与由抛物算子生成的分数阶Poisson型算子相关的微分变换算子的有界性  

Boundedness of differential transforms for fractional Poisson type operators generated by parabolic operators

作　　者：张超[1] Chao Zhang

机构地区：[1]浙江工商大学统计与数学学院,杭州310018

出　　处：《中国科学：数学》2022年第11期1283-1306,共24页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11971431和11401525);浙江省自然科学基金(批准号:LY18A010006);浙江省一流学科A类(浙江工商大学统计学)资助项目。

摘　　要：本文考虑如下类型级数:T^(α)_(N)f(x,t)=∑N_(2)j=N_(1)v_(j)P^(α)_(a_(j))f(x,t)),(x,t)∈R^(N+1),N=(N_(1),N_(2))∈Z^(2),α>0的收敛性,其中,{P_(τ)^(α)}τ>0为由抛物算子L:=■_(t)-△生成的分数阶Poisson型算子,△为Laplace算子,{v_(j)}_(j∈Z)为有界实数序列,{a_(j)}_(j∈Z)为递增实数序列.本文将主要证明算子T_(N)^(α)及其极大算子T^(*)f(x,t)=sup_(N∈Z^(2))|T_(N)^(α)f(x,t)|在L^(p)(R^(n+1))空间和BMO(R^(n+1))空间上的有界性.本文还证明了极大算子T^(*)对于具有局部支撑的函数f的局部增长性与奇异积分算子的局部增长性具有相同的阶.另外,本文还证明了,如果{v_(j)}_(j∈Z)∈l^(p)(Z),则极大算子T^(*)的局部增长性介于奇异积分与Hardy-Littlewood极大算子的局部增长性之间.In this paper,we analyze the convergence of the following type of series T^(α)_(N)f(x,t)=∑N_(2)j=N_(1)v_(j)P^(α)_(a_(j))f(x,t)),(x,t)∈R^(N+1),N=(N_(1),N_(2))∈Z^(2),α>0,where{P_(τ)^(α)}τ>0 are the fractional Poisson-type operators generated by the parabolic operator L=■_(t)-△with△being the classical Laplacian,{v_(j)}_(j∈Z)a bounded real sequence and{a_(j)}_(j∈Z)an increasing real sequence.Our analysis will consist of the boundedness,in L^(p)(R^(n+1)and in BMO(R^(n+1)),of the operator TαNand its maximal operator T^(*)f(x,t)=sup_(N∈Z^(2))|.It is also shown that the local size of the maximal differential transform operators is the same as the order of a singular integral for functions f having local support.Moreover,{v_(j)}_(j∈Z)∈l^(p)(Z),we get an intermediate size between the local sizes of singular integrals and the Hardy-Littlewood maximal operator.

关 键 词：微分变换 分数阶Poisson型算子 极大算子 

分 类 号：O177[理学—数学]