Helson Inequality, Hankel Operators, and Weak Factorization on Paley-Wiener spaces of Convex Domains

Abstract

For any convex set $\Omega \subset {\mathbb{R}}^n$ 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 ${\omega }_{\Omega }(x)=m(\Omega \cap (2x-\Omega ))$. 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 $1\le p<\infty ,$ extending the already known $1\le p\le 2$ range.

arXiv comment17 pages