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

Let $k\ge 2,n\ge 1$ 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 $\mathrm{\parallel }f{\mathrm{\parallel }}_{U^1}=\mathrm{\mid }{\int }_{{\mathbb{R}}^n}f(x)dx\mathrm{\mid }$. 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 $k\ge 2$ there exist $A(k,n)<\mathrm{\infty }$ and $p_k=2^k\mathrm{/}(k+1)$ such that $| f|{U^{k}} \leq A(k,n)|f|{p{k}}$,forall$f \in L^{p_{k}}($\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 $\mathrm{\parallel }f{\mathrm{\parallel }}_{U^k}\mathrm{/}\mathrm{\parallel }f{\mathrm{\parallel }}_{p_k}$ is nearly maximal, then $f$ is close in $L^{p_k}$ norm to an extremizer.