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

Abstract: For any convex set $Ω\subset\mathbb{R}<sup>n$ that does not contain affine lines, we prove the inequality $$\int_Ω\frac{|\hat{f}(x)|^{2}{ω_Ω(x)}\,dx\leq} C(n)|f|<em>{L^{1}^2,\quad} \supp\hat{f}\subsetΩ,$$ where $ω</em>Ω(x)=m(Ω\cap(2x-Ω))$. As a consequence, we derive a weak factorization for $$\PW^1(Ω)={f\in L<sup>1(\mathbb{R}<sup>n):\supp\hat{f}\subsetΩ}.$$ Furthermore, we establish a complete characterization of Schatten class Hankel operators for polyhedra for all $1\leq p&lt;\infty,$ extending the already known $1\leq p\leq 2$ range.