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Regularity Properties of Bilinear Maximal Function and Its Fractional Variant
Results in Mathematics ( IF 1.1 ) Pub Date : 2020-06-02 , DOI: 10.1007/s00025-020-01215-2 Feng Liu , Suying Liu , Xiao Zhang
Results in Mathematics ( IF 1.1 ) Pub Date : 2020-06-02 , DOI: 10.1007/s00025-020-01215-2 Feng Liu , Suying Liu , Xiao Zhang
We introduce and study the bilinear fractional maximal operator $$\begin{aligned} {\mathfrak {M}}_\alpha (f,g)(x)=\sup \limits _{r>0}\frac{1}{|B(O,r)|^{1-\frac{\alpha }{n}}}\int _{B(O,r)}|f(x+y)g(x-y)|dy,\ \ \ \ \ x\in {\mathbb {R}}^n, \end{aligned}$$ M α ( f , g ) ( x ) = sup r > 0 1 | B ( O , r ) | 1 - α n ∫ B ( O , r ) | f ( x + y ) g ( x - y ) | d y , x ∈ R n , where $$O=(0,0,\ldots ,0)\in {\mathbb {R}}^n$$ O = ( 0 , 0 , … , 0 ) ∈ R n and $$0\le \alpha
中文翻译：
双线性极大函数及其分数变体的正则性
我们介绍并研究了双线性分数极大算子 $$\begin{aligned} {\mathfrak {M}}_\alpha (f,g)(x)=\sup \limits _{r>0}\frac{1} {|B(O,r)|^{1-\frac{\alpha }{n}}}\int _{B(O,r)}|f(x+y)g(xy)|dy,\ \ \ \ \ x\in {\mathbb {R}}^n, \end{aligned}$$ M α ( f , g ) ( x ) = sup r > 0 1 | B ( O , r ) | 1 - α n ∫ B ( O , r ) | f ( x + y ) g ( x - y ) | dy , x ∈ R n ，其中 $$O=(0,0,\ldots ,0)\in {\mathbb {R}}^n$$ O = ( 0 , 0 , … , 0 ) ∈ R n 和$$0\le \alpha
更新日期：2020-06-02
中文翻译：
双线性极大函数及其分数变体的正则性
我们介绍并研究了双线性分数极大算子 $$\begin{aligned} {\mathfrak {M}}_\alpha (f,g)(x)=\sup \limits _{r>0}\frac{1} {|B(O,r)|^{1-\frac{\alpha }{n}}}\int _{B(O,r)}|f(x+y)g(xy)|dy,\ \ \ \ \ x\in {\mathbb {R}}^n, \end{aligned}$$ M α ( f , g ) ( x ) = sup r > 0 1 | B ( O , r ) | 1 - α n ∫ B ( O , r ) | f ( x + y ) g ( x - y ) | dy , x ∈ R n ，其中 $$O=(0,0,\ldots ,0)\in {\mathbb {R}}^n$$ O = ( 0 , 0 , … , 0 ) ∈ R n 和$$0\le \alpha
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