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

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.