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

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(k, n) 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.