Functions of nearly maximal Gowers-Host-Kra norms on Euclidean spaces

Abstract: Let $k\geq 2, n\geq 1$ be integers. Let $f: \mathbb{R}^{n} \to \mathbb{C}$. The $k$th Gowers-Host-Kra norm of $f$ is defined recursively by \begin{equation*} | f|{U^{k}}^{2^{k}} =\int{\mathbb{R}^{n}} | T^{h}f \cdot \bar{f} |{U^{k-1}}^{2^{k-1}} \, dh \end{equation*} with $T^{h}f(x) = f(x+h)$ and $|f|{U^1} = | \int_{\mathbb{R}^{n}} f(x)\, dx |$. These norms were introduced by Gowers in his work on Szemer\'edi's theorem, and by Host-Kra in ergodic setting. It's shown by Eisner and Tao that for every $k\geq 2$ there exist $A(k,n)< \infty$ and $p_{k} = 2^{k}/(k+1)$ such that $| f|{U^{k}} \leq A(k,n)|f|{p_{k}}$, for all $f \in L^{p_{k}}(\mathbb{R}^{n})$. The optimal constant $A(k,n)$ and the extremizers for this inequality are known. In this exposition, it is shown that if the ratio $| f |{U^{k}}/|f|{p_{k}}$ is nearly maximal, then $f$ is close in $L^{p_{k}}$ norm to an extremizer.