solidot新版网站常见问题,请点击这里查看。
消息
本文已被查看4704次
Functions of nearly maximal Gowers-Host-Kra norms on Euclidean spaces. (arXiv:1711.04900v2 [math.CA] UPDATED)
来源于:arXiv
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 $\| f\|_{U^{k}} \leq
A(k,n)\|f\|_{p_{k}}$, with $p_{k} = 2^{k}/(k+1)$ 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. 查看全文>>