西南大学学报 (自然科学版)  2019, Vol. 41 Issue (8): 41-47.  DOI: 10.13718/j.cnki.xdzk.2019.08.007
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  • 分数次Hardy算子的交换子在Lipschitz空间上的端点估计    [PDF全文]
    郭庆栋, 周疆     
    新疆大学 数学与系统科学学院, 乌鲁木齐 830046
    摘要:主要研究分数次Hardy算子和Lipschitz函数生成的交换子在Lipschitz空间上的端点估计.分数次积分算子的方法不适用于分数次Hardy算子,将给出新的方法,同时也将考虑分数次极大算子的交换子的结果.
    关键词分数次Hardy算子的交换子    Lipschitz空间    端点估计    分数次极大算子    

    f是定义在$\mathbb{R}_{+}$上的非负可积函数,经典的Hardy算子定义如下:

    $ H f(x)=\frac{1}{x} \int_{0}^{x} f(y) \mathrm{d} y \quad x>0 $

    文献[1]在证明Hilbert双重级数定理的过程中得到了如下著名的Hardy积分不等式:

    $ {\left\| {Hf} \right\|_{{L^p}\left( {{\mathbb{R}_ + }} \right)}} \leqslant \frac{p}{{p - 1}}{\left\| f \right\|_{{L^p}\left( {{\mathbb{R}_ + }} \right)}} $

    此后Hardy算子逐渐被人们广泛关注[2-5].文献[6]定义了如下n维Hardy算子的形式,并得到了与1维结果平行的n维积分不等式:

    定义1  设1<p<∞,fLloc($\mathbb{R}$n). n维Hardy算子被定义为

    $ \mathscr{H} f(x)=\frac{1}{|x|^{n}} \int_{|y|<|x|} f(y) \mathrm{d} y \quad x \in \mathbb{R}^{n} \backslash\{0\} $

    随着Hardy算子研究的逐渐深入,Hardy算子与Lipschitz(简记为Lipα)或BMO等函数生成的交换子也被人们广泛研究,Hardy算子对刻画函数空间的性质有着非常重要的意义. Hardy算子的交换子定义为

    $ [b, \mathscr{H}] f=b \mathscr{H} f-\mathscr{H}(f b) $

    其中b是定义在$\mathbb{R}$n上的局部可积函数.本文把[b$\mathscr{H} $]简记为$\mathscr{H} $b,可知当n=1时,$\mathscr{H} $b=Hb.文献[7]证明了HbLp($\mathbb{R}$n)(简记为Lp)空间上的有界性,其中b属于单边二进中心BMO函数,1<p<∞.文献[8]首次定义了如下经典的分数次Hardy算子,并得到分数次Hardy算子及其交换子刻画Lp空间的相应结果.最近关于分数次Hardy算子及其交换子的相关结论可参考文献[9-15].

    定义2  设0<βnfLloc($\mathbb{R}$n).分数次Hardy算子以及分数次极大算子的交换子的定义分别为:

    $ \mathscr{H}_{\beta, b}(f)(x)=\frac{1}{|x|^{n-\beta}} \int_{|y|<|x|}(b(x)-b(y)) f(y) \mathrm{d} y \quad x \in \mathbb{R}^{n} \backslash\{0\} $
    $ \mathscr{M}_{\beta, b}(f)(x)=\sup \limits_{x \in B} \frac{1}{|B|^{1-\frac{\beta}{n}}} \int_{B}|b(x)-b(y)||f(y)| \mathrm{d} y $

    文献[16]得到了分数次积分算子从Morrey空间到Lipα空间上的端点估计.受此启发,本文将考虑分数次Hardy算子的交换子从Morrey空间到Lipα空间上的端点估计,同时针对分数次极大算子的交换子在端点处的情形进行研究.虽然分数次极大算子的交换子与分数次Hardy算子的交换子有控制关系,但是两者的处理方法不同.本文的主要目的是给出分数次Hardy算子的交换子以及分数次极大算子的交换子的端点估计.

    定义3  设0<α<1.如果f是定义在$\mathbb{R}$n上的函数,且f满足

    $ \|f\|_{\mathrm{Lip}_{\alpha}}=\sup \limits_{x, y \in \mathbb{R}^{n}, x \neq y} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}}<\infty $

    则称f属于Lipschitz空间.

    定义4  设1<qp<∞.如果函数fLlocq($\mathbb{R}$n)满足

    $ {\left\| f \right\|_{M_q^p}} = \mathop {\sup }\limits_{x \in B} \frac{1}{{|B{|^{\frac{1}{q} - \frac{1}{p}}}}}{\left( {\int_B | f(y){|^q}{\rm{d}}y} \right)^{\frac{1}{q}}} < \infty $

    则称f属于Morrey空间Mqp,其中B表示$\mathbb{R}$n中的一个开球.

    本文中的C通常表示与空间维数等有关的常数,每次出现时其值可能并不相同. B表示以x为中心,$\mathbb{R}$>0为半径的球体.对于$\mathbb{R}$n中的可测子集E,用|E|表示E的Lebesgue测度.

    定理1  设$0<\alpha<1, 0<\beta<\frac{n}{p}$和1<qp<∞,如果$0<\alpha+\beta-\frac{n}{p}<\min \left\{1, n-\frac{n}{p}\right\}$f(x)∈Mqpb∈Lipα$\mathscr{H} $βb(f)(0)=0,则$\mathscr{H} $βb(f)∈$\operatorname{Lip}_{\alpha+\beta-\frac{n}{p}}$,且存在常数C>0,使得

    $ \left\|\mathscr{H}_{\beta, b}(f)\right\|_{\operatorname{Lip}_{a+\beta-\frac{n}{p}}} \leqslant C\|f\|_{M_{q}^{p}} $

    定理2  设0<α<1,0<βn和1<qp<∞,如果$\alpha+\beta=\frac{n}{p}$,和b∈Lipα,则$\mathscr{M} $βb(f)∈L,且存在常数C>0,使得

    $ \left\|\mathscr{M}_{\beta, b}(f)\right\|_{L^{\infty}} \leqslant C\|f\|_{M_{q}^{p}} $

    定理3  设$0 <\alpha<1,0<\beta<\frac{n}{p}$和1<qp<∞,如果$0<\alpha+\beta-\frac{n}{p}<\min \left\{1, n-\frac{n}{p}\right\}$$f \in M_q^p$,和b∈Lipα,则$\mathscr{M} $βb(f)∈$\operatorname{Lip}_{\alpha+\beta-\frac{n}{p}}$,且存在常数C>0,使得

    $ \left\|\mathscr{M}_{\beta, b}(f)\right\|_{\operatorname{Lip}_{\alpha+\beta-\frac{n}{p}}} \leqslant C\|f\|_{M_{q}^{p}} $

    定理1的证明  对任意固定的点x1x2$\mathbb{R}$n,不妨假设|x1|≥|x2|.下面分两种情况讨论:

    情况1   |x1|≥2|x2|.

    根据|x1|≥2|x2|,可得|x1|≤2(|x1|-|x2|),则有

    $ \begin{array}{l} \left| {{{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_1}} \right) - {{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_2}} \right)} \right| \le \\ \frac{1}{{{{\left| {{x_1}} \right|}^{n - \beta }}}}\int_{|y| < \left| {{x_1}} \right|} {\left| {\left( {b\left( {{x_1}} \right) - b(y)} \right)f(y)} \right|{\rm{d}}y} + \frac{1}{{{{\left| {{x_2}} \right|}^{n - \beta }}}}\int_{|y| < \left| {{x_2}} \right|} {\left| {\left( {b\left( {{x_2}} \right) - b(y)} \right)f(y)} \right|{\rm{d}}y} \le \\ C\left( {{{\left| {{x_1}} \right|}^{a + \beta - \frac{n}{p}}} + {{\left| {{x_2}} \right|}^{a + \beta - \frac{n}{\rho }}}} \right){\left\| f \right\|_{M_q^p}} \le C{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)^{a + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^p}} \end{array} $

    情况2   |x1|<2|x2|.

    首先令Ω=B(0,a)\B(0,b),且假设Bi(i=1,2,…)是一列互不相交的极大球体族,其中每个球的半径为$\frac{a-b}{2}$,中心属于集合$\left\{t_{0} :\left|t_{0}\right|=\frac{a+b}{2}\right\}$,则∪iBiΩ.

    易知Ω⊂∪i3Bi成立.否则,令$\tilde{t}$Ω$\tilde{t}$⊈∪i3Bi,点t0在直线${\iota _{O, {\mathcal{\tilde z}}}}$上满足$\left|t_{0}\right|=\frac{a+b}{2}$,则

    $ \left|\tilde{t}-t_{0}\right| \leqslant \frac{a-b}{2} \quad\left|t_{i}-\tilde{t}\right| \geqslant \frac{3(a-b)}{2} $

    其中tiBi的中心,因此

    $ \left|t_{i}-t_{0}\right| \geqslant\left|t_{i}-\tilde{t}\right|-\left|\tilde{t}-t_{0}\right| \geqslant a-b $

    此时,球$B_{0}=B\left(t_{0}, \frac{a-b}{2}\right)$被包含在极大球体族∪iBi中,与极大球体族的定义矛盾.

    可得

    $ \begin{array}{l} \left| {{{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_1}} \right) - {{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_2}} \right)} \right| \le \\ \left| {{{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_1}} \right) - {{\left( {\frac{{\left| {{x_2}} \right|}}{{\left| {{x_1}} \right|}}} \right)}^{n - \beta }}{{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_2}} \right)} \right| + \left| {{{\left( {\frac{{\left| {{x_2}} \right|}}{{\left| {{x_1}} \right|}}} \right)}^{n - \beta }}{{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_2}} \right) - {{\mathscr{H}}_{\beta ,b}}(f)\left( {{x_2}} \right)} \right| = \\ {I_1} + {I_2} \end{array} $

    下面处理I1.令

    $ a=\sqrt{\left|x_{1}\right|^{2}-|y|^{2}} \quad b=\sqrt{\left|x_{2}\right|^{2}-|y|^{2}} $
    $ \mathit{\Omega } = B(0,a)\backslash B(0,b) $

    $ a+b \leqslant\left|x_{1}\right|+\left|x_{2}\right| \quad a^{2}-b^{2}=\left|x_{1}\right|^{2}-\left|x_{2}\right|^{2} $

    可得a-b≥|x1|-|x2|以及

    $ |\mathit{\Omega }| = C\left( {{a^n} - {b^n}} \right) \le C{\left( {{a^2} - {b^2}} \right)^{\frac{n}{2}}} \le C{\left( {{{\left| {{x_1}} \right|}^2} - {{\left| {{x_2}} \right|}^2}} \right)^{\frac{n}{2}}} \le C{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)^n} $

    根据

    $ 0<\alpha+\beta-\frac{n}{p}<\min \left\{1, n-\frac{n}{p}\right\} \quad \beta \leqslant \frac{n}{p} $

    $ \begin{array}{l} {I_1} \le |\frac{1}{{{{\left| {{x_1}} \right|}^{n - \beta }}}}\int_{\left| {{x_2}} \right| \le |y| < \left| {{x_1}} \right|} {\left( {b\left( {{x_1}} \right) - b(y)} \right)} f(y){\rm{d}}y + \frac{1}{{{{\left| {{x_1}} \right|}^{n - \beta }}}}\int_{|y| < \left| {{x_2}} \right|} {\left( {b\left( {{x_1}} \right) - b(y)} \right)} f(y){\rm{d}}y - \\ \;\;\;\;\;\;\frac{1}{{{{\left| {{x_1}} \right|}^{n - \beta }}}}\int_{|y| < \left| {{x_2}} \right|} {\left( {b\left( {{x_2}} \right) - b(y)} \right)} f(y){\rm{d}}y| \le \\ \;\;\;\;\;\;C\left[ {\frac{1}{{{{\left| {{x_1}} \right|}^{n - \alpha - \beta }}}}\int_{\left| {{x_2}} \right| \le |y| < \left| {{x_1}} \right|} | f(y)|{\rm{d}}y + \frac{{{{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^\alpha }}}{{{{\left| {{x_1}} \right|}^{n - \beta }}}}\int_{|y| < \left| {{x_2}} \right|} | f(y)|{\rm{d}}y} \right] \le \\ \;\;\;\;\;\;C\left[ {\frac{1}{{{{\left| {{x_1}} \right|}^{n - \alpha - \beta }}}}\int_{B(0,a)\backslash B(0,b)} | f(y)|{\rm{d}}y + {{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^{a + \beta - \frac{n}{p}}}{{\left\| f \right\|}_{M_q^p}}} \right] \le \\ \;\;\;\;\;\;C\left[ {\frac{1}{{{{\left| {{x_1}} \right|}^{n - \alpha - \beta }}}}\sum\limits_i {\int_{3{B_i}} | } f(y)|{\rm{d}}y + {{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^{\alpha + \beta - \frac{n}{p}}}{{\left\| f \right\|}_{M_q^p}}} \right] \le \\ \;\;\;\;\;\;C\left[ {\frac{{|\mathit{\Omega }|{{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^{ - \frac{n}{p}}}}}{{{{\left| {{x_1}} \right|}^{n - \alpha - \beta }}}}{{\left\| f \right\|}_{M_q^p}} + {{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^{\alpha + \beta - \frac{n}{p}}}{{\left\| f \right\|}_{M_q^p}}} \right] \le \\ \;\;\;\;\;\;C\left[ {\frac{{{{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^{n - \frac{n}{p}}}}}{{{{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^{n - \alpha - \beta }}}}{{\left\| f \right\|}_{M_q^p}} + {{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)}^{\alpha + \beta - \frac{n}{p}}}{{\left\| f \right\|}_{M_q^p}}} \right] \le \\ \;\;\;\;\;\;C{\left( {\left| {{x_1}} \right| - \left| {{x_2}} \right|} \right)^{\alpha + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^p}} \end{array} $

    下面处理I2.首先根据微分中值定理,在直线${\iota_{|{x_2}{\rm{|}}, |{x_1}{\rm{|}}}}$上存在一点θ,使得

    $ \frac{1}{\left|x_{2}\right|^{n}}-\frac{1}{\left|x_{1}\right|^{n}}=C \frac{\left|x_{1}\right|-\left|x_{2}\right|}{|\theta|^{n+1}} \leqslant C \frac{\left|x_{1}\right|-\left|x_{2}\right|}{\left|x_{2}\right|^{n+1}} $

    再由|x2|>|x1|-|x2|可得

    $ I_{2} \leqslant C \frac{\left|x_{1}\right|-\left|x_{2}\right|}{\left|x_{2}\right|^{n+1-\alpha-\beta}} \int_{|y|<\left|x_{2}\right|}|f(y)| \mathrm{d} y \leqslant C\left(\left|x_{1}\right|-\left|x_{2}\right|\right)^{a+\beta-\frac{n}{p}}\|f\|_{M_{q}^{\rho}} $

    综合情况1与情况2,可得

    $ \left|\mathscr{H}_{\beta, b}(f)\left(x_{1}\right)-\mathscr{H}_{\beta, b}(f)\left(x_{2}\right)\right| \leqslant C\left(\left|x_{1}\right|-\left|x_{2}\right|\right)^{\alpha+\beta-\frac{n}{p}}\|f\|_{M_{q}^{p}} $

    现在考虑x1$\mathbb{R}$n\{0},x2=0时的情况,根据$\mathscr{H} $βb(f)(0)=0,可得

    $ \left|\mathscr{H}_{\beta, b}(f)\left(x_{1}\right)-\mathscr{H}_{\beta, b}(f)(0)\right|=\left|\mathscr{H}_{\beta, b}(f)\left(x_{1}\right)\right| \leqslant C\left|x_{1}\right|^{\alpha+\beta-\frac{n}{p}}\|f\|_{M_{q}^{p}} $

    因此,我们完成了定理1的全部证明.

    定理2的证明  对任意的点xB,根据Hölder不等式以及$\alpha+\beta=\frac{n}{p}$,有

    $ \frac{1}{|B|^{1-\frac{\beta}{n}}} \int_{B}|b(x)-b(y)||f(y)| \mathrm{d} y \leqslant C \frac{1}{|B|^{1-\frac{a+\beta}{n}}} \int_{B}|f(y)| \mathrm{d} y \leqslant C\|f\|_{M_{q}^{p}} $ (1)

    根据(1)式易知定理2成立.

    定理3的证明  对任意固定的点xy$\mathbb{R}$n,不妨假设$\mathscr{M} $βb(f)(x)≥$\mathscr{M} $βb(f)(y).通过计算可得$\mathscr{M} $βb(f)(x)≤CfMqp,而fMqp,则有$\mathscr{M} $βb(f)(x)≤∞.因此,对任意的ε>0,存在方体B1=B(zr)∋x,使得

    $ \frac{1}{\left|B_{1}\right|^{1-\frac{\beta}{n}}} \int_{B_{1}}|b(x)-b(t)||f(t)| \mathrm{d} t+\varepsilon>\mathscr{M}_{\beta, b}(f)(x) $ (2)

    存在方体B2=B(zr+|x-y|)∋y,使得

    $ \frac{1}{\left|B_{2}\right|^{1-\frac{\beta}{n}}} \int_{B_{2}}|b(y)-b(t)||f(t)| \mathrm{d} t \leqslant \mathscr{M}_{\beta, b}(f)(y) $ (3)

    根据(2),(3)式可得

    $ \begin{array}{l} \left| {{{\mathscr{M}}_{\beta ,b}}(f)(x) - {{\mathscr{M}}_{\beta ,b}}(f)(y)} \right| \le \\ {r^\beta }\left| {\frac{1}{{\left| {{B_1}} \right|}}\int_{{B_1}} {\left| {b(x) - b(t)} \right|\left| {f(t)} \right|{\rm{d}}t} - \frac{1}{{\left| {{B_2}} \right|}}\int_{{B_2}} {\left| {b(y) - b(t)} \right|\left| {f(t)} \right|{\rm{d}}t} } \right| + \varepsilon \end{array} $

    下面我们将分两种情况讨论:

    情况1   r≤|x-y|.

    $ \begin{array}{l} \left| {{{\mathscr{M}}_{\beta ,b}}(f)(x) - {{\mathscr{M}}_{\beta ,b}}(f)(y)} \right| \le \\ C{r^{\alpha + \beta }}\left| {\frac{1}{{\left| {{B_1}} \right|}}\int_{{B_1}} {\left| {f(t)} \right|{\rm{d}}t} } \right| + \left| {\frac{1}{{\left| {{B_2}} \right|}}\int_{{B_2}} {\left| {f(t)} \right|{\rm{d}}t} } \right| + \varepsilon \le \\ C{r^{\alpha + \beta }}{\left\| f \right\|_{M_q^p}}\left( {{{\left| {{B_1}} \right|}^{\frac{{ - 1}}{p}}} + {{\left| {{B_2}} \right|}^{\frac{{ - 1}}{p}}}} \right) + \varepsilon \le \\ C{(|x - y|)^{\alpha + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^p}} + \varepsilon \end{array} $

    情况2   r≥|x-y|.

    Ω=B(zr+|x-y|)\B(zr),且假设Ri(i=1,2,…)是一列互不相交的极大球体族.类似定理1中的情况2,容易得∪i$\mathbb{R}$iΩ⊂∪i3$\mathbb{R}$i.因此

    $ \begin{array}{l} \left| {\frac{1}{{{{\left| {{B_1}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} | b(x) - b(t)||f(t)|{\rm{d}}t - \frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_2}} | b(y) - b(t)||f(t)|{\rm{d}}t} \right| \le \\ \left| {\frac{1}{{{{\left| {{B_1}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} | b(x) - b(y)||f(t)|{\rm{d}}t} \right| + \\ \left| {\frac{1}{{{{\left| {{B_1}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} | b(y) - b(t)||f(t)|{\rm{d}}t - \frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_2}} | b(y) - b(t)||f(t)|{\rm{d}}t} \right| = \\ {I_1} + {I_2} \end{array} $

    对于I1,根据$\beta \leqslant \frac{n}{p}$,有

    $ \begin{array}{l} {I_1} \le C\frac{{{{(|x - y|)}^\alpha }}}{{{{\left| {{B_1}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} | f(t)|{\rm{d}}t \le C\frac{{{{(|x - y|)}^\alpha }}}{{{{\left| {{B_1}} \right|}^{1 - \frac{\beta }{n}}}}}{\left| {{B_1}} \right|^{1 - \frac{1}{p}}}{\left\| f \right\|_{M_q^p}} \le \\ \;\;\;\;\;\;C{(|x - y|)^{a + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^\rho }} \end{array} $

    对于I2,进一步处理为

    $ \begin{array}{l} {I_2} \le \left| {\frac{1}{{{{\left| {{B_1}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} | b(y) - b(t)||f(t)|{\rm{d}}t - \frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} | b(y) - b(t)||f(t)|{\rm{d}}t} \right| + \\ \;\;\;\;\;\;\left| {\frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} | b(y) - b(t)||f(t)|{\rm{d}}t - \frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_2}} | b(y) - b(t)||f(t)|{\rm{d}}t} \right| = \\ \;\;\;\;\;\;{J_1} + {J_2} \end{array} $

    类似于定理1中情况2的证明,根据

    $ 0<\alpha+\beta-\frac{n}{p}<\min \left\{1, n-\frac{n}{p}\right\} $

    $ \begin{array}{l} {J_1} \le C\left| {\frac{{|x - y|}}{{{{\left| {{B_1}} \right|}^{1 + \frac{{1 - \beta }}{n}}}}}\int_{{B_1}} {\left| {b(y) - b(t)} \right|\left| {f(t)} \right|} {\rm{d}}t} \right| \le \\ \;\;\;\;\;\;\;C\left| {\frac{{|x - y|}}{{{{\left| {{B_1}} \right|}^{1 + \frac{{1 - \beta - \alpha }}{n}}}}}\int_{{B_1}} {\left| {f(t)} \right|} {\rm{d}}t} \right| \le \\ \;\;\;\;\;\;\;C{\left( {\left| {x - y} \right|} \right)^{\alpha + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^p}} \end{array} $

    对于J2,可处理为

    $ \begin{array}{l} {J_2} \le C\frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{{\alpha + \beta }}{n}}}}}\int_\mathit{\Omega } {\left| {f(t)} \right|{\rm{d}}t} \le \\ \;\;\;\;\;\;\;C\frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{{\alpha + \beta }}{n}}}}}\int_{\bigcup\nolimits_i {3{R_i}} } {\left| {f(t)} \right|{\rm{d}}t} \le C\frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{{\alpha + \beta }}{n}}}}}\sum\limits_i {{{\left| {{R_i}} \right|}^{1 - \frac{1}{p}}}{{\left\| f \right\|}_{M_q^p}}} \le \\ \;\;\;\;\;\;\;C\frac{{\left| \mathit{\Omega } \right|{{\left| {x - y} \right|}^{ - \frac{n}{p}}}}}{{{{\left| {{B_2}} \right|}^{1 - \frac{{\alpha + \beta }}{n}}}}}{\left\| f \right\|_{M_q^p}} \le C{\left| {x - y} \right|^{\alpha + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^p}} \end{array} $

    易知

    $ \begin{array}{l} \left| {\frac{1}{{{{\left| {{B_1}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_1}} {\left| {b(x) - b(t)} \right|\left| {f(t)} \right|{\rm{d}}t} - \frac{1}{{{{\left| {{B_2}} \right|}^{1 - \frac{\beta }{n}}}}}\int_{{B_2}} {\left| {b(y) - b(t)} \right|\left| {f(t)} \right|{\rm{d}}t} } \right| \le \\ C{(|x - y|)^{\alpha + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^p}} \end{array} $

    $ \begin{array}{l} \left| {{{\mathscr{M}}_{\beta ,b}}(f)(x) - {{\mathscr{M}}_{\beta ,b}}(f)(y)} \right| \le \\ {r^\beta }\left| {\frac{1}{{\left| {{B_1}} \right|}}\int_{{B_1}} {\left| {b(x) - b(t)} \right|\left| {f(t)} \right|{\rm{d}}t} - \frac{1}{{\left| {{B_2}} \right|}}\int_{{B_2}} {\left| {b(y) - b(t)} \right|\left| {f(t)} \right|{\rm{d}}t} } \right| + \varepsilon \le \\ C{(|x - y|)^{\alpha + \beta - \frac{n}{p}}}{\left\| f \right\|_{M_q^p}} + \varepsilon \end{array} $

    由情况1与情况2,可得

    $ \left|\mathscr{M}_{\beta, b}(f)(x)-\mathscr{M}_{\beta, b}(f)(y)\right| \leqslant C(|x-y|)^{\alpha+\beta-\frac{n}{p}}\|f\|_{M_{q}^{p}} $

    至此,我们完成了定理3的全部证明.

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    Endpoint Estimates of the Commutator of Fractional Hardy Operator in the Lipschitz Space
    GUO Qing-dong, ZHOU Jiang     
    School of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
    Abstract: In this paper, we mainly study the endpoint estimates of the commutators generated by fractional Hardy operators with Lipschitz functions in the Lipschitz space. In addition, the method of fractional integral operator is not suitable for the fractional Hardy operator, for which this paper gives a new method. The results of commutators of fractional maximal operators are also considered.
    Key words: commutator of fractional Hardy operators    Lipschitz space    endpoint estimate    fractional maximal operator    
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