本文探討 Hardy 上的多個等價半距，有時我們無法直接得出最大函數 (maximal
function) 之間的距等價關係，而且不容易看出這些函數彼此之間的關聯，這時
我們可以先引進限制條件較強的輔助函數，再逐步減少輔助函數的限制條件，
雖然因為引進的輔助函數的表示法會讓過程看起來很瑣碎，但最終藉由輔助函
數之間大小關係的幫助，我們可以得到原來函數之間所想要的互相等價的結論。

We study the quasi-norm equivalence of several maximal functions which
serves the definition of Hardy space. We have difficulty that we can not
obtain the result directly and need to introduce some auxiliary unctions first. These auxiliary functions have nicer properties than original maximal functions and play the role to make connections between original maximal functions. By the help of auxiliary functions, we can obtain the equivalence of original maximal functions.

Contents
1 Introduction 1
2 Settings and Definitions 1
2.1 Definition of Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . 1
2.2 Definition of a class of Maximal Functions . . . . . . . . . . . . . . 2
3 Quasi-norm Equivalence of Sever Maximal Function 3
3.1 Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Reference 28

References
[1] Grafakos, Loukas. (2014), “Modern Fourier Analysis”.Springer. 55-90
[2] Grafakos, Loukas. (2014), “Classical Fourier Analysis”.Springer. p127
[3] Elias M. Stein. (1993), “Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory Integrals”. Princeton University Press. 87-125
[4] Wheeden, Zygmund. (2015), “Measure and Integral: An Introduction to Real Analysis Second Edition”. CRC Press. Chapter 9.