为后面的叙述方便, 设θ(x)(>0)为可测函数, ρ≥1, 定义如下函数空间:

$\begin{array}{l} {L^\rho }\left( {0, \infty } \right):\; = \left\{ {{{\left\| h \right\|}_\rho }:\; = } \right.\\ \;\;\;\;\left. {{{\left\{ {\int_0^\infty {{{\left| {h\left( x \right)} \right|}^\rho }{\rm{d}}x} } \right\}}^{\frac{1}{\rho }}} < \infty } \right\}, \end{array}$

和

$\begin{array}{l} L_\theta ^\rho \left( {0, \infty } \right):\; = \left\{ {{{\left\| h \right\|}_{\rho, \theta }}:\; = } \right.\\ \;\;\;\left. {{{\left\{ {\int_0^\infty {\theta \left( x \right){{\left| {h\left( x \right)} \right|}^\rho }{\rm{d}}x} } \right\}}^{\frac{1}{\rho }}} < \infty } \right\}. \end{array}$

设$f, g \ge 0, f, g, \in {L^2}\left( {0, \infty } \right), {\left\| f \right\|_2}, {\left\| g \right\|_2} > 0$, 则有下面的Hilbert积分不等式^[1]

$\int_0^\infty {\int_0^\infty {\frac{{f\left( x \right)g\left( y \right)}}{{x + y}}{\rm{d}}x{\rm{d}}y < {\rm{\pi }}{{\left\| f \right\|}_2}{{\left\| g \right\|}_2}, } } $

(1)

这里的常数因子π是最佳值.在与式(1)相同的条件下, 还有下面基本Hilbert型积分不等式^[2]:

$\int_0^\infty {\int_0^\infty {\frac{{\left| {\ln \frac{x}{y}} \right|f\left( x \right)g\left( y \right)}}{{x + y}}{\rm{d}}x{\rm{d}}y < {c_0}{{\left\| f \right\|}_2}{{\left\| g \right\|}_2}, } } $

(2)

这里的常数因子${c_0}\left( { = \sum\limits_{k = 1}^\infty {\frac{{8{{\left( {-1} \right)}^{k-1}}}}{{{{\left( {2k-1} \right)}^2}}} = 7.327\;{7^ + }} } \right)$是最佳值.近年来, 人们在Hilbert型积分不等式研究中的主要成就:一方面是将以前的基本核进行组合, 得到一些混合核的积分不等式, 同时进行参量化研究, 综合、推广和改进已有结果^[3-6].另一方面, 构造一些新的积分核, 发现新的Hilbert型积分不等式^[7-10].这些所获得的不等式在分析学和偏微分方程理论等领域有重要应用.本文引入Γ-函数、推广的ζ-函数等刻划常数因子, 利用权函数方法和实分析的技巧, 建立一个联系特殊函数的多参数Hilbert型积分不等式, 给出它的等价式, 证明了它们的常数因子是最佳的, 并通过选取特殊参数值, 得到一些有意义的结果.

1 有关引理

本文将用到以下特殊函数^[11]:

Γ-函数：

$\Gamma \left( z \right) = \int_0^\infty {{{\rm{e}}^{-u}}{u^{z-1}}{\rm{d}}u, \left( {z > 0} \right)}, $

(3)

黎曼ζ-函数:

$\zeta \left( s \right) = \sum\limits_{k = 1}^\infty {\frac{1}{{{k^s}}}\left( {{\mathop{\rm Re}\nolimits} \left( s \right) > 1} \right)}, $

(4)

推广的ζ-函数:

$\zeta \left( {s, a} \right) = \sum\limits_{k = 0}^\infty {\frac{1}{{{{\left( {k + a} \right)}^s}}}}, $

(5)

这里Re(s)>1, a不等于零和负整数.显然, ζ(s, 1)=ζ(s).

引理1.1   设Re(s)>1, $\frac{a}{2}$与$\frac{{a + 1}}{2}$均不等于零和负整数, 则有求和公式

$\sum\limits_{k = 0}^\infty {\frac{{{{\left( {- 1} \right)}^k}}}{{{{\left( {k + a} \right)}^s}}}} = \frac{1}{{{2^s}}}\left[{\zeta \left( {s, \frac{a}{2}} \right)-\zeta \left( {s, \frac{{a + 1}}{2}} \right)} \right].$

(6)

证明

$\begin{array}{l} \sum\limits_{k = 0}^\infty {\frac{{{{\left( {- 1} \right)}^k}}}{{{{\left( {k + a} \right)}^s}}} = \sum\limits_{k = 0}^\infty {\frac{1}{{{{\left( {2k + a} \right)}^s}}}- \sum\limits_{k = 0}^\infty {\frac{1}{{{{\left( {2k + 1 + a} \right)}^s}}}} } } \\ = \frac{1}{{{2^s}}}\left[{\sum\limits_{k = 0}^\infty {\frac{1}{{{{\left( {k + \frac{a}{2}} \right)}^s}}}}-\sum\limits_{k = 0}^\infty {\frac{1}{{{{\left( {k + \frac{{a + 1}}{2}} \right)}^s}}}} } \right]\\ = \frac{1}{{{2^s}}}\left[{\zeta \left( {s, \frac{a}{2}} \right)-\zeta \left( {s, \frac{{a + 1}}{2}} \right)} \right]. \end{array}$

引理1.2   设$p > 1, \frac{1}{p} + \frac{1}{q} = 1, \alpha > 0, \frac{{\beta + 1}}{4}$与$\frac{{\beta + 3}}{4}$均不为零和负整数, 定义如下权函数：

$\begin{array}{l} \omega \left( {\alpha, \beta, x} \right) = \int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}}} \\ \frac{{{y^{-\frac{{\beta + 1}}{2}}}}}{{{x^{-\frac{{p\left( {\beta + 1} \right)}}{{2q}}}}}}{\rm{d}}y{\rm{, }}x \in \left( {0, + \infty } \right), \\ \omega \left( {\alpha, \beta, y} \right) = \int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}}} \\ \frac{{{y^{-\frac{{\beta + 1}}{2}}}}}{{{x^{ - \frac{{q\left( {\beta + 1} \right)}}{{2p}}}}}}{\rm{d}}x, y \in \left( {0, + \infty } \right), \;则 \end{array}$

$\begin{array}{l} \omega \left( {\alpha, \beta, x} \right) = C\left( {\alpha, \beta } \right){x^{\frac{{p\left( {\beta + 1} \right)}}{2}-1}}, \\ \omega \left( {\alpha, \beta, y} \right) = C\left( {\alpha, \beta } \right){y^{\frac{{q\left( {\beta + 1} \right)}}{2}-1}}, \end{array}$

其中

$\begin{array}{l} C\left( {\alpha, \beta } \right) = \frac{1}{{{2^\alpha }}}\left[{\zeta \left( {\alpha + 1, \frac{{\beta + 1}}{4}} \right)-} \right.\\ \;\;\;\;\;\;\left. {\zeta \left( {\alpha + 1, \frac{{\beta + 3}}{4}} \right)} \right]\Gamma \left( {\alpha + 1} \right). \end{array}$

(7)

证明令$\frac{y}{x} = u$, 由Fubini定理¹²和引理1.1有

$\begin{array}{l} \omega \left( {\alpha, \beta, x} \right) = \int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}}} \frac{{{y^{- \frac{{\beta + 1}}{2}}}}}{{{x^{- \frac{{p\left( {\beta + 1} \right)}}{{2q}}}}}}{\rm{d}}y\\ = {x^{\frac{{p\left( {\beta + 1} \right)}}{2}- 1}}\int_0^\infty {\frac{{{{\left| {\ln \;u} \right|}^\alpha }{{\left( {\min \left\{ {1, u} \right\}} \right)}^\beta }{u^{ - \frac{{\beta + 1}}{2}}}}}{{1 + u}}{\rm{d}}u} \\ = 2{x^{\frac{{p\left( {\beta + 1} \right)}}{2} - 1}}\int_0^1 {\frac{{{{\left| {\ln u} \right|}^\alpha }{u^{\frac{{\beta - 1}}{2}}}}}{{1 + u}}{\rm{d}}u} \\ = 2{x^{\frac{{p\left( {\beta + 1} \right)}}{2} - 1}}\int_0^\infty {\frac{{{{\rm{e}}^{ - \left( {\frac{{\beta + 1}}{2}} \right)t}}{t^\alpha }}}{{1 + {{\rm{e}}^{ - t}}}}{\rm{d}}t} \\ = 2{x^{\frac{{p\left( {\beta + 1} \right)}}{2} - 1}}\sum\limits_{k = 0}^\infty {{{\left( { - 1} \right)}^k}\int_0^\infty {{{\rm{e}}^{ - \left( {k + \frac{{\beta + 1}}{2}} \right)t}}} {t^\alpha }{\rm{d}}t} \\ = 2{x^{\frac{{p\left( {\beta + 1} \right)}}{2} - 1}}\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {k + \frac{{\beta + 1}}{2}} \right)}^{\alpha + 1}}}}\int_0^\infty {{{\rm{e}}^{ - t}}{t^\alpha }{\rm{d}}t} } \\ = \frac{1}{{{2^\alpha }}}\left[{\zeta \left( {\alpha + 1, \frac{{\beta + 1}}{4}} \right)-\zeta \left( {\alpha + 1, \frac{{\beta + 3}}{4}} \right)} \right] \times \\ \Gamma \left( {\alpha + 1} \right){x^{\frac{{p\left( {\beta + 1} \right)}}{2} -1}} = C\left( {\alpha, \beta } \right){x^{\frac{{p\left( {\beta + 1} \right)}}{2} -1}}. \end{array}$

同理可证$\omega \left( {\alpha, \beta, y} \right) = C\left( {\alpha, \beta } \right){y^{\frac{{q\left( {\beta + 1} \right)}}{2}-1}}$.

引理1.3   设$p > 1, \frac{1}{p} + \frac{1}{q} = 1, \alpha > 0, \beta >- 1, \varepsilon + \sqrt[3]{\varepsilon } < \frac{{q\left( {\beta + 1} \right)}}{2}$, 且0 < ε可以充分地小, 定义如下函数：

$\begin{array}{l} \tilde f\left( x \right) = \left\{ \begin{array}{l} 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \left( {0, 1} \right)\\ {x^{\frac{{- \frac{{p\left( {\beta + 1} \right)}}{2}- \varepsilon }}{p}}}, \;\;\;\;x \in \left[{1, \infty } \right) \end{array} \right., \\ \tilde g\left( y \right) = \left\{ \begin{array}{l} 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;y \in \left( {0, 1} \right)\\ {y^{\frac{{- \frac{{q\left( {\beta + 1} \right)}}{2}- \varepsilon }}{q}}}, \;\;\;\;y \in \left[{1, \infty } \right) \end{array} \right., \end{array}$

则有

$\begin{array}{l} \tilde J\varepsilon = {\left[{\int_0^\infty {{x^{\frac{{p\left( {\beta-1} \right)}}{2}-1}}} {{\tilde f}^p}\left( x \right){\rm{d}}x} \right]^{\frac{1}{p}}} \times \\ \;\;\;\;\;\;\;\;{\left[{\int_0^\infty {{y^{\frac{{q\left( {\beta + 1} \right)}}{2}-1}}{{\tilde g}^q}\left( y \right){\rm{d}}y} } \right]^{\frac{1}{q}}}\varepsilon = 1, \end{array}$

(8)

$\begin{array}{l} \tilde I\varepsilon = \varepsilon \int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }\tilde f\left( x \right)\tilde g\left( y \right)}}{{x + y}}} } {\rm{d}}x{\rm{d}}y\\ \;\;\;\;\;\;\;\; > C\left( {\alpha, \beta } \right)\left( {1-o\left( 1 \right)} \right)\left( {\varepsilon \to {0^ + }} \right). \end{array}$

(9)

证明   容易得到

$\begin{array}{l} \tilde J\varepsilon = {\left[{\int_0^\infty {{x^{\frac{{p\left( {\beta-1} \right)}}{2}-1}}} {{\tilde f}^p}\left( x \right){\rm{d}}x} \right]^{\frac{1}{p}}} \times \\ \;\;\;\;\;\;\;\;{\left[{\int_0^\infty {{y^{\frac{{q\left( {\beta + 1} \right)}}{2}-1}}{{\tilde g}^q}\left( y \right){\rm{d}}y} } \right]^{\frac{1}{q}}}\varepsilon \\ \;\;\;\;\;\; = {\left[{\int_1^\infty {{x^{-\left( {1 + \varepsilon } \right)}}{\rm{d}}x} } \right]^{\frac{1}{p}}}{\left[{\int_1^\infty {{y^{-\left( {1 + \varepsilon } \right)}}{\rm{d}}y} } \right]^{\frac{1}{q}}}\varepsilon \\ \;\;\;\;\;\; = 1. \end{array}$

因为$F\left( t \right) = \frac{{{t^{\frac{{\beta + 1}}{2}- \frac{{\varepsilon + \sqrt[3]{\varepsilon }}}{q}}}{{\left| {\ln t} \right|}^\alpha }}}{{1 + t}}$在(0, 1]内连续, 且用洛比达法则得$\mathop {\lim }\limits_{t \to {0^ + }} F\left( t \right) = \mathop {\lim }\limits_{t \to {0^ + }} \frac{{{t^{\frac{{\beta + 1}}{2}- \frac{{\varepsilon + \sqrt[3]{\varepsilon }}}{q}}}{{\left| {\ln \;t} \right|}^\alpha }}}{{1 + t}} = 0$, 故存在M>0, 使F(t)≤M, 则有

$\begin{array}{l} \tilde I\varepsilon = \varepsilon \int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \;\left\{ {x, y} \right\}} \right)}^\beta }\tilde f\left( x \right)\tilde g\left( y \right)}}{{x + y}}} {\rm{d}}x{\rm{d}}y} \\ = \varepsilon \int_1^\infty {{x^{\frac{{- \frac{{p\left( {\beta + 1} \right)}}{2}- \varepsilon }}{p}}}{\rm{d}}x\left[{\int_1^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}} \times } } \right.} \\ \;\;\;\;\;\;\;\left. {{y^{\frac{{-\frac{{q\left( {\beta + 1} \right)}}{2}-\varepsilon }}{q}}}{\rm{d}}y} \right]\\ \;\; = \varepsilon \int_1^\infty {{x^{ - 1 - \varepsilon }}{\rm{d}}x\left[{\int_0^1 {\frac{{{{\left| {\ln \;t} \right|}^\alpha }{t^{\frac{{\beta-1}}{2}-\frac{\varepsilon }{q}}}}}{{1 + t}}} } \right.} {\rm{d}}t + \\ \;\;\;\left. {\int_0^1 {\frac{{{{\left| {\ln \;t} \right|}^\alpha }{t^{\frac{{\beta-1}}{2} + \frac{\varepsilon }{q}}}}}{{1 + t}}{\rm{d}}t - \int_0^{{x^{ - 1}}} {\frac{{{{\left| {\ln \;t} \right|}^\alpha }{t^{\frac{{\beta - 1}}{2} - \frac{\varepsilon }{q}}}}}{{1 + t}}{\rm{d}}t} } } \right]\\ \;\;=\int_{0}^{\infty }{\frac{\left( {{\text{e}}^{-\ \frac{u\left( \beta +1 \right)}{2}+\frac{\varepsilon }{q}}}\text{+}{{\text{e}}^{-\ \frac{u\left( \beta +1 \right)}{2}-\frac{\varepsilon }{q}}} \right){{u}^{\alpha }}}{1+{{\text{e}}^{-u}}}}\text{d}u-\\ \;\;\varepsilon \int_1^\infty {{x^{ - 1 - \varepsilon }}{\rm{d}}x\int_0^{{x^{ - 1}}} {\frac{{{{\left| {\ln \;t} \right|}^\alpha }{t^{\frac{{\beta - 1}}{2} - \frac{\varepsilon }{q}}}}}{{1 + t}}{\rm{d}}t} } \\ \;\; = \sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {k + \frac{{\beta + 1}}{2} + \frac{\varepsilon }{q}} \right)}^{\alpha + 1}}}}} \int_0^\infty {{{\rm{e}}^{ - u}}{u^\alpha }{\rm{d}}u + } \\ \;\;\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {k + \frac{{\beta + 1}}{2} - \frac{\varepsilon }{q}} \right)}^{\alpha + 1}}}}\int_0^\infty {{{\rm{e}}^{ - u}}{u^\alpha }{\rm{d}}u - } } \\ \;\;\;\;\varepsilon \int_1^\infty {{x^{ - 1 - \varepsilon }}{\rm{d}}x\int_0^{{x^{ - 1}}} {\frac{{{{\left| {\ln \;t} \right|}^\alpha }{t^{\frac{{\beta - 1}}{2} - \frac{\varepsilon }{q}}}}}{{1 + t}}{\rm{d}}t} } \\ > C\left( {\alpha, \beta } \right) + {o_1}\left( 1 \right) - \\ \;\;\;M\varepsilon \int_1^\infty {{x^{ - 1}}{\rm{d}}x\int_0^{{x^{ - 1}}} {{t^{ - 1 + \frac{{\sqrt[3]{\varepsilon }}}{q}}}{\rm{d}}t} } \\ = C\left( {\alpha, \beta } \right) + {o_1}\left( 1 \right) - M{q^2}\sqrt[3]{\varepsilon }\\ = C\left( {\alpha, \beta } \right)\left( {1 -o\left( 1 \right)} \right)\left( {\varepsilon \to {o^ + }} \right). \end{array}$

2 主要结论

定理2.1   设$\begin{array}{l} p > 1, \frac{1}{p} + \frac{1}{q} = 1, \alpha > 0, \beta >-1, \varphi \left( x \right) = {x^{\frac{{p\left( {\beta + 1} \right)}}{2}-1}}, \psi \left( y \right) = {y^{\frac{{q\left( {\beta + 1} \right)}}{2}-1}}, f, g > 0, f \in L_\varphi ^p\left( {0, \infty } \right), g \in L_\psi ^q\left( {0, \infty } \right) \end{array}$, 则有

$\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }f\left( x \right)g\left( y \right)}}{{x + y}}} } {\rm{d}}x{\rm{d}}y\\ \;\;\;\;\;\;\;\;\;\;\;\;\; < C\left( {\alpha, \beta } \right){\left\| f \right\|_{p, \varphi }}{\left\| g \right\|_{q, \psi }}, \end{array}$

(10)

这里的常数因子C(α, β)(同式(7))是式(10)的最佳值.

证明   由Hölder不等式¹³和引理2及Fubini定理有

$\begin{array}{l} I: = \int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }f\left( x \right)g\left( y \right)}}{{x + y}}} } {\rm{d}}x{\rm{d}}y\\ = \int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }f\left( x \right)g\left( y \right)}}{{x + y}} \times } } \\ \;\;\;\;\;\;\;\;\left[{\frac{{{y^{-\frac{{\beta + 1}}{{2p}}}}}}{{{x^{-\frac{{\beta + 1}}{{2q}}}}}}} \right]\left[{\frac{{{x^{-\frac{{\beta + 1}}{{2q}}}}}}{{{y^{-\frac{{\beta + 1}}{{2p}}}}}}} \right]{\rm{d}}x{\rm{d}}y\\ \le {\left[{\int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \;\frac{x}{y}} \right|}^a}{{\left( {\min \;\left\{ {x, y} \right\}} \right)}^\beta }{f^p}\left( x \right)}}{{x + y}}\frac{{{y^{-\frac{{\beta + 1}}{2}}}{\rm{d}}x{\rm{d}}y}}{{{x^{-\frac{{p\left( {\beta + 1} \right)}}{{2q}}}}}}} } } \right]^{\frac{1}{p}}} \times \\ {\left[{\int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \frac{s}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }{g^q}\left( y \right)}}{{x + y}}\frac{{{x^{-\frac{{\beta + 1}}{2}}}}}{{{y^{-\frac{{q\left( {\beta + 1} \right)}}{{2p}}}}}}{\rm{d}}x{\rm{d}}y} } } \right]^{\frac{1}{q}}}\\ = {\left\{ {\int_0^\infty {\omega \left( {\alpha, \beta, x} \right){f^p}\left( x \right){\rm{d}}x} } \right\}^{\frac{1}{p}}} \times \\ \;\;{\left\{ {\int_0^\infty {\omega \left( {\alpha, \beta, y} \right){g^q}\left( y \right){\rm{d}}y} } \right\}^{\frac{1}{q}}}\\ = C\left( {\alpha, \beta } \right){\left\| f \right\|_{p, \varphi }}{\left\| g \right\|_{q, \psi }}, \end{array}$

(11)

若式(11)取等号, 则存在不全为零的实数A和B, 使$A\frac{{{y^{-\frac{{\beta + 1}}{2}}}}}{{{x^{-\frac{{p\left( {\beta + 1} \right)}}{{2q}}}}}}{f^p}\left( x \right) = B\frac{{{x^{-\frac{{\beta + 1}}{2}}}}}{{{y^{ - \frac{{q\left( {\beta + 1} \right)}}{{2p}}}}}}{g^q}\left( y \right), {\rm{a}}{\rm{.e}}$.于$\left( {0, \infty } \right) \times \left( {0, \infty } \right)$, 于是有常数C，使$A{x^{\frac{{p\left( {\beta + 1} \right)}}{2}}}{f^p}\left( x \right) = B{y^{\frac{{q\left( {\beta + 1} \right)}}{2}}}{g^q}\left( y \right) = C, {\rm{a}}{\rm{.e}}{\rm{.}}$.于$\left( {0, \infty } \right) \times \left( {0, \infty } \right)$, 不妨设A≠0, 则有${x^{\frac{{p\left( {\beta + 1} \right)}}{2}-1}}{f^p}\left( x \right) = \frac{C}{{Ax}}, {\rm{a}}{\rm{.e}}{\rm{.}}$于$\left( {0, \infty } \right)$, 这与${\rm{0 < }}{\left\| f \right\|_{p, \varphi }} < \infty $矛盾, 故式(11)取严格不等号.若C(α, β)不是式(10)的最佳常数因子, 则存在正数K < C(α, β), 使式(10)的常数因子换成K后仍成立, 于是由式(8)和(9)有C(α, β)(1-o(1)) < K, 让ε→0⁺得: K≥C(α, β), 这与K < C(α, β)矛盾, 故C(α, β)是式(10)的最佳常数因子.

定理2.2   在与定理2.1相同的条件下, 我们还有

$\begin{array}{l} {\int_0^\infty y ^{\frac{{- \frac{{q\left( {\beta + 1} \right)}}{2} + 1}}{{q- 1}}}}{\rm{d}}y{\left[{\int_0^\infty {\frac{{{{\left| {\ln \;\frac{x}{y}} \right|}^\alpha }{{\left( {\min \;\left\{ {x, y} \right\}} \right)}^\beta }f\left( x \right)}}{{x + y}}{\rm{d}}x} } \right]^p}\\ \;\;\;\;\; < {C^p}\left( {\alpha, \beta } \right)\left\| f \right\|_{p, \varphi }^p, \end{array}$

(12)

这里的常数因子C^p(α, β)是式(12)的最佳值，且式(12)与(10)等价.

证明   设置如下有界可测函数

${\left[{f\left( x \right)} \right]_n}:\min \left\{ {n, f\left( x \right)} \right\} = \left\{ \begin{array}{l} f\left( x \right), \;\;\;f\left( x \right) < n\\ n, \;\;\;\;\;\;\;\;\;f\left( x \right) \ge n \end{array} \right..$

因${\rm{0 < }}{\left\| f \right\|_{p, \varphi }} < \infty $, 存在n₀∈N, 使得当n≥n₀时, 有$0 < \int_{\frac{1}{n}}^n {{x^{\frac{{p\left( {\beta + 1} \right)}}{2}- 1}}\left[{f\left( x \right)} \right]_n^p{\rm{d}}x < \infty } $, 置${g_n}\left( y \right): = {y^{\frac{{- \frac{{q\left( {\beta + 1} \right)}}{2} + 1}}{{q- 1}}}}{\left[{\int_{\frac{1}{n}}^n {\frac{{{{\left| {\ln \;\frac{x}{y}} \right|}^\alpha }{{\left( {\min \;\left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}}{{\left[{f\left( x \right)} \right]}_n}{\rm{d}}x} } \right]^{\frac{p}{q}}}\left( {\frac{1}{n} < y < n, n \ge {n_0}} \right)$, 则当n≥n₀时, 由式(10)有

$\begin{array}{l} 0 < \int_{\frac{1}{n}}^n {{y^{\frac{{q\left( {\beta + 1} \right)}}{2}- 1}}g_n^q\left( y \right){\rm{d}}y} \\ \;\;\;\;\; = \int_{\frac{1}{n}}^n {{y^{\frac{{- \frac{{q\left( {\beta + 1} \right)}}{2} + 1}}{{q- 1}}}}\left[{\int_{\frac{1}{n}}^n {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}} \times } } \right.} \\ \;\;\;\;\;\;\;\;\;{\left. {\;{{\left[{f\left( x \right)} \right]}_n}{\rm{d}}x} \right]^p}{\rm{d}}y\\ \;\;\;\; = \int_{\frac{1}{n}}^n {\int_{\frac{1}{n}}^n {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}}{{\left[{f\left( x \right)} \right]}_n}{g_n}\left( y \right){\rm{d}}x{\rm{d}}y} } \\ \;\;\;\; < C\left( {\alpha, \beta } \right){\left\{ {\int_{\frac{1}{n}}^n {{x^{\frac{{p\left( {\beta + 1} \right)}}{2} - 1}}\left[{f\left( x \right)} \right]_n^p{\rm{d}}x} } \right\}^{\frac{1}{p}}} \times \\ \;\;\;\;\;\;{\left\{ {\int_{\frac{1}{n}}^n {{y^{\frac{{q\left( {\beta + 1} \right)}}{2} -1}}g_n^q\left( y \right){\rm{d}}y} } \right\}^{\frac{1}{q}}}, \end{array}$

(13)

$\begin{array}{l} 0 < \int_{\frac{1}{n}}^n {{y^{\frac{{q\left( {\beta + 1} \right)}}{2}-1}}g_n^q\left( y \right){\rm{d}}y} \\ \;\;\; < {C^p}\left( {\alpha, \beta } \right)\left\| f \right\|_{p, \varphi }^p < \infty, \end{array}$

(14)

即${\rm{0 < }}{\left\| f \right\|_{p, \varphi }} < \infty $.当 $n \to \infty $时, 应用式(10), 则式(13)取严格不等号, 式(14)亦然, 故有式(12).

反之, 由带权Hölder不等式有

$\begin{array}{l} I = \int_0^\infty {\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}}} } f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y\\ \;\; = \int_0^\infty {\left[{{y^{\frac{{-\frac{{q\left( {\beta + 1} \right)}}{2} + 1}}{{p\left( {q-1} \right)}}}}\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }f\left( x \right){\rm{d}}x}}{{x + y}}} } \right]} \; \times \\ \;\;\;\;\;\left[{{y^{\frac{{\frac{{q\left( {\beta + 1} \right)}}{2}-1}}{{p\left( {q-1} \right)}}}}g\left( y \right)} \right]{\rm{d}}y\\ \le \left\{ {\int_0^\infty {{y^{\frac{{ - \frac{{q\left( {\beta + 1} \right)}}{2} + 1}}{{\left( {q - 1} \right)}}}}{\rm{d}}y \times } } \right.\\ {\left. {{{\left[{\int_0^\infty {\frac{{{{\left| {\ln \frac{x}{y}} \right|}^\alpha }{{\left( {\min \left\{ {x, y} \right\}} \right)}^\beta }}}{{x + y}}f\left( x \right)} {\rm{d}}x} \right]}^p}} \right\}^{\frac{1}{p}}}{\left\| g \right\|_{q, \psi }}\\ < C\left( {\alpha, \beta } \right){\left\| f \right\|_{p, \varphi }}{\left\| g \right\|_{q, \psi }}. \end{array}$

上不等式即为式(10), 因此式(10)和式(12)等价.

若式(12)中的常数因子不是最佳的, 则由式(12)得到式(10)的常数因子也不是最佳的, 故常数因子C^p(α, β)是式(12)的最佳值.

我们在式(10)和(12)中选取符合定理条件的参数α, β以及共轭指数对(p, q)的合适值, 并借助Maple数学软件的计算, 可以得到一些有意义的不等式.

如取α=1, β=0, p=q=2, 计算式(7)得$C\left( {1, 0} \right) = {c_0} = \frac{{{{\rm{\pi }}^2}}}{2} + 4{\rm{catalan}}-\frac{1}{2}\Psi \left( {1, \frac{3}{4}} \right) = 7.327\;724\;{76^ + }\left( {其中\Psi \left( {n, z} \right)为n次\Gamma 函数} \right)$, 则有式(2)和它的等价式:

$\int_0^\infty {{\rm{d}}y{{\left[{\int_0^\infty {\frac{{\left( {\ln \frac{x}{y}} \right)f\left( x \right)}}{{x + y}}{\rm{d}}x} } \right]}^2} < c_0^2\left\| f \right\|_2^2.} $

(15)

这里的常数因子c₀²是式(15)的最佳值.

如取α=2, β=1, p=q=2, 计算式(7)得$C\left( {2, 1} \right) = 3\zeta \left( 3 \right) = 3.606\;170\;{709^ + }$, 这时$\varphi \left( x \right) = x$, 设$f, g \in L_\varphi ^2\left( {0, \infty } \right), {\left\| f \right\|_{2, \varphi }}, {\left\| g \right\|_{2, \varphi }} > 0$, 则有下列等价式:

$\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{{{\left( {\ln \frac{x}{y}} \right)}^2}\min \left\{ {x, y} \right\}}}{{x + y}}f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y} } \\ \;\;\;\;\;\;\; < 3\zeta \left( 3 \right){\left\| f \right\|_{2, \varphi }}{\left\| g \right\|_{2, \varphi }}, \; \end{array}$

(16)

$\begin{array}{l} \int_0^\infty {{y^{- 1}}{\rm{d}}y{{\left[{\int_0^\infty {\frac{{{{\left( {\ln \frac{x}{y}} \right)}^2}\min \left\{ {x, y} \right\}}}{{x + y}}f\left( x \right){\rm{d}}x} } \right]}^2}} \\ \;\;\;\;\;\;\;\;\;\; < 9{\zeta ^2}\left( 3 \right)\left\| f \right\|_{2, \varphi }^2.\; \end{array}$

(17)

这里的常数因子3ζ(3), 9ζ²(3)分别是式(16), (17)的最佳值.

如取$\alpha = \beta = \frac{1}{2}, p = q = 2$, 计算式(7)有$C\left( {\frac{1}{2}, \frac{1}{2}} \right) = \frac{{\sqrt {2{\rm{\pi }}} }}{4}\left[{\zeta \left( {\frac{3}{2}, \frac{3}{8}} \right)-\zeta \left( {\frac{3}{2}, \frac{7}{8}} \right)} \right] = 2.206\;556\;{861^ + }$, 这时φ(x)=x, 设$f, g \in L_\varphi ^2\left( {0, \infty } \right), {\left\| f \right\|_{2, \varphi }}, {\left\| g \right\|_{2, \varphi }} > 0$, 则有下列等价式:

$\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{\sqrt {\left| {\ln \frac{x}{y}} \right|\min \left\{ {x, y} \right\}} }}{{x + y}}f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y} } \\ < \frac{{\sqrt {2{\rm{\pi }}} }}{4}\left[{\zeta \left( {\frac{3}{2}, \frac{3}{8}} \right)-\zeta \left( {\frac{3}{2}, \frac{7}{8}} \right)} \right]{\left\| f \right\|_{2, \varphi }}{\left\| g \right\|_{2, \varphi }}, \end{array}$

(18)

$\begin{array}{l} \int_0^\infty {\frac{1}{{\sqrt y }}{\rm{d}}y{{\left[{\int_0^\infty {\frac{{\sqrt {\left| {\ln \frac{x}{y}} \right|\min \left\{ {x, y} \right\}} }}{{x + y}}} f\left( x \right){\rm{d}}x} \right]}^2}} \\ \;\; < \frac{{\rm{\pi }}}{8}{\left[{\zeta \left( {\frac{3}{2}, \frac{3}{8}} \right)-\zeta \left( {\frac{3}{2}, \frac{7}{8}} \right)} \right]^2}\left\| f \right\|_{2, \varphi }^2. \end{array}$

(19)

这里的常数因子$\frac{{\sqrt {2{\rm{\pi }}} }}{4}\left[{\zeta \left( {\frac{3}{2}, \frac{3}{8}} \right)-\zeta \left( {\frac{3}{2}, \frac{7}{8}} \right)} \right], \frac{{\rm{\pi }}}{8}{\left[{\zeta \left( {\frac{3}{2}, \frac{3}{8}} \right)-\zeta \left( {\frac{3}{2}, \frac{7}{8}} \right)} \right]^2}$分别是式(18), (19)的最佳值.

如取α=1, β=-12, p=q=2, 计算式(7)有$C\left( {1, \;\;-\;\;\frac{1}{2}} \right) = \frac{{\Psi \left( {1, \frac{1}{8}} \right)-\Psi \left( {1, \frac{5}{8}} \right)}}{2} = 30.993\;475\;{13^ + }$, 这时$\varphi \left( x \right) = \frac{1}{{\sqrt x }}$, 设$f, g \in L_\varphi ^2\left( {0, \infty } \right), {\left\| f \right\|_{2, \varphi }}, {\left\| g \right\|_{2, \varphi }} > 0$, 则有下列等价式:

$\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{\left| {\ln \frac{x}{y}} \right|}}{{\left( {x + y} \right)\sqrt {\min \left\{ {x, y} \right\}} }}} f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y} \\ < \frac{{\Psi \left( {1, \frac{1}{8}} \right)-\Psi \left( {1, \frac{5}{8}} \right)}}{2}{\left\| f \right\|_{2, \varphi }}{\left\| g \right\|_{2, \varphi }}, \end{array}$

(20)

$\begin{array}{l} \int_0^\infty {\sqrt y {\rm{d}}y{{\left[{\int\limits_0^\infty {\frac{{\left| {\ln \frac{x}{y}} \right|}}{{\left( {x + y} \right)\sqrt {\min \left\{ {x, y} \right\}} }}f\left( x \right){\rm{d}}x} } \right]}^2}} \\ \;\; < \frac{1}{4}{\left[{\Psi \left( {1, \frac{1}{8}} \right)-\Psi \left( {1, \frac{5}{8}} \right)} \right]^2}\left\| f \right\|_{2, \varphi }^2. \end{array}$

(21)

这里的常数因子$\frac{{\Psi \left( {1, \frac{1}{8}} \right)- \Psi \left( {1, \frac{5}{8}} \right)}}{2}, \frac{1}{4}{\left[{\Psi \left( {1, \frac{1}{8}} \right)-\Psi \left( {1, \frac{5}{8}} \right)} \right]^2}$分别是式(20), (21)的最佳值.