime }({t^{{\lambda _1}}})|{t^{{\lambda _1} - 1}}{\rm{d}}t = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{{\lambda _1}(\lambda + 1)}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)|,} \end{array}$

${{\omega _1}(x) = \int_0^{ + \infty } {{\varphi ^\lambda }} ({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{\varphi ^\prime }({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{y^{{\lambda _2} - 1}}{\rm{d}}y = {x^{ - {\lambda _1}}}{W_1},}$

${{\omega _2}(y) = \int_0^{ + \infty } {{\varphi ^\lambda }} ({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{\varphi ^\prime }({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{x^{{\lambda _1} - 1}}{\rm{d}}x = {y^{ - {\lambda _2}}}{W_2}.}$

证明先证φ′(t)≥0的情形，此时φ(t)在(0, +∞)上递增.作变换t^λ₂=u，有

$\begin{array}{*{20}{l}} {{W_1} = \frac{1}{{{\lambda _2}}}\int_0^{ + \infty } {{\varphi ^\lambda }} (u){\varphi ^\prime }(u){u^{({\lambda _2} - 1)/{\lambda _2} + 1/{\lambda _2} - 1}}{\rm{d}}t = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{{\lambda _2}}}\int_0^{ + \infty } {{\varphi ^\lambda }} (u){\varphi ^\prime }(u){\rm{d}}u = \frac{1}{{{\lambda _2}}}\int_0^{ + \infty } {{\varphi ^\lambda }} (u){\rm{d}}\varphi (u) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{{\lambda _2}(\lambda + 1)}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)|.} \end{array}$

同理可证W₂的情形.作变换x^λ₁y^λ₂=t^λ₂，有

$\begin{array}{*{20}{c}} {{\omega _1}(x) = {x^{ - {\lambda _1}/{\lambda _2}}}\int_0^{ + \infty } {{\varphi ^\lambda }} ({t^{{\lambda _2}}})|{\varphi ^\prime }({t^{{\lambda _2}}})|{{({x^{ - {\lambda _1}/{\lambda _2}}}t)}^{{\lambda _2} - 1}}{\rm{d}}t = }\\ {{x^{ - {\lambda _1}({\lambda _2} - 1)/{\lambda _2} - {\lambda _1}/{\lambda _2}}}\int_0^{ + \infty } {{\varphi ^\lambda }} ({t^{{\lambda _2}}}){\varphi ^\prime }({t^{{\lambda _2}}}){t^{{\lambda _2} - 1}}{\rm{d}}t = {x^{ - {\lambda _1}}}{W_1}.} \end{array}$

同理可证ω₂(y)的情形.

类似地可证明φ′(t)≤0的情形.证毕.

下面给出本文的主要结论.

定理1 设 $\frac{1}{p} + \frac{1}{q} = 1\; (p > 1)$ ，λ₁>>0，λ₂>>0，φ(t)非负可导，φ′(t)保持定号，φ^λ+1(+∞)及φ^λ+1(0)存在, 则

(ⅰ)当且仅当 $\frac{{{\lambda _1} + \alpha {\lambda _2}}}{p} = \frac{{{\lambda _2} + \beta {\lambda _1}}}{q}$ 时，存在常数M>0，使得

$\begin{array}{*{20}{l}} {\int_0^{ + \infty } {\int_0^{ + \infty } {{\varphi ^\lambda }} } ({x^\lambda }{y^\lambda })|{\varphi ^\prime }({x^\lambda }{y^\lambda })|f(x)g(y){\rm{d}}x{\rm{d}}y \le }\\ {\quad M{{\left\| f \right\|}_{p,\alpha }}{{\left\| g \right\|}_{q,\beta }},} \end{array}$

(2)

其中, f(x)∈L_p^α(0, +∞)，g(y)∈L_q^β(0, +∞).

(ⅱ)当 $\frac{{{\lambda _1} + \alpha {\lambda _2}}}{p} = \frac{{{\lambda _2} + \beta {\lambda _1}}}{q}$ 时，式(2)的最佳常数因子为：

${\rm{inf}}{\kern 1pt} {\kern 1pt} M = \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)|.$

证明记K(x, y)=φ^λ(x^λ₁y^λ₂)φ′(x^λ₁y^λ₂)， $\frac{{{\lambda _1} + \alpha {\lambda _2}}}{p} = \frac{{{\lambda _2} + \beta {\lambda _1}}}{q}{\rm{ = }}c$ .

(ⅰ)设式(2)成立.若c < 0，取ε=－c/(2λ₁λ₂)>0，令

${f(x) = \left\{ {\begin{array}{*{20}{l}} {{x^{( - \alpha - 1 - {\lambda _1}\varepsilon )/p}}}&{(x \ge 1),}\\ 0&{(0 < x < 1),} \end{array}} \right.}$

${g(y) = \left\{ {\begin{array}{*{20}{l}} {{y^{( - \beta - 1 + {\lambda _2}\varepsilon )/q}}}&{(0 < y \le 1),}\\ 0&{(y > 1),} \end{array}} \right.}$

可得

$\begin{array}{*{20}{l}} {M{{\left\| f \right\|}_{p,\alpha }}{{\left\| g \right\|}_{q,\beta }} = M{{\left( {\int_1^{ + \infty } {{x^{ - 1 - {\lambda _1}\varepsilon }}} {\rm{d}}x} \right)}^{1/p}}{{\left( {\int_0^1 {{y^{ - 1 + {\lambda _2}\varepsilon }}} {\rm{d}}y} \right)}^{1/q}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{M}{\varepsilon } \cdot \frac{1}{{\lambda _1^{1/p}\lambda _2^{1/q}}} = - \frac{{2M}}{c}\lambda _1^{1/q}\lambda _2^{1/p},} \end{array}$

$\begin{array}{l} \begin{array}{*{20}{c}} {\int_0^{ + \infty } {\int_0^{ + \infty } {{\varphi ^\lambda }} } ({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{\varphi ^\prime }({x^{{\lambda _1}}}{y^{{\lambda _2}}})|f(x)g(y){\rm{d}}x{\rm{d}}y = }\\ \ \ \ \ {\int_0^1 {{y^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}} \left( {\int_1^{ + \infty } K (x,y){x^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}{\rm{d}}x} \right){\rm{d}}y = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {\int_0^1 {{y^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}} \left( {\int_1^{ + \infty } K ({y^{{\lambda _2}/{\lambda _1}}}x,1){x^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}{\rm{d}}x} \right){\rm{d}}y = }\\ {\int_0^1 {{y^{ - \frac{{\beta + 1 - {\lambda _2}\varepsilon }}{q} + \frac{{{\lambda _2}}}{{{\lambda _1}}} \cdot \frac{{\alpha + 1 + {\lambda _1}\varepsilon }}{p} - \frac{{{\lambda _2}}}{{{\lambda _1}}}}}} \left( {\int_{{y^\lambda }{2^\prime }{\lambda _1}}^{ + \infty } K (t,1){t^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}{\rm{d}}t} \right){\rm{d}}y \ge }\\ {\int_0^1 {{y^{ - 1 + c/{\lambda _1} + {\lambda _2}\varepsilon }}} \left( {\int_1^{ + \infty } K (t,1){t^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}{\rm{d}}t} \right){\rm{d}}y = }\\ {\int_0^1 {{y^{ - 1 + c/(2{\lambda _1})}}} {\rm{d}}y\int_1^{ + \infty } K (t,1){t^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}{\rm{d}}t,} \end{array} \end{array}$

则有

$\begin{array}{*{20}{l}} {\int_0^1 {{y^{ - 1 + c/(2{\lambda _1})}}} {\rm{d}}y\int_1^{ + \infty } K (t,1){t^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}{\rm{d}}t \le }\\ {\quad - \frac{{2M}}{c}\lambda _1^{1/q}\lambda _2^{1/p} < + \infty .} \end{array}$

(3)

由于 $\frac{c}{{2{\lambda _1}}} < 0$ ，则 $\int_0^1 {{y^{ - 1 + c/\left({2{\lambda _1}} \right)}}} {\rm{d}}y = + \infty$ ，故式(3)是一个矛盾.从而知c≥0.

若c>0，取ε=c/(2λ₁λ₂)>0，令

${f(x) = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {x^{( - \alpha - 1 + {\lambda _1}\varepsilon )/p}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (0 < x \le 1),\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x > 1), \end{array} \end{array}} \right.}$

$g(y) = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {y^{( - \beta - 1 - {\lambda _2}\varepsilon )/q}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (y \ge 1),\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (0 < y < 1), \end{array} \end{array}} \right.$

可得

$M{\left\| f \right\|_{p,\alpha }}{\left\| g \right\|_{q,\beta }} = \frac{{2M}}{c}\lambda _1^{1/q}\lambda _2^{1/p},$

$\begin{array}{l} \begin{array}{*{20}{c}} {\int_0^{ + \infty } {\int_0^{ + \infty } {{\varphi ^\lambda }} } ({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{\varphi ^\prime }({x^{{\lambda _1}}}{y^{{\lambda _2}}})|f(x)g(y){\rm{d}}x{\rm{d}}y = }\\ \ \ \ \ {\int_1^{ + \infty } {{y^{ - (\beta + 1 + {\lambda _2}\varepsilon )/q}}} \left( {\int_0^1 K (x,y){x^{ - (\alpha + 1 - {\lambda _1}\varepsilon )/p}}{\rm{d}}x} \right){\rm{d}}y = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_1^{ + \infty } {{y^{ - (\beta + 1 + {\lambda _2}\varepsilon )/q}}} \left( {\int_0^1 K ({y^{{\lambda _2}/{\lambda _1}}}x,1){x^{ - (\alpha + 1 - {\lambda _1}\varepsilon )/p}}{\rm{d}}x} \right){\rm{d}}y = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_1^{ + \infty } {{y^{ - \frac{{\beta + 1 + {\lambda _2}\varepsilon }}{q} + \frac{\lambda }{{{\lambda _1}}} \cdot \frac{{\alpha + 1 - {\lambda _1}\varepsilon }}{p} - \frac{{{\lambda _2}}}{{{\lambda _1}}}}}} \left( {\int_0^{{y^\lambda }{2^\lambda }1} K (t,1){t^{ - \frac{{\alpha + 1 - {\lambda _1}\varepsilon }}{p}}}{\rm{d}}t} \right){\rm{d}}y \ge \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {\int_1^{ + \infty } {{y^{ - 1 + c/{\lambda _1} - {\lambda _2}\varepsilon }}} \left( {\int_0^1 K (t,1){t^{ - (\alpha + 1 - {\lambda _1}\varepsilon )/p}}{\rm{d}}t} \right){\rm{d}}y = }\\ {\int_1^{ + \infty } {{y^{ - 1 + c/(2{\lambda _1})}}} {\rm{d}}y\int_0^1 K (t,1){t^{ - (\alpha + 1 - {\lambda _1}\varepsilon )/p}}{\rm{d}}t,} \end{array} \end{array}$

则有

$\int_1^{ + \infty } {{y^{ - 1 + c/(2{\lambda _1})}}} {\rm{d}}y\int_0^1 K (t,1){t^{ - (\alpha + 1 - {\lambda _1}\varepsilon )/p}}{\rm{d}}t \le \frac{{2M}}{c}\lambda _1^{1/q}\lambda _2^{1/p} < + \infty .$

(4)

由于 $\frac{c}{{2{\lambda _1}}} > 0$ ，则 $\int_1^{ + \infty } {{y^{ - 1 + c/\left({2{\lambda _1}} \right)}}} {\rm{d}}y = + \infty$ ，故式(4)是一个矛盾.所以c≤0.

综上可知c=0，即 $\frac{{{\lambda _1} + \alpha {\lambda _2}}}{p} = \frac{{{\lambda _2} + \beta {\lambda _1}}}{q}.$ .

反之，设 $\frac{{{\lambda _1} + \alpha {\lambda _2}}}{p} = \frac{{{\lambda _2} + \beta {\lambda _1}}}{q}$ ，则 $\frac{{\alpha + 1}}{q} + \frac{{{\lambda _1}}}{{{\lambda _2}}}\left({\frac{{\beta + 1}}{q} - 1} \right) = \alpha, \frac{{\beta + 1}}{p} + \frac{{{\lambda _2}}}{{{\lambda _1}}}\left({\frac{{\alpha + 1}}{p} - 1} \right) = \beta$ .于是, 由Hölder不等式及引理1，有

$\begin{array}{l} \int_0^{ + \infty } {\int_0^{ + \infty } {{\varphi ^\lambda }} } ({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{\varphi ^\prime }({x^{{\lambda _1}}}{y^{{\lambda _2}}})|f(x)g(y){\rm{d}}x{\rm{d}}y = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_0^{ + \infty } {\int_0^{ + \infty } {\left( {\frac{{{x^{(\alpha + 1)/(pq)}}}}{{{y^{(\beta + 1)/(pq)}}}}f(x)} \right)\left( {\frac{{{y^{(\beta + 1)/(pq)}}}}{{{x^{(\alpha + 1)/(pq)}}}}g(y)} \right)K(x,y)} } {\rm{d}}x{\rm{d}}y \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{{\left( {\int_0^{ + \infty } {\int_0^{ + \infty } {\frac{{{x^{(\alpha + 1)/q}}}}{{{y^{(\beta + 1)/q}}}}} } {f^p}(x)K(x,y){\rm{d}}x{\rm{d}}y} \right)}^{1/p}} \times }\\ {{{\left( {\int_0^{ + \infty } {\int_0^{ + \infty } {\frac{{{y^{(\beta + 1)/p}}}}{{{x^{(\alpha + 1)/p}}}}} } {g^q}(y)K(x,y){\rm{d}}x{\rm{d}}y} \right)}^{1/q}} = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{{\left( {\int_0^{ + \infty } {{x^{(\alpha + 1)/q}}} {f^p}(x){\omega _1}(x){\rm{d}}x} \right)}^{1/p}} \times }\\ {{{\left( {\int_0^{ + \infty } {{y^{(\beta + 1)/p}}} {g^q}(y){\omega _2}(y){\rm{d}}y} \right)}^{1/q}} = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {W_1^{1/p}W_1^{1/q}{{\left( {\int_0^{ + \infty } {{x^{\frac{{\alpha + 1}}{q} + \frac{\lambda }{{{\lambda _2}}}\left( {\frac{{\beta + 1}}{q} - 1} \right)}}} {f^p}(x){\rm{d}}x} \right)}^{1/p}} \times }\\ {{{\left( {\int_0^{ + \infty } {{y^{\frac{{\beta + 1}}{p} + \frac{\lambda }{{{\lambda _1}}}\left( {\frac{{\alpha + 1}}{p} - 1} \right)}}} {g^q}(y)dy} \right)}^{1/q}} = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {W_1^{1/p}W_1^{1/q}{{\left( {\int_0^{ + \infty } {{x^\alpha }} {f^p}(x){\rm{d}}x} \right)}^{1/p}}{{\left( {\int_0^{ + \infty } {{y^\beta }} {g^q}(y){\rm{d}}y} \right)}^{1/q}} = }\\ {\frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)|{{\left\| f \right\|}_{p,\alpha }}{{\left\| g \right\|}_{q,\beta }}.} \end{array} \end{array}$

任取 $M \ge \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}\left| {{\varphi ^{\lambda + 1}}(+ \infty) - {\varphi ^{\lambda + 1}}(0)} \right|$ ，式(2)都成立.

(ⅱ)当 $\frac{{{\lambda _1} + \alpha {\lambda _2}}}{p} = \frac{{{\lambda _2} + \beta {\lambda _1}}}{q}$ 时，设式(2)的最佳常数因子为M₀，则由(i)的证明可知：

$\begin{array}{l} \begin{array}{*{20}{r}} {{M_0} \le \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)|,}\\ {\int_0^{ + \infty } {\int_0^{ + \infty } {{\varphi ^\lambda }} } ({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{\varphi ^\prime }({x^{{\lambda _1}}}{y^{{\lambda _2}}})|f(x)g(y){\rm{d}}x{\rm{d}}y \le } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {M_0}{\left\| f \right\|_{p,\alpha }}{\left\| g \right\|_{q,\beta }}. \end{array}$

取充分小的ε>0及足够大的自然数n，令

${f(x) = \left\{ {\begin{array}{*{20}{l}} {{x^{( - \alpha - 1 - {\lambda _1}\varepsilon )/p}}}&{(x \ge 1),}\\ 0&{(0 < x < 1),} \end{array}} \right.}$

${g(y) = \left\{ {\begin{array}{*{20}{l}} {{y^{( - \beta - 1 + {\lambda _2}\varepsilon )/q}}}&{(0 < y \le n),}\\ 0&{(y > n),} \end{array}} \right.}$

可得

${M_0}{\left\| f \right\|_{p,\alpha }}{\left\| g \right\|_{q,\beta }} = \frac{{{M_0}}}{{\varepsilon \lambda _1^{1/p}\lambda _2^{1/q}}}{n^{{\lambda _2}\varepsilon /q}},$

$\begin{array}{l} \begin{array}{*{20}{l}} {\int_0^{ + \infty } {\int_0^{ + \infty } {{\varphi ^\lambda }} } ({x^{{\lambda _1}}}{y^{{\lambda _2}}})|{\varphi ^\prime }({x^{{\lambda _1}}}{y^{{\lambda _2}}})|f(x)g(y){\rm{d}}x{\rm{d}}y = }\\ {\quad \int_1^{ + \infty } {{x^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}} \left( {\int_1^n K (x,y){y^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}{\rm{d}}y} \right){\rm{d}}x = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {\int_1^{ + \infty } {{x^{ - (\alpha + 1 + {\lambda _1}\varepsilon )/p}}} \left( {\int_0^n K (1,{x^{{\lambda _1}/{\lambda _2}}}y){y^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}{\rm{d}}y} \right){\rm{d}}x = }\\ {\int_1^{ + \infty } {{x^{ - \frac{{\alpha + 1 + {\lambda _1}\varepsilon }}{p} + \frac{{{\lambda _1}}}{{{\lambda _2}}} \cdot \frac{{\beta + 1 - {\lambda _2}\varepsilon }}{q} - \frac{{{\lambda _1}}}{{{\lambda _2}}}}}} \left( {\int_0^{n{x^{{\lambda _1}/{\lambda _2}}}} K (1,u){u^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}{\rm{d}}u} \right){\rm{d}}x \ge } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {\int_1^{ + \infty } {{x^{ - 1 - {\lambda _1}\varepsilon }}} {\rm{d}}x\int_0^n K (1,u){u^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}{\rm{d}}u = }\\ {\frac{1}{{{\lambda _1}\varepsilon }}\int_0^n {{\varphi ^\lambda }} ({u^{{\lambda _2}}})|{\varphi ^\prime }({u^{{\lambda _2}}})|{u^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}{\rm{d}}u,} \end{array} \end{array}$

则有

$\frac{1}{{{\lambda _1}}}\int_0^n {{\varphi ^\lambda }} ({u^{{\lambda _2}}})|{\varphi ^\prime }({u^{{\lambda _2}}})|{u^{ - (\beta + 1 - {\lambda _2}\varepsilon )/q}}{\rm{d}}u \le \frac{{{M_0}}}{{\lambda _1^{1/p}\lambda _2^{1/q}}}{n^{({\lambda _2}\varepsilon )/q}}.$

先令ε→0⁺，再令n→+∞，得

$\frac{1}{{{\lambda _1}}}\int_0^{ + \infty } {{\varphi ^\lambda }} ({u^{{\lambda _2}}})|{\varphi ^\prime }({u^{{\lambda _2}}})|{u^{ - (\beta + 1)/q}}{\rm{d}}u \le \frac{{{M_0}}}{{\lambda _1^{1/p}\lambda _2^{1/q}}}.$

再由引理1可得

$\frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)| \le {M_0}.$

于是可知式(2)的最佳常数因子为:

${M_0} = \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}|{\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)|.$

证毕.

在定理1中，取φ(t)=arccot(t)，则 $\varphi '\left(t \right) = - \frac{1}{{1 + {t^2}}} < 0$ .根据λ>0，有

${\varphi ^{\lambda + 1}}( + \infty ) - {\varphi ^{\lambda + 1}}(0)| = \left| {0 - {{\left( {\frac{\pi }{2}} \right)}^{\lambda + 1}}} \right| = {\left( {\frac{\pi }{2}} \right)^{\lambda + 1}}.$

于是可得：

推论1 设 $\frac{1}{p} + \frac{1}{q} = 1\; (p > 1)$ ，λ>0，λ₁>>0，λ₂>>0，则

(ⅰ)当且仅当 $\frac{{{\lambda _1} + \alpha {\lambda _2}}}{p} = \frac{{{\lambda _2} + \beta {\lambda _1}}}{q}$ 时，存在常数M>0，使

$\begin{array}{l} \int_0^{ + \infty } {\int_0^{ + \infty } {\frac{{ {\rm{arccot}}{ ^\lambda }({x^{{\lambda _1}}}{y^{{\lambda _2}}})}}{{1 + {{({x^{{\lambda _1}}}{y^{{\lambda _2}}})}^2}}}} } f(x)g(y){\rm{d}}x{\rm{d}}y \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} M{\left\| f \right\|_{p,\alpha }}{\left\| g \right\|_{q,\beta }}, \end{array}$

(5)

其中，f(x)∈L_p^α(0, +∞)，g(y)∈L_q^β(0, +∞).

(ⅱ)当式(5)成立时，其最佳常数因子为：

${\rm{inf}}{\kern 1pt} {\kern 1pt} M = \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}{\left( {\frac{\pi }{2}} \right)^{\lambda + 1}}.$

设双曲余切函数 $t = \frac{{{{\rm{e}}^x} + {{\rm{e}}^{ - x}}}}{{{{\rm{e}}^x} - {{\rm{e}}^{ - x}}}}\; (x > 0, t > 1)$ 的反双曲余切函数为x=arcoth(t) (t>1, x>0)，则 $\frac{{{\rm{d}}x}}{{{\rm{d}}t}} = \frac{1}{{1 - {t^2}}}$ < 0 (t>1)，arcoth(a)= $\frac{1}{2}\ln \frac{{a + 1}}{{a - 1}}(a > 1)$ ，arcoth(+∞)=0.于是，在定理1中取φ(t)=arcoth(t+a)，可得

推论2 设 $\frac{1}{p} + \frac{1}{q} = 1\; (p > 1)$ ，λ₁>0，λ₂>0，a>1，则

(ⅰ)当且仅当λ₁+αλ₂p=λ₂+βλ₁q时，存在常数M>0，使得

$\begin{array}{*{20}{l}} {\int_0^{ + \infty } {\int_0^{ + \infty } {\frac{{ {\rm{arcoth}}{ ^\lambda }({x^{{\lambda _1}}}{y^{{\lambda _2}}} + a)}}{{{{({x^{{\lambda _1}}}{y^{{\lambda _2}}} + a)}^2} - 1}}} } f(x)g(y){\rm{d}}x{\rm{d}}y \le }\\ {\quad M{{\left\| f \right\|}_{p,\alpha }}{{\left\| g \right\|}_{q,\beta }},} \end{array}$

(6)

其中，f(x)∈L_p^α(0, +∞)，g(y)∈L_q^β(0, +∞).

(ⅱ)当式(6)成立时，其最佳常数因子为：

${\rm{inf}}{\kern 1pt} {\kern 1pt} M = \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}{\left( {\frac{1}{2}{\rm{ln}}\frac{{a + 1}}{{a - 1}}} \right)^{\lambda + 1}}.$

2. 在算子理论中的应用

设K(x, y)≥0, 定义奇异积分算子：

${{T_1}(f)(y) = \int_0^{ + \infty } K (x,y)f(x){\rm{d}}x\quad (y > 0),}$

${{T_2}(g)(x) = \int_0^{ + \infty } K (x,y)g(y){\rm{d}}y\quad (x > 0),}$

则根据Hilbert型不等式的基本理论，式(2)等价于：

${\left\| {{T_1}(f)} \right\|_{p,\beta (1 - p)}} \le M{\left\| f \right\|_{p,\alpha }},{\left\| {{T_2}(g)} \right\|_{q,\alpha (1 - q)}} \le M{\left\| g \right\|_{q,\beta }}.$

则由推论1和推论2，可得

命题1 设 $\frac{1}{p} + \frac{1}{q} = 1\; (p > 1)$ ，λ>0，λ₁>0，λ₂>0.定义奇异积分算子：

$\begin{array}{l} {T_1}(f)(y) = \int_0^{ + \infty } {\frac{{ {\rm{arccot}}{ ^\lambda }({x^{{\lambda _1}}}{y^{{\lambda _2}}})}}{{1 + {{({x^{{\lambda _1}}}{y^{{\lambda _2}}})}^2}}}} f(x){\rm{d}}x\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (f(x) \in L_p^\alpha (0, + \infty )), \end{array}$

$\begin{array}{l} {T_2}(g)(x) = \int_0^{ + \infty } {\frac{{ {\rm{arccot}}{ ^\lambda }({x^{{\lambda _1}}}{y^{{\lambda _2}}})}}{{1 + {{({x^{{\lambda _1}}}{y^{{\lambda _2}}})}^2}}}} g(y){\rm{d}}y\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (g(y) \in L_q^\beta (0, + \infty )), \end{array}$

则

(ⅰ)当且仅当(λ₁+αλ₂)/p=(λ₂+βλ₁)/q时，T₁是L_p^α(0, +∞)到L_p^β(1－p)(0, +∞)的有界线性算子，T₂是L_q^β(0, +∞)到L_q^α(1－q)(0, +∞)的有界线性算子.

(ⅱ)当(λ₁+αλ₂)/p=(λ₂+βλ₁)/q时，T₁与T₂的范数为：

$\left\| {{T_1}} \right\| = \left\| {{T_2}} \right\| = \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}{\left( {\frac{\pi }{2}} \right)^{\lambda + 1}}.$

命题2 设 $\frac{1}{p} + \frac{1}{q} = 1\; (p > 1)$ ，λ>0，λ₁>0，λ₂>0，a>1, 定义奇异积分算子：

$\begin{array}{l} {T_1}(f)(y) = \int_0^{ + \infty } {\frac{{ {\rm{arcoth}}{ ^\lambda }({x^{{\lambda _1}}}{y^{{\lambda _2}}} + a)}}{{{{({x^{{\lambda _1}}}{y^{{\lambda _2}}} + a)}^2} - 1}}} f(x){\rm{d}}x\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (f(x) \in L_p^\alpha (0, + \infty )), \end{array}$

$\begin{array}{l} {T_2}(g)(x) = \int_0^{ + \infty } {\frac{{ {\rm{arcoth}}{ ^\lambda }({x^{{\lambda _1}}}{y^{{\lambda _2}}} + a)}}{{{{({x^{{\lambda _1}}}{y^{{\lambda _2}}})}^2} - 1}}} g(y){\rm{d}}y\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (g(y) \in L_q^\beta (0, + \infty )), \end{array}$

则

(ⅰ)当且仅当(λ₁+αλ₂)/p=(λ₂+βλ₁)/q时，T₁是L_p^α(0, +∞)到L_p^β(1－p)(0, +∞)的有界线性算子，T₂是L_q^β(0, +∞)到L_q^α(1－q)(0, +∞)的有界线性算子.

(ⅱ)当(λ₁+αλ₂)/p=(λ₂+βλ₁)/q时，T₁与T₂的算子范数为：

$\left\| {{T_1}} \right\| = \left\| {{T_2}} \right\| = \frac{1}{{(\lambda + 1)\lambda _1^{1/q}\lambda _2^{1/p}}}{\left( {\frac{1}{2}{\rm{ln}}\frac{{a + 1}}{{a - 1}}} \right)^{\lambda + 1}}.$

参考文献(15)

[1]	RASSIAS M T, YANG B C. Equivalent propertions of a Hilbert-type integral inequality with the best constant factor related to the hurwitz zeta function[J]. Annals of Functional Analysis, 2018, 9(2):282-295. http://www.researchgate.net/publication/321147479_Equivalent_properties_of_a_Hilbert-type_integral_inequality_with_the_best_constant_factor_related_to_the_Hurwitz_zeta_function
[2]	HONG Y. On Hardy-Hilbert integral inequalities with some parameters[J]. Journal of Inequalities in Pure and Applied Mathematics, 2005, 6(4):92/1-10. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=Open J-Gate000001682473
[3]	洪勇.具有齐次核的Hilbert型积分不等式的构造特征及其应用[J].吉林大学学报(理学版), 2017, 55(2):189-194. http://www.cqvip.com/QK/95191A/20172/671625594.html HONG Y. Structural characteristics and applications of Hilbert-type integral inequalities with homogeneous kernel[J]. Journal of Jilin University(Science Edition), 2017, 55(2):189-194. http://www.cqvip.com/QK/95191A/20172/671625594.html
[4]	钟建华.一个核为零齐次的Hilbert级数型不等式及其逆[J].华南师范大学学报(自然科学版), 2011, 43(2):33-37. http://journal-n.scnu.edu.cn/article/id/502 ZHONG J H. A Hilbert-type series inequality and its reverses with the homogeneous kernels of zero degree[J]. Journal of South China Normal University(Natural Science Edition), 2011, 43(2):33-37. http://journal-n.scnu.edu.cn/article/id/502
[5]	曾志红, 杨必成.关于一个参量化的全平面Hilbert不等式[J].华南师范大学学报(自然科学版), 2017, 49(5):100-103. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=hnsfdx201705018 ZENG Z H, YANG B C. A parametric Hilbert's integral inequality in the whore plane[J]. Journal of South China Normal University(Natural Science Edition), 2017, 49(5):100-103. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=hnsfdx201705018
[6]	洪勇.涉及多个函数的Hilbert型积分不等式[J].数学学报(中文版), 2006, 49(1):39-44. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=sxxb201405001 HONG Y. On Hardy-type integral inequalities with some functions[J]. Acta Mathematical Sinica(Chinese Series), 2006, 49(1):39-44. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=sxxb201405001
[7]	洪勇.一类带齐次核的奇异重积分算子的范数及其应用[J].数学年刊(中文版), 2011, 32(5):599-606. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=sxnk201105008 HONG Y. On the singular multiple integral operator with homogeneous kernel and application[J]. Chinese Annals of Mathematics, 2011, 32(5):599-606. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=sxnk201105008
[8]	YANG B C, CHEN Q. Two kinds of Hilbert-type integral inequalities in the whole plane[J]. Journal of Inequalities and Applications, 2015(1):1-11. doi: 10.1186/s13660-014-0545-8
[9]	LIU T, YANG B C, HE L P. On a multidimensional Hilbert-type integral inequality with logarithm function[J]. Mathematical Inequalities and Application, 2015, 18(4):1219-1234. http://www.researchgate.net/publication/284896618_On_a_multidimensional_Hilbert-type_integral_inequality_with_logarithm_function
[10]	RASSIAS M T, YANG B C. A Hilbert-type integral inequality in the whole plane reated to the hyper geometric function and the beta function[J]. Journal of Mathematical Analysis and Applications, 2015, 428:1286-1308. doi: 10.1016/j.jmaa.2015.04.003
[11]	XU L Z, GAO Y K. Note on Hardy-Riesz's extension of Hilbert's inequality[J]. Chinese Quarterly Journal Mathe-matics, 1991, 6(1):75-77. http://en.cnki.com.cn/Article_en/CJFDTOTAL-SXJK199101010.htm
[12]	XIE Z T, ZENG Z. A new Hilbert-type inequality in whole plane with the homogeneous kernel of degree 0[J]. i-manager's Journal on Mathematics, 2013, 3(1):13-19.
[13]	KUANG J C. On new extensions of Hilbert's integral inequality[J]. Journal of Mathematical Analysis and Applications, 1999, 235:608-614. doi: 10.1006/jmaa.1999.6373
[14]	MA Q W, YANG B C, HE L P. A half-discrete Hilbert-type inequality in the whole plane with multiparameters[J]. Journal of Function Spaces, 2016: 6059065/1-9.
[15]	LIAO J Q, YANG B C. On a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of exponential function[J]. Journal of Inequalities and Applications, 2016(1):1-21. http://europepmc.org/articles/PMC4980859/