Weighted Estimates for the Hardy‐Littlewood Maximal Operator and Dirac Deltas
Weighted Estimates for the Hardy‐Littlewood Maximal Operator and Dirac Deltas
复制标题
Hardy-Littlewood 极大算子和狄拉克 Delta 的加权估计
DOI：
10.1112/blms/22.4.367
复制
发表时间：
1990
影响因子：
0.9
通讯作者：
C. Vitanza
中科院分区：
文献类型：
--
作者：
D. Termini;C. Vitanza
H. Carlsson在他的论文b[1]中给出了Hardy-Littlewood极大算子的弱类型（1,1）的一个新的证明（定义见下）。他的方法是基于M. de Guzman[4]的结果，该结果表明，如果弱类型（1,1）在delta的线性组合上已知，则对于所有L^ IR”)函数也是如此（参见[2]获得相关结果）。我们这里的目的是将卡尔森的结果推广到一个加权的情况。准确地说，我们证明（定理（2.1））Hardy-Littlewood极大算子对于Ax权值（定义见下文，例如[3,5]）是弱类型（1,1），通过证明它在函数的线性组合上是弱类型（1,1）。我们强调定理（2.1）是众所周知的，我们方法的新颖之处在于使用了de Guzman的技术。事实上，我们对定理的证明遵循[1]的模式，而对于定理4.1的加权版本的证明，则遵循de Guzman的证明模式。1在[4]。
In his paper [1] H. Carlsson gave a new proof of the weak type (1, 1) of the Hardy-Littlewood maximal operator (see below for definitions). His approach is based on a result of M. de Guzman [4] stating that if the weak type (1, 1) is known on the linear combinations of deltas the same is true for all L^ IR") functions (see also [2] for a related result).Our purpose here is to extend Carlsson's result to a weighted situation. Precisely, we show (Theorem (2.1)) that the Hardy-Littlewood maximal operator is of weak type (1, 1) with respect to Ax weights (see below for definitions and, for example,[3, 5]) by showing that it is of weak type (1, 1) on the linear combinations of deltas. We stress that Theorem (2.1) is well known, the novelty of our approach being in the use of de Guzman's technique. In fact our proof follows the pattern of [1] for the theorem and de Guzman's proof for what concerns the proof of a weighted version of Theorem 4.1. 1 in [4].