局部平均分解以及最佳加權上界

Abstract

本篇論文主要是研究奇異積分算子的最佳加權上界問題，而此問題相當於研究奇異積分算子在 L2 加權的有界性，而本篇主要介紹的手法是 A. Lerner 在 2013 發表論文中所使用的局部平均分解。藉由對於函數跟函數局部中位數的誤差估計，可被局部平均震盪函數的組合以及 dyadic local sharp maximal function 所控制，且他們分別又可以被控制，進而得到我們的結論。 同時我們也會介紹 D. Cruz-Uribe,J.Martell,C.Pérez 他們的一篇文章，使用局部加權來處理 Haar shift operator 的最佳加權上界問題。

In this paper,we mainly study the sharp A2 bound of singular integral operators, and this problem is equivalent to study the L2 weighted bound of the singular integral operators. The main method introduced in this thesis is the local mean decomposition invented by A.Lerner in his 2013 paper. All the estimates can be controlled by the combination of the local mean oscillation and the dyadic local sharp maximal function,and they can be respectively controlled. In addition, we will also introduce the work of D.Cruz-Uribe, J.Martell, C.Pérez, in which the sharp weighted bound of Haar shift operators are studied.