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Functions of Nearly Maximal Gowers–Host–Kra Norms on Euclidean Spaces

  • A. Martina NeumanEmail author
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Abstract

Let \(k\ge 2, n\ge 1\) be integers. Let \(f: {\mathbb {R}}^{n} \rightarrow {\mathbb {C}}\). The kth Gowers–Host–Kra norm of f is defined recursively by
$$\begin{aligned} \Vert f\Vert _{U^{k}}^{2^{k}} =\int _{{\mathbb {R}}^{n}} \Vert T^{h}f \cdot {\bar{f}} \Vert _{U^{k-1}}^{2^{k-1}} \, \text {d}h \end{aligned}$$
with \(T^{h}f(x) = f(x+h)\) and \(\Vert f\Vert _{U^1} = | \int _{{\mathbb {R}}^{n}} f(x)\, \text {d}x |\). These norms were introduced by Gowers (Geom Funct Anal 11:465–588, 2001) in his work on Szemerédi’s theorem, and by Host and Kra (in Ann Math 161:398–488, 2005) in ergodic setting. These norms are also discussed extensively in Tao and Vu (in Additive combinatorics, Cambridge University Press, 2016). It is shown by Eisner and Tao (in J Anal Math 117:133–186, 2012) that for every \(k\ge 2\) there exist \(A(k,n)< \infty \) and \(p_{k} = 2^{k}/(k+1)\) such that \(\Vert f\Vert _{U^{k}} \le A(k,n)\Vert f\Vert _{p_{k}}\), for all \(f \in L^{p_{k}}({\mathbb {R}}^{n})\). The optimal constant A(kn) and the extremizers for this inequality are known [9]. In this dissertation, it is shown that if the ratio \(\Vert f \Vert _{U^{k}}/\Vert f\Vert _{p_{k}}\) is nearly maximal, then f is close in \(L^{p_{k}}\) norm to an extremizer.

Keywords

Gowers–Host–Kra norms on Euclidean spaces Brascamp–Lieb inequalities Stability problems Brunn–Minkowski inequality 

Mathematics Subject Classification

42B35 

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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