各向异性变指标Hardy-Lorentz空间上的Fourier变换及其应用  

Fourier transform of variable anisotropic Hardy-Lorentz spaces and its applications

作　　者：刘军 张明东 Jun Liu;Mingdong Zhang

机构地区：[1]中国矿业大学数学学院,徐州221116

出　　处：《中国科学：数学》2023年第4期577-590,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12001527);江苏省自然科学基金(批准号:BK20200647);中国博士后科学基金(批准号:2021M693422)资助项目。

摘　　要：设p(·)是R^(n)上的一个满足全局log-H?lder连续性条件的可测函数,其本性上确界p_(+)和下确界p_(-)满足0<p_(-)≤p_(+)≤1.另设q∈(0,1],A是一个伸缩矩阵,Hp_(A)^(()p(·)),^(q)(R^(n))表示通过径向主极大函数定义的各向异性变指标Hardy-Lorentz空间.本文利用Hp_(A)^(()p(·)),^(q)(R^(n))的原子分解,证明了该空间中的元素f的Fourier变换■在缓增分布意义下等于R^(n)上的一个连续函数F.进一步地,本文得到了上述函数F的一个点态控制,即它被f的Hp_(A)^(()p(·)),^(q)(R^(n))范数和相关于A的转置矩阵的齐次拟模的乘积点态控制.作为应用,本文还获得了F在原点的高阶收敛性以及Hp_(A)^(()p(·)),^(q)(R^(n))上的Hardy-Littlewood不等式.本文推广了Taibleson和Weiss关于经典Euclid空间上Hardy空间H^(p)(R^(n))的相应结果,即使对于各向同性的常指标Hardy-Lorentz空间H^(p),q(R^(n)),上述结果也是新的.Let p(·)be a measurable function on Rnsatisfying the globally log-H?lder continuous condition.Its essential supremum p_+and infimum p_-satisfy 0 p_-≤p_+≤1.Let Hp_A~(p(·)),~q(R~n)be the variable anisotropic Hardy-Lorentz spaces defined via the radial grand maximal function,where q∈(0,1]and A is an expansive matrix.In this article,by using the atomic decomposition of Hp_A~(p(·)),~q(R~n),we prove that the Fourier transform of f∈Hp_A~(p(·)),~q(R~n)equals a continuous function F on Rnin the sense of tempered distributions.Moreover,the function F can be pointwisely controlled by the product of the Hp_A~(p(·)),~q(R~n)norm of f and the homogeneous quasi-norm associated with the transpose matrix of A.As applications,we obtain a higher order of convergence for the function F at the origin,and an analogue of Hardy-Littlewood inequalities in the present setting of Hp_A~(p(·)),~q(R~n).All these results are new even for the isotropic Hardy-Lorentz spaces H~(p,q)(R~n)and they generalize the corresponding conclusions of Taibleson and Weiss on classical Hardy spaces H~p(R~n).

关 键 词：各向异性Euclid空间 变指标Hardy(-Lorentz)空间 FOURIER变换 HARDY-LITTLEWOOD不等式 

分 类 号：O177[理学—数学]