本文考慮一種特殊的算子T:Lp→Lq。我們透過理解它的擬最佳函數的性質來
尋找最佳化函數。這類算子是一般化Radon變換的基本的例子。我們首先引進了重要的幾何物件---- 拋物球，使得稍後的估計更為簡易。接著，這些性質幫助我們處理不同型的擬最佳化函數。最後，一個好的擬最佳化序列將會趨近最佳化函數。

We study the quasi-extremizers of a specified linear operator T:Lp →Lq in order to find an extremizer. This operator is a basic example of the generalized Radon transforms. We first includes some properties of paraballs for a better situation of approximation. Then these properties help us to deal with different type of quasi-extremizers. Finally, a good quasiextremizing sequence appraoches to an extremizer, by its inner product
form.

0 Introduction 1
1 Settings and Definitions 2
2 Geometric Part 2
2.1 The Symmetric Group Gd . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 The conjugation ϕ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Paraballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Dual Paraballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Quasi-Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Analytic Part 10
4 Levels Estimate by Many Paraballs 11
5 Estimate 13
6 Quasi-Extremals 15
6.1 Represention of a (1 − δ)- quasiextremized function . . . . . . . . . 15
6.2 The ρ-M statement and The Bound . . . . . . . . . . . . . . . . 16
6.3 A proof of lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 A Sketch of Lemma 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . 20
7 Estimation using paraballs 20
8 Existence of an Extremizer 21
9 Further Discussion 22
Reference 22

[1] Michael Christ, Quasiextremals for a Radon-like transform, math.CA arXiv:
1106.0722
[2] Michael Christ, On extremals for a Radon-like transform, math.CA arXiv:
1106.0728
[3] Michael Christ, Near-extremizers of Young’s Inequality for Rd, math.CA
arXiv:1112.4875