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

For any convex set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ that does not contain affine lines, we prove the inequality $${\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }\widehat{f}(x){\mathrm{\mid }}^2}{{\omega }_{\mathrm{\Omega }}(x)}dx\le C(n)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1}^2,\text{supp}\widehat{f}\subset \mathrm{\Omega },$$ where ${\omega }_{\mathrm{\Omega }}(x)=m(\mathrm{\Omega }\cap (2x-\mathrm{\Omega }))$. As a consequence, we derive a weak factorization for $${\text{PW}}^1(\mathrm{\Omega })=\{f\in L^1({\mathbb{R}}^n):\text{supp}\widehat{f}\subset \mathrm{\Omega }\}\mathrm{.}$$ Furthermore, we establish a complete characterization of Schatten class Hankel operators for polyhedra for all $1\le p<\mathrm{\infty },$ extending the already known $1\le p\le 2$ range.