AI 中文总结
本文针对不含仿射直线的凸集证明Helson不等式，由此导出Paley-Wiener空间的弱分解，并将多面体上Schatten类Hankel算子的刻画从$1\leq p\leq 2$推广到所有$1\leq p<\infty$。
AI 中文摘要
对于任意不含仿射直线的凸集$\Omega\subset\mathbb{R}^n$，我们证明不等式$$\int_\Omega\frac{|\hat{f}(x)|^2}{\omega_\Omega(x)}\\,dx\leq C(n)\\|f\\|_{L^1}^2,\quad \supp\hat{f}\subset\Omega,$$其中$\omega_\Omega(x)=m(\Omega\cap(2x-\Omega))$。作为推论，我们得到$\PW^1(\Omega)=\{f\in L^1(\mathbb{R}^n):\supp\hat{f}\subset\Omega\}$的弱分解。此外，我们建立了多面体上Schatten类Hankel算子的完整刻画，对所有$1\leq p<\infty$成立，将已知的$1\leq p\leq 2$范围进行了推广。
英文摘要
For any convex set $Ω\subset\mathbb{R}^n$ that does not contain affine lines, we prove the inequality $$\int_Ω\frac{|\hat{f}(x)|^2}{ω_Ω(x)}\,dx\leq C(n)\|f\|_{L^1}^2,\quad \supp\hat{f}\subsetΩ,$$ where $ω_Ω(x)=m(Ω\cap(2x-Ω))$. As a consequence, we derive a weak factorization for $$\PW^1(Ω)=\{f\in L^1(\mathbb{R}^n):\supp\hat{f}\subsetΩ\}.$$ Furthermore, we establish a complete characterization of Schatten class Hankel operators for polyhedra for all $1\leq p<\infty,$ extending the already known $1\leq p\leq 2$ range.
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