Hardy--Littlewood Constants Beyond Polarization

Abstract

The polynomial Hardy--Littlewood inequalities bound coefficient norms of homogeneous polynomials on ${\ell }_p$ balls by their supremum norms, with constants independent of the dimension. Their $p=\infty$ endpoint is the Bohnenblust--Hille inequality. A longstanding obstruction to connecting their constants is the polarization loss in the usual multilinear approach. We overcome this obstruction by proving $D_{m,p_m}\le (1+o(1))D_m$ whenever $p_m/m^2\to \infty$, where $D_{m,p}$ and $D_m$ are the optimal complex polynomial Hardy--Littlewood and Bohnenblust--Hille constants. At the opposite boundary, we determine the optimal endpoint constant exactly, $D_{m,m}=m$. We then prove the full first-order asymptotic $D_{m,p_m}\sim m$ whenever $p_m\ge m$ and $p_m-m\to 0$, and the exact polynomial growth $D_{m,p_m}=m^{1+o(1)}$ on the wider scale $p_m-m=o((m/\log m{)}^{1/3})$.

43 pages. The paper has been refocused on the complex setting, and the exposition has been revised throughout. No substantial changes to the main results