Sparse Kahane--Salem--Zygmund Forms and Weighted Hardy--Littlewood Inequalities Across the Critical Endpoint

We study sparse Kahane–Salem–Zygmund constructions and weighted Hardy–Littlewood inequalities for homogeneous polynomials. For supports of cardinality $n^{d+o(1)}$, we determine the sharp power of $n$ governing the smallest norm of a unimodular $m$-linear form on ${\mathrm{\ell }}_{p_1}^n\times \cdots \times {\mathrm{\ell }}_{p_m}^n$; in the diagonal case, this yields the missing polynomial growth exponent in the coefficient-versus-supremum norm problem for $2\le p\le m$ and $2\le r\le \mathrm{\infty }$. We then introduce a diagonal weighted Hardy–Littlewood functional which, on $s=p\ge m$, agrees exactly with the classical Hardy–Littlewood coefficient norm with the same optimal constant. We determine the optimal diagonal weight exponent for $2\le p\le m$ and $1\le q\le 2$, on the full critical line $p=m$, and on a sharp part of the region $q>2$; at $q=\mathrm{\infty }$ the optimal weight exponent is obtained for every $2\le p\le m$. The sparse coefficient estimates provide the matching dimensional obstructions.