  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.
