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

RESUMO

For any convex set $\Omega\subset\mathbb{R}^n$ that does not contain affine lines, we prove the inequality $$\int_\Omega\frac{|\hat{f}(x)|^2}{\omega_\Omega(x)}\,dx\leq C(n)\|f\|_{L^1}^2,\quad \supp\hat{f}\subset\Omega,$$ where $\omega_\Omega(x)=m(\Omega\cap(2x-\Omega))$. As a consequence, we derive a weak factorization for $$\PW^1(\Omega)=\{f\in L^1(\mathbb{R}^n):\supp\hat{f}\subset\Omega\}.$$ 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.