西南师范大学学报(自然科学版)   2019, Vol. 44 Issue (12): 24-30.  DOI: 10.13718/j.cnki.xsxb.2019.12.005
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  • 双线性分数次积分算子交换子在Triebel-Lizorkin空间上有界的充分必要条件    [PDF全文]
    房成龙     
    伊犁师范大学 数学与统计学院, 新疆 伊宁 835000
    摘要:首先讨论了双线性分数次积分算子与Lipschitz函数生成的线性交换子在Triebel-Lizorkin空间上的有界性.然后证明了b1=b2为Lipschitz函数的等价条件是双线性分数次积分算子交换子从乘积Lebesgue空间到Lebesgue空间(或Triebel-Lizorkin空间)有界.
    关键词双线性分数次积分    交换子    Lipschitz函数    Triebel-Lizorkin空间    有界性    

    20世纪70年代,文献[1-2]发现Calderón-Zygmund奇异积分算子交换子的研究可以归结为一类双线性奇异积分算子的研究,并获得了Calderón-Zygmund奇异积分算子交换子在Lebesgue空间上的有界性,之后许多学者开始研究交换子[3-6].

    β>0,若函数f满足

    $ {\left\| f \right\|_{{{\mathit{\dot \Lambda }}_\beta }}} = \mathop {\sup }\limits_{x,h \in {\mathbb{R}^n},h \ne 0} \frac{{|f\left( {x + h} \right) - f(x)|}}{{|h{|^\beta }}} < \infty $

    则称f属于Lipschitz空间$\dot \varLambda _β $.对方体Q$f_{Q}=\frac{1}{|Q|} \int_{Q} f(x) \mathrm{d} x$.

    文献[5]证明了

    $ b \in {{\mathit{\dot \Lambda }}_\beta } \Leftrightarrow C_f^\alpha :{L^p}\left( {{\mathbb{R}^n}} \right) \to F_q^{\beta ,\infty }有界 \Leftrightarrow C_f^\alpha :{L^p}\left( {{\mathbb{R}^n}} \right) \to {L^r}\left( {{\mathbb{R}^n}} \right)有界 $ (1)

    其中Cfα是分数次积分算子与Lipschitz函数生成的交换子,

    $ 0 < \beta < 1\;\;\;\;1 < p < q < \infty $
    $ \frac{1}{p} - \frac{1}{q} = \frac{\beta }{n}\;\;\;\;\frac{1}{p} - \frac{1}{r} = \frac{{\alpha + \beta }}{n} $

    等价关系(1)的证明用到了文献[6]中讨论的Triebel-Lizorkin空间Fpβ,∞的一个重要性质:

    $ {\left\| f \right\|_{\dot F_p^{\beta ,\infty }}} \approx {\left\| {\mathop {\sup }\limits_Q \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {f - {f_Q}} \right|} {\rm{d}}x} \right\|_p} $

    受文献[5]结果的启发,一个自然的问题产生了:双线性分数次积分算子交换子的有界性是否可以刻画Lipschitz空间?本文在第二部分给出了肯定的回答.

    定义1[7]   设0 < α < 2n,双线性分数次积分算子Iα定义为

    $ {I_\alpha }\left( {{f_1},{f_2}} \right)(x) = \int_{{\mathbb{R}^n}} {\int_{{\mathbb{R}^n}} {\frac{{{f_1}\left( {{y_1}} \right){f_2}\left( {{y_2}} \right)}}{{{{\left( {\left| {x - {y_1}} \right| + \left| {x - {y_2}} \right|} \right)}^{2n - \alpha }}}}} } {\text{d}}{y_1}{\text{d}}{y_2} $

    文献[7]获得了IαLp1×Lp2Lq的有界性,其中

    $ 1 < {p_1},{p_2} < q < \infty $
    $ \frac{1}{{{p_1}}} + \frac{1}{{{p_2}}} - \frac{1}{q} = \frac{\alpha }{n}\;\;\;\;0 < \alpha < 2n $

    另外,有关多线性分数次积分算子的研究可见文献[8].

    本文主要讨论两种类型的交换子,下面给出其定义.

    定义2  设bjLloc1(j=1,2),Iα是双线性分数次积分算子.

    (a) 线性交换子$\left[\varSigma \vec{b}, I_{a}\right]$定义为

    $ \left[ {\Sigma \vec b,{I_a}} \right]\left( {{f_1},{f_2}} \right)\left( x \right) = \int_{{\mathbb{R}^n}} {\int_{{\mathbb{R}^n}} {\frac{{\left[ {\left( {{b_1}(x) - {b_1}\left( {{y_1}} \right)} \right) + \left( {{b_2}(x) - {b_2}\left( {{y_2}} \right)} \right)} \right]{f_1}\left( {{y_1}} \right){f_2}\left( {{y_2}} \right)}}{{{{\left( {\left| {x - {y_1}} \right| + \left| {x - {y_2}} \right|} \right)}^{2n - \alpha }}}}{\text{d}}{y_1}{\text{d}}{y_2}} } $

    (b) 迭代交换子$\left[Π \vec{b}, I_{a}\right]$定义为

    $ \left[ {\Pi \vec b,{I_a}} \right]\left( {{f_1},{f_2}} \right)\left( x \right) = \int_{{\mathbb{R}^n}} {\int_{{\mathbb{R}^n}} {\frac{{\left( {{b_1}(x) - {b_1}\left( {{y_1}} \right)} \right) + \left( {{b_2}(x) - {b_2}\left( {{y_2}} \right)} \right){f_1}\left( {{y_1}} \right){f_2}\left( {{y_2}} \right)}}{{{{\left( {\left| {x - {y_1}} \right| + \left| {x - {y_2}} \right|} \right)}^{2n - \alpha }}}}{\text{d}}{y_1}{\text{d}}{y_2}} } $

    文献[9]证明了b1b2$\dot \varLambda _β$时,从乘积Lebesgue空间到Lebesgue空间(或Triebel-Lizorkin空间)的算子$\left[\varPi \vec{b}, I_{a}\right]$是有界的.

    文献[10]验证了文献[9]的结果对一种广义高阶交换子也成立.

    文献[11]证明了

    $ {b_1},{b_2} \in {{\mathit{\dot \Lambda }}_\beta } \Leftrightarrow \left[ {\Sigma \vec b,{I_\alpha }} \right]:{L^{{p_1}}}\left( {{\mathbb{R}^n}} \right) \times {L^{{p_2}}}\left( {{\mathbb{R}^n}} \right) \to {L^q}\left( {{\mathbb{R}^n}} \right)有界 $

    其中

    $ 0 < \alpha < 2n\;\;\;\;0 < \beta < 1 $

    $ \alpha + \beta < 2n\;\;\;\;1 < {p_1},{p_2} < \infty $
    $ \frac{1}{{{p_1}}} + \frac{1}{{{p_2}}} = \frac{1}{p}\;\;\;\frac{1}{p} - \frac{1}{q} = \frac{{\alpha + \beta }}{n} $

    受文献[5-11]中结果的启发,本文的第一部分证明了b1b2$\dot \varLambda _β$时,线性交换子$\left[\varSigma \vec{b}, I_{a}\right]$在Triebel-Lizorkin空间上有界;第二部分验证了$\left[\varSigma \vec{b}, I_{a}\right]$$\left[\Pi \vec{b}, I_{a}\right]$在Lebesgue空间、Triebel-Lizorkin空间上的有界性可以刻画Lipschitz空间.

    对于一个集合AχA表示其特征函数. C表示常数,每次出现时有可能其值并不相同.当C>0时,ACBA$\mathop < \limits_ \sim $B来表示;同时,AB表示A$\mathop < \limits_ \sim $BB$\mathop < \limits_ \sim $A.

    1 分数次积分算子的线性交换子在Triebel-Lizorkin空间上的有界性

    引理1  设0 < β < 1,则:

    (ⅰ)若1 < q≤∞,则

    $ {\left\| f \right\|_{{{\mathit{\dot \Lambda }}_\beta }}} \approx \mathop {\sup }\limits_Q \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {f - {f_Q}} \right|} {\text{d}}x \approx \mathop {\sup }\limits_Q \frac{1}{{{{\left| Q \right|}^{\frac{\beta }{n}}}}}{\left( {\frac{1}{{\left| Q \right|}}\int_Q {{{\left| {f - {f_Q}} \right|}^q}} {\text{d}}x} \right)^{\frac{1}{q}}} $

    (ⅱ)若1 < p < ∞,则

    $ {\left\| f \right\|_{\dot F_p^{\beta ,\infty }}} \approx {\left\| {\mathop {\sup }\limits_Q \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {f - {f_Q}} \right|} {\text{d}}x} \right\|_p} $

    (ⅰ)的证明见文献[12],且q=∞显然成立. (ⅱ)的证明见文献[6].

    引理2  设$1<p<q<\infty, \frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}$. hQ是一个定义在方体Q上的函数.若0≤γ,则

    $ {\left\| {\mathop {\sup }\limits_Q \frac{1}{{{{\left| Q \right|}^{1 + \frac{\gamma }{n}}}}}\int_Q {\left| {{h^Q}} \right|} {\text{d}}x} \right\|_q} \leqslant C{\left\| {\mathop {\sup }\limits_Q \frac{1}{{{{\left| Q \right|}^{1 + \frac{\gamma }{n} + \frac{\alpha }{n}}}}}\int_Q {\left| {{h^Q}} \right|} {\text{d}}x} \right\|_p} $

    其中常数Cpqαn有关.

    定理1  设0 < β < 1,1 < r < ∞,1 < p1p2 < q < ∞. $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}, \frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{q}=\frac{\alpha}{n}, \frac{1}{p_{1}}+\frac{1}{p_{2}}-$$\frac{1}{r}=\frac{\alpha+\beta}{n}$.若b1b2$\dot \varLambda _β$,则$\left[\varSigma \vec{b}, I_{a}\right]$Lp1($\mathbb R$nLp2($\mathbb R$n)→Fqβ,∞有界.

      通过引理1(ⅱ)和引理2,可得

    $ \begin{array} [c]{l} {\left\| {\left[ {\varSigma \vec b,{I_\alpha }} \right]\left( {{g_1},{g_2}} \right)} \right\|_{F_q^{\beta ,\infty }}} \lesssim \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left[ {\varSigma \vec b,{I_\alpha }} \right]\left( {{g_1},{g_2}} \right) - {{\left( {\left[ {\varSigma \vec b,{I_a}} \right]\left( {{g_1},{g_2}} \right)} \right)}_Q}} \right|{\text{d}}x} } \right\|_q} \lesssim \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left[ {\varSigma b - {{\vec b}_Q},{I_\alpha }} \right]\left( {{g_1},{g_2}} \right) - {{\left( {\left[ {\varSigma b - {{\vec b}_Q},{I_a}} \right]\left( {{g_1},{g_2}} \right)} \right)}_Q}} \right|{\text{d}}x} } \right\|_q} \lesssim \hfill \\ \left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left[ {\varSigma b - {{\vec b}_Q},{I_\alpha }} \right]\left( {{g_1},{g_2}} \right) - {I_\alpha }\left( {g_1^\infty ,\left( {b - {b_Q}} \right)g_2^\infty } \right)\left( {{x_Q}} \right) - } \right.} } \right. \hfill \\ {\left. {\left. {{I_\alpha }\left( {\left( {b - {b_Q}} \right)g_1^\infty ,g_2^\infty } \right)\left( {{x_Q}} \right)} \right|{\text{d}}x} \right\|_q} \lesssim \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left( {b - {b_Q}} \right){I_\alpha }\left( {{g_1},{g_2}} \right)} \right|{\text{d}}x} } \right\|_q} + \hfill \\ \left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{{{\left| Q \right|}^{1 + \frac{\alpha }{n} + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {\left( {b - {b_Q}} \right){g_1},{g_2}} \right) + {I_\alpha }\left( {{g_1},\left( {b - {b_Q}} \right){g_2}} \right) - } \right.} } \right. \hfill \\ {\left. {\left. {{I_\alpha }\left( {\left( {b - {b_Q}} \right)g_1^\infty ,g_2^\infty } \right)\left( {{x_Q}} \right) - {I_\alpha }\left( {g_1^\infty ,\left( {b - {b_Q}} \right)g_2^\infty } \right)\left( {{x_Q}} \right)} \right|{\text{d}}x} \right\|_p} \lesssim \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left( {b - {b_Q}} \right){I_\alpha }\left( {{g_1},{g_2}} \right)} \right|} {\text{d}}x} \right\|_q} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {\left( {b - {b_Q}} \right)g_1^0,g_2^0} \right)} \right|} {\text{d}}x} \right\|_q} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {\left( {b - {b_{{1_Q}}}} \right)g_1^0,g_2^\infty } \right)} \right|} {\text{d}}x} \right\|_q} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {\left( {b - {b_{{1_Q}}}} \right)g_1^\infty ,g_2^0} \right)} \right|} {\text{d}}x} \right\|_q} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {g_1^0,\left( {b - {b_Q}} \right)g_2^0} \right)} \right|} {\text{d}}x} \right\|_q} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {g_1^0,\left( {b - {b_Q}} \right)g_2^\infty } \right)} \right|} {\text{d}}x} \right\|_q} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {g_1^\infty ,\left( {b - {b_Q}} \right)g_2^0} \right)} \right|} {\text{d}}x} \right\|_q} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{\frac{a}{n} + \frac{\beta }{n}}}}}\mathop {\sup }\limits_{y \in Q} \left| {{I_a}\left( {\left( {b - {b_Q}} \right)g_1^\infty ,g_2^\infty } \right)(y) - {I_a}\left( {\left( {b - {b_Q}} \right)g_1^\infty ,g_2^\infty } \right)\left( {{x_Q}} \right)} \right|{\text{d}}x} \right\|_p} + \hfill \\ {\left\| {\mathop {\sup }\limits_{Q \ni \cdot } \frac{1}{{|Q{|^{\frac{\alpha }{\mu } + \frac{\beta }{n}}}}}\mathop {\sup }\limits_{y \in Q} \left| {{I_\alpha }\left( {g_1^\infty ,\left( {b - {b_Q}} \right)g_2^\infty } \right)(y) - {I_\alpha }\left( {g_1^\infty ,\left( {b - {b_Q}} \right)g_2^\infty } \right)\left( {{x_Q}} \right)} \right|{\text{d}}x} \right\|_p} = \hfill \\ {D_1} + {D_2} + {D_3} + {D_4} + {D_5} + {D_6} + {D_7} + {D_8} + {D_9} \hfill \\ \end{array} $

    首先估计D1.任意固定xQ,通过引理1(ⅰ),有

    $ \begin{array} [c]{l} \frac{1}{{|Q{|^{\frac{\beta }{n}}}}}\frac{1}{{|Q|}}\int_Q {\left| {\left( {b - {b_Q}} \right){I_\alpha }\left( {{g_1},{g_2}} \right)} \right|} {\text{d}}x \lesssim \hfill \\ \frac{1}{{|Q{|^{\frac{\beta }{n}}}}}\mathop {\sup }\limits_{y \in Q} \left| {\left( {b(y) - {b_Q}} \right)} \right|\left( {\frac{1}{{|Q|}}\int_Q {\left| {{I_\alpha }\left( {{g_1},{g_2}} \right)} \right|} {\text{d}}x} \right) \lesssim \hfill \\ {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}M\left( {{I_\alpha }\left( {{g_1},{g_2}} \right)} \right)(x) \hfill \\ \end{array} $

    根据Iα的有界性[7],得

    $ {D_1} \lesssim {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {{I_\alpha }\left( {{g_1},{g_2}} \right)} \right\|_q} \lesssim {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {{g_1}} \right\|_{{p_1}}}{\left\| {{g_2}} \right\|_{{p_2}}} $

    再估计D2.运用Minkowski不等式和Iα的有界性,有

    $ \begin{array} [c]{l} {\left\| {\frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {{I_\alpha }\left( {\left( {b - {b_Q}} \right)g_1^0,g_2^0} \right)} \right|} {\text{d}}x} \right\|_{{L^q}}} \lesssim \hfill \\ \frac{1}{{|Q{|^{1 + \frac{\beta }{n}}}}}\int_Q {{{\left\| {{I_\alpha }\left( {\left( {b - {b_Q}} \right)g_1^0,g_2^0} \right)} \right\|}_{{L^q}}}} {\text{d}}x \lesssim \hfill \\ \frac{1}{{|Q{|^{\frac{\beta }{n}}}}}{\left( {\int_Q {{{\left| {\left( {b - {b_Q}} \right)g_1^0} \right|}^{{p_1}}}} {\text{d}}x} \right)^{\frac{1}{{{p_1}}}}}{\left( {\int_Q {{{\left| {g_2^0} \right|}^{{p_2}}}{\text{d}}x} } \right)^{\frac{1}{{{p_2}}}}} \leqslant \hfill \\ \frac{1}{{|Q{|^{\frac{\beta }{n}}}}}\mathop {\sup }\limits_{y \in Q} \left| {b(y) - {b_Q}} \right|{\left\| {{g_1}} \right\|_{{p_1}}}{\left\| {{g_2}} \right\|_{{p_2}}} \lesssim \hfill \\ {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {{g_1}} \right\|_{{p_1}}}{\left\| {{g_2}} \right\|_{{p_2}}} \hfill \\ \end{array} $

    因此

    $ {D_2} \lesssim {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {{g_1}} \right\|_{{p_1}}}{\left\| {{g_2}} \right\|_{{p_2}}} $

    同理得

    $ {D_m} \lesssim {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {{g_1}} \right\|_{{p_1}}}{\left\| {{g_2}} \right\|_{{p_2}}}\;\;\;\;m = 3,4,5,6,7 $

    由于

    $ \begin{array} [c]{l} \frac{1}{{{{\left| Q \right|}^{\frac{a}{n} + \frac{\beta }{n}}}}}\left| {{I_a}\left( {\left( {b - {b_Q}} \right)g_1^\infty ,g_2^\infty } \right)\left( y \right) - {I_a}\left( {\left( {b - {b_Q}} \right)g_1^\infty ,g_2^\infty } \right)\left( {{x_Q}} \right)} \right| \lesssim \hfill \\ \frac{1}{{{{\left| Q \right|}^{\frac{\alpha }{n} + \frac{\beta }{n}}}}}\int_{{{\left( {2Q} \right)}^c}} {\int_{{{\left( {2Q} \right)}^c}} {\frac{{\left| {b\left( {{y_1}} \right) - {b_Q}} \right|\left| {{g_1}\left( {{y_1}} \right)} \right|\left| {{g_2}\left( {{y_2}} \right)} \right|}}{{{{\left( {\left| {x - {y_1}} \right| + \left| {x - {y_2}} \right|} \right)}^{2n - \alpha }}}}{\text{d}}{y_1}{\text{d}}{y_2} \lesssim } } \hfill \\ \frac{1}{{{{\left| Q \right|}^{\frac{\alpha }{n} + \frac{\beta }{n}}}}}\int_{{{\left( {2Q} \right)}^c}} {\frac{{\left| {{g_2}\left( {{y_2}} \right)} \right|}}{{{{\left| {x - {y_2}} \right|}^{n - \frac{\alpha }{2}}}}}{\text{d}}{y_2}} \times \frac{1}{{{{\left| Q \right|}^{\frac{\alpha }{n} + \frac{\beta }{n}}}}}\int_{{{\left( {2Q} \right)}^c}} {\frac{{\left| {{g_1}\left( {{y_1}} \right)} \right|}}{{{{\left| {x - {y_1}} \right|}^{n - \frac{\alpha }{2}}}}}} \left| {b\left( {{y_1}} \right) - {b_Q}} \right|{\text{d}}{y_1} \lesssim \hfill \\ \frac{1}{{{{\left| Q \right|}^{\frac{\alpha }{n} + \frac{\beta }{n}}}}}\int_{{{\left( {2Q} \right)}^c}} {\frac{{\left| {{g_2}\left( {{y_2}} \right)} \right|}}{{{{\left| {x - {y_2}} \right|}^{n - \frac{\alpha }{2}}}}}{\text{d}}{y_2}} \times \frac{1}{{{{\left| Q \right|}^{\frac{\alpha }{n} + \frac{\beta }{n}}}}}\int_{{{\left( {2Q} \right)}^c}} {\frac{{\left| {{g_1}\left( {{y_1}} \right)} \right|}}{{{{\left| {x - {y_1}} \right|}^{n - \frac{\alpha }{2}}}}}} \left| {b\left( {{y_1}} \right) - {b_Q}} \right|{\text{d}}{y_1} \lesssim \hfill \\ \sum\limits_{m = 2}^\infty {\frac{{{2^{ - (m - 1) \cdot n - \frac{\alpha }{2} - m + \frac{{{m_\alpha }}}{2}}}}}{{|Q{|^{\frac{\alpha }{n} + \frac{\beta }{n} - \frac{\alpha }{{2n}}}}}}} {\smallint _{{2^m}Q\backslash {2^{m - 1}}Q}}{\left| {{2^m}Q} \right|^{ - 1}}\left| {{g_2}\left( {{y_2}} \right)} \right|{\text{d}}{y_2} \times \hfill \\ \sum\limits_{k = 2}^\infty {\frac{{{2^{ - (k - 1) \cdot n - \frac{\alpha }{2} - m + \frac{{{m_\alpha }}}{2}}}}}{{|Q{|^{\frac{\alpha }{n} + \frac{\beta }{n} - \frac{\alpha }{{2n}}}}}}} \int_{{2^k}Q\backslash {2^{k - 1}}Q} {{{\left| {{2^k}Q} \right|}^{ - 1}}\left( {\left| {b\left( {{y_1}} \right) - b_2^k} \right| + \left| {b_{{2^k}}^k - {b_Q}} \right|} \right)\left| {{g_1}\left( {{y_1}} \right)} \right|{\text{d}}{y_1}} \lesssim \hfill \\ \left( {\sum\limits_{m = 2}^\infty {\frac{{{2^{ - (k - 1) \cdot n - \frac{\alpha }{2} - m + \frac{{{m_\alpha }}}{2}}}}}{{|Q{|^{\frac{\alpha }{{2n}} + \frac{\beta }{n}}}}}} M\left( {{g_2}} \right)(x)} \right) \times \hfill \\ \left( {\sum\limits_{k = 2}^\infty {\frac{{{2^{ - (k - 1) \cdot n - \frac{\alpha }{2} - m + \frac{{{m_\alpha }}}{2}}}}}{{|Q{|^{\frac{\alpha }{{2n}} + \frac{\beta }{n}}}}}} {{\left| {{2^k}Q} \right|}^{\frac{\beta }{n}}}{{\left\| b \right\|}_{{{\mathit{\dot \Lambda }}_\beta }}}M\left( {{g_1}} \right)(x)} \right) \leqslant \hfill \\ {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{ \in ^{ - \frac{\alpha }{n} - \frac{\beta }{n}}}M\left( {{g_1}} \right)(x)M\left( {{g_2}} \right)(x) \lesssim \hfill \\ {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}M\left( {{g_1}} \right)(x)M\left( {{g_2}} \right)(x) \hfill \\ \end{array} $

    因此

    $ \begin{gathered} {D_8} \lesssim {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {M\left( {{g_1}} \right)M\left( {{g_2}} \right)} \right\|_p} \lesssim {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {M\left( {{g_1}} \right)} \right\|_{{p_1}}}{\left\| {M\left( {{g_2}} \right)} \right\|_{{p_2}}} \lesssim \hfill \\ \;\;\;\;\;\;\;\;{\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {{g_1}} \right\|_{{p_1}}}{\left\| {{g_2}} \right\|_{{p_2}}} \hfill \\ \end{gathered} $

    同理得

    $ {D_9} \lesssim {\left\| b \right\|_{{{\mathit{\dot \Lambda }}_\beta }}}{\left\| {{g_1}} \right\|_{{p_1}}}{\left\| {{g_2}} \right\|_{{p_2}}} $

    结合前面的估计,定理1得证.

    2 刻画Lipschitz空间

    定理2  设0 < β < 1,1 < r < ∞,1 < pj < q < ∞(j=1,2). $ \frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}, \frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{q}=\frac{\alpha}{n}, $$\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{r}=\frac{\alpha+\beta}{n}$.若b1=b2Lloc1,则下面结论等价:

    (ⅰ) bi$\dot \varLambda _β$(i=1,2);

    (ⅱ) $\left[\varSigma \vec{b}, I_{a}\right]$Lp1($\mathbb R^n$Lp2($\mathbb R^n$)→Fqβ,∞有界;

    (ⅲ) $\left[\varSigma \vec{b}, I_{a}\right]$Lp1($\mathbb R^n$Lp2($\mathbb R^n$)→Lr($\mathbb R^n$)有界;

    (ⅳ) $\left[\Pi \vec{b}, I_{a}\right]$Lp1($\mathbb R^n$Lp2($\mathbb R^n$)→Fqβ,∞有界;

    (ⅴ) $\left[\Pi \vec{b}, I_{a}\right]$Lp1($\mathbb R^n$Lp2($\mathbb R^n$)→Lr($\mathbb R^n$)有界.

      根据定理1、文献[11]的定理2.6及文献[9]的定理1和定理3,只需说明(ⅱ)⇒(ⅰ),(ⅳ)⇒(ⅰ)和(ⅴ)⇒(ⅰ)成立即可.

    选择x0$\mathbb R^n$t>0,令

    $ Q = Q\left( {{x_0},t} \right)\;\;\;\;{Q^0} = Q\left( {{x_0} + {z_1}t,t} \right) $

    xQy1y2Q0.根据文献[13]中第3页的证明,可得

    $ {\left( {{{\left| {\frac{{{y_1} - x}}{t}} \right|}^2} + {{\left| {\frac{{{y_2} - x}}{t}} \right|}^2}} \right)^{n - \frac{\alpha }{2}}} = {\delta ^{ - 2n + \alpha }}\sum\limits_{m = 0}^\infty {{a_m}} {{\text{e}}^{{\text{i}}\frac{\delta }{t}{v_m}\left( {{y_1} - x,{y_2} - x} \right)}} $

    (ⅱ)⇒(ⅰ):令

    $ s\left( x \right) = \overline {{\rm sgn}\left( {b\left( x \right) - {b_{{Q^0}}}} \right)} $

    $ \begin{array} [c]{l} \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left( {{b_i}(x) - {b_{{i_Q}}}} \right)} \right|} {\text{d}}x \lesssim \hfill \\ \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left( {{b_i}(x) - {b_{{i_{{Q^0}}}}}} \right)} \right|} {\text{d}}x \lesssim \hfill \\ \frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\frac{1}{{{{\left| {{Q^0}} \right|}^2}}}\int_Q s (x)\left( {\int_{{Q^0}} {\int_{{Q^0}} {\left( {\left( {{b_1}(x) - {b_1}\left( {{y_1}} \right)} \right) + \left( {{b_2}(x) - {b_2}\left( {{y_2}} \right)} \right)} \right)} } {\text{d}}{y_1}{\text{d}}{y_2}} \right){\text{d}}x = \hfill \\ \frac{1}{{{t^{3n + \beta }}}}\int_Q s \left( x \right)\left( {\int_{{Q^0}} {\int_{{Q^0}} {\frac{{\left( {{b_1}(x) - {b_1}\left( {{y_1}} \right)} \right) + \left( {{b_2}(x) - {b_2}\left( {{y_2}} \right)} \right)}}{{{{\left( {{{\left| {x - {y_1}} \right|}^2} + {{\left| {x - {y_2}} \right|}^2}} \right)}^{n - \frac{\alpha }{2}}}}}} } \cdot } \right. \hfill \\ \left. {{{\left( {{{\left| {x - {y_1}} \right|}^2} + {{\left| {x - {y_2}} \right|}^2}} \right)}^{n - \frac{\alpha }{2}}}{\text{d}}{y_1}{\text{d}}{y_2}} \right){\text{d}}x = \hfill \\ \frac{{{t^{n - \frac{\alpha }{2}}}}}{{{t^{3n + \beta }}}}\int_Q s (x)\left( {\int_{{Q^0}} {\int_{{Q^0}} {\frac{{\left( {{b_1}(x) - {b_1}\left( {{y_1}} \right)} \right) + \left( {{b_2}(x) - {b_2}\left( {{y_2}} \right)} \right)}}{{{{\left( {{{\left| {x - {y_1}} \right|}^2} + {{\left| {x - {y_2}} \right|}^2}} \right)}^{n - \alpha /2}}}} \cdot } } } \right. \hfill \\ \left. {{{\left( {\frac{{{{\left| {{y_1} - x} \right|}^2} + {{\left| {{y_2} - x} \right|}^2}}}{t}} \right)}^{n - \frac{\alpha }{2}}}{\text{d}}{y_1}{\text{d}}{y_2}} \right){\text{d}}x = \hfill \\ \frac{{{\delta ^{ - 2n + \alpha }}}}{{{t^{2n + \frac{\alpha }{2} + \beta }}}}\sum\limits_{m = 0}^\infty {{a_m}} \int_Q s (x)\left( {\int_{{Q^0}} {\int_{{Q^0}} {\frac{{\left( {{b_1}(x) - {b_1}\left( {{y_1}} \right)} \right) + \left( {{b_2}(x) - {b_2}\left( {{y_2}} \right)} \right)}}{{{{\left( {{{\left| {x - {y_1}} \right|}^2} + {{\left| {x - {y_2}} \right|}^2}} \right)}^{n - \frac{d}{2}}}}}} } \cdot } \right. \hfill \\ \left. {{{\text{e}}^{{\text{i}}\frac{\delta }{t}{v_m}\left( {{y_1},{y_2}} \right)}}{\text{d}}{y_1}{\text{d}}{y_2}} \right){{\text{e}}^{ - {\text{i}}\frac{\delta }{t}{v_m}(x,x)}}{\text{d}}x \lesssim \hfill \\ \frac{{{\delta ^{ - 2n + \alpha }}}}{{{t^{2n + \frac{\alpha }{2} + \beta }}}}\left| {\sum\limits_{m = 0}^\infty {{a_m}} \int_{{\mathbb{R}^n}} {\left[ {\varSigma \vec b,{I_a}} \right]} \left( {{\chi _{{Q^0}}}{{\text{e}}^{i\frac{\delta }{t}v_m^1{y_1}}},{\chi _{{Q^0}}}{{\text{e}}^{i\frac{\delta }{t}v_m^2{y_2}}}} \right)(x)\left( {{\chi _Q}(x){{\text{e}}^{ - {\text{i}}\frac{\delta }{t}{v_m}(x,x)}}{s^2}(x)} \right){\text{d}}x} \right| \lesssim \hfill \\ \sum\limits_{m = 0}^\infty {\left| {{a_m}} \right|{{\left\| {\left[ {\varSigma \vec b,{I_\alpha }} \right]\left( {\chi _Q^0{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^1{y_1}}},\chi _Q^{\text{o}}{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^2{y_2}}}} \right)} \right\|}_{\dot F_p^{\beta ,\infty }}}{{\left\| {{\chi _Q}} \right\|}_{\dot F_{p'}^{ - \beta ,1}}} \lesssim } \hfill \\ {\left\| {\left[ {\varSigma \vec b,{I_\alpha }} \right]} \right\|_{{L^{{p_1}}} \times {L^{{p_2}}} \to F_p^{\beta ,\infty }}} \hfill \\ \end{array} $

    由于$\left[\varSigma \vec{b}, I_{a}\right]$Lp1($\mathbb R^n$Lp2($\mathbb R^n$) → $\dot F_p^{β,∞}$有界.因此,(ⅱ)⇒(ⅰ)得证.

    (ⅳ)⇒(ⅰ):类似于(ⅱ)⇒(ⅰ),得

    $ \begin{array} [c]{l} {\left( {\frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left( {{b_i}(x) - {b_{{i_Q}}}} \right)} \right|} {\text{d}}x} \right)^2} \lesssim \hfill \\ \frac{1}{{{{\left| Q \right|}^{2 + \frac{{2\beta }}{n}}}}}\int_Q {\left| {\left( {b(x) - {b_Q}} \right)\left( {b(x) - {b_Q}} \right)} \right|} {\text{d}}x \lesssim \hfill \\ \frac{1}{{{{\left| Q \right|}^{2 + \frac{{2\beta }}{n}}}}}d\int_Q {\left| {\left( {b(x) - {b_{{Q^0}}}} \right)\left( {b(x) - {b_{{Q^0}}}} \right)} \right|} {\text{d}}x \lesssim \hfill \\ \frac{1}{{{{\left| Q \right|}^{2 + \frac{{2\beta }}{n}}}}}\frac{1}{{{{\left| {{Q^0}} \right|}^2}}}\int_Q {\left| {\int_{{Q^0}} {\int_{{Q^0}} {\left( {b(x) - b\left( {{y_1}} \right)} \right)} } \left( {b(x) - b\left( {{y_2}} \right)} \right){\text{d}}{y_1}{\text{d}}{y_2}} \right|} {\text{d}}x \lesssim \hfill \\ \frac{{{\delta ^{ - 2n + \alpha }}}}{{{t^{3n + \frac{\alpha }{2} + 2\beta }}}}\left| {\sum\limits_{m = 0}^\infty {{a_m}} \int_{{\mathbb{R}^n}} {\left[ {\Pi \vec b,{I_a}} \right]} \left( {{\chi _{{Q^0}}}{{\text{e}}^{i\frac{\delta }{t}v_m^1{y_1}}},{\chi _{{Q^0}}}{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^2{y_2}}}} \right)(x){\chi _Q}(x){{\text{e}}^{ - {\text{i}}\frac{\delta }{t}{v_m}(x,x)}}{s^2}(x){\text{d}}x} \right| \lesssim \hfill \\ \sum\limits_{m = 0}^\infty {\left| {{a_m}} \right|} {\left\| {\left[ {\Pi \vec b,{I_a}} \right]\left( {{\chi _{{Q^0}}}{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^1{y_1}}},{\chi _{{Q^0}}}{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^2{y_2}}}} \right)} \right\|_{\dot F_p^{\beta ,\infty }}}{\left\| {{\chi _Q}} \right\|_{\dot F_{p'}^{ - \beta ,1}}} \lesssim \hfill \\ {\left\| {\left[ {\Pi \vec b,{I_\alpha }} \right]} \right\|_{{L^{{p_1}}} \times {L^{{p_2}}} \to \dot F_p^{\beta ,\infty }}} \hfill \\ \end{array} $

    (ⅴ)⇒(ⅰ):类似于(ⅳ)⇒(ⅰ),可得

    $ \begin{array} [c]{l} {\left( {\frac{1}{{{{\left| Q \right|}^{1 + \frac{\beta }{n}}}}}\int_Q {\left| {\left( {{b_i}(x) - {b_{{i_Q}}}} \right)} \right|} {\text{d}}x} \right)^2} \lesssim \hfill \\ \left| {\sum\limits_{m = 0}^\infty {{a_m}} \int_{{R^n}} {\left[ {\Pi \vec b,{I_\alpha }} \right]} \left( {{\chi _{{Q^0}}}{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^1{y_1}}},{\chi _{{Q^0}}}{{\text{e}}^{\frac{{i\vec 0}}{t}v_m^2{y_2}}}} \right)(x){\chi _Q}(x){{\text{e}}^{ - i\frac{\delta }{t}{v_m}(x,x)}}{s^2}(x){\text{d}}x} \right| \lesssim \hfill \\ \sum\limits_{m = 0}^\infty {\left| {{a_m}} \right|} {\left\| {\left[ {\Pi \vec b,{I_\alpha }} \right]\left( {{\chi _{{Q^0}}}{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^1{y_1}}},{\chi _{{Q^0}}}{{\text{e}}^{{\text{i}}\frac{\delta }{t}v_m^2{y_2}}}} \right)} \right\|_{{L^r}}}{\left\| {{\chi _Q}} \right\|_{{L^{r'}}}} \lesssim \hfill \\ {\left\| {\left[ {\Pi \vec b,{I_\alpha }} \right]} \right\|_{{L^{{p_1}}} \times {L^{{p_2}}} \to {L^r}}} \hfill \\ \end{array} $

    定理2证毕.

    参考文献
    [1]
    COIFMAN R R, METER Y. On Commutators of Singular Integrals and Bilinear Singular Integrals[J]. Trans Amer Math Soc, 1975, 212: 315-331. DOI:10.1090/S0002-9947-1975-0380244-8
    [2]
    COIFMAN R R, ROCHBERG R, WEISS G. Factorization Theorems for Hardy Spaces in Several Variables[J]. AnnMath, 1976, 103(3): 611-635.
    [3]
    刘荣辉, 周疆. P-Adic域上的多线性分数次Hardy算子交换子的估计[J]. 西南师范大学学报(自然科学版), 2018, 43(2): 6-13.
    [4]
    郭庆栋, 周疆. 分数次Hardy算子的交换子在Lipschitz空间上的端点估计[J]. 西南大学学报(自然科学版), 2019, 41(8): 41-47.
    [5]
    PALUSZYNSKI M. Characterization of the Besov Spaces via the Commutator Operator of Coifman, Rochberg and Weiss[J]. IndianaUnivMathJ, 1995, 44(1): 1-17.
    [6]
    SEEGER A. A Note on Triebel-Lizorkin Spaces[J]. Approximation and Function Spaces, 1989, 22: 391-400.
    [7]
    KENIG C E, STEIN E M. Multilinear Estimates and Fractional Integration[J]. MathResLett, 1999, 6(1): 1-15.
    [8]
    周盼, 周疆. 多线性分数次积分算子在Morrey型空间上新的端点估计[J]. 西南大学学报(自然科学版), 2017, 39(12): 74-80.
    [9]
    默会霞, 张志英. 多线性分数次积分与Lipschitz函数生成的交换子的有界性[J]. 数学物理学报(A辑), 2011, 31(5): 1447-1458.
    [10]
    MO H X, YU D Y, ZHOU H P. Generalized Higher Commutators Generated by the Multilinear Fractional Integrals and Lipschitz Functions[J]. TurkJ Math, 2014, 38: 851-861. DOI:10.3906/mat-1209-48
    [11]
    王松柏, 潘继斌, 江寅生. 多线性分数次积分算子有界的充分必要条件[J]. 数学物理学报(A辑), 2015, 35(6): 1106-1114.
    [12]
    DEVORE R A, SHARPLEY R C. Maximal Functions Measuring Smoothness[M]. New York: MemAmerMathSoc, 1984.
    [13]
    CHAFFEE L. Characterizations of BMO Through Commutators of Bilinear Singular Integral Operators[J]. Mathematics Subject Classification, 2018, 2018: 1-7.
    Sufficient and Necessary Conditions for Commutators of Bilinear Fractional Integral Operators to Be Bounded on Triebel-Lizorkin Spaces
    FANG Cheng-long     
    School of Mathematics and Statistical, Yili Normal University, Yining Xinjiang 835000, China
    Abstract: In this paper, we first discuss the boundedness of linear commutators generated by bilinear fractional integral operators and Lipschitz functions on Triebel-Lizorkin spaces. Then it is proved that b1=b2 is Lipschitz function and is equivalent to the boundedness of commutator by bilinear fractional integral operators from product Lebesgue spaces to Lebesgue spaces or Triebel-Lizorkin spaces.
    Key words: bilinear fractional integral    commutators    Lipschitz function    Triebel-Lizorkin space    boundedness    
    X