Polynomial Bohnenblust--Hille Bounds and Hardy--Littlewood Inequalities for Finite Local Systems
The complex polynomial Bohnenblust--Hille inequality states that, for a $d$-homogeneous polynomial $P:\mathbb C^N\to\mathbb C$, $P(z)=\sum_{|α|=d}c_αz^α$, the coefficient norm $(\sum_{|α|=d}|c_α|^{p_d})^{1/p_d}$ is bounded by $B_d^{\mathrm{pol}}\|P\|_\infty$, where $p_d=2d/(d+1)$ and $B_d^{\mathrm{pol}}$ is independent of $N$. Here $\|P\|_\infty$ is the supremum norm on the unit polydisc. This inequality has a Boolean analogue for the Fourier--Walsh coefficients of functions of degree at most $d$, with the supremum taken over $\{-1,1\}^N$. In this setting, Defant, Mastyło and Pérez obtained subexponential bounds, and Ivanisvili established polynomial growth. Further polynomial estimates were obtained by Slote, Tseng and Volberg in the Boolean setting and by Slote and Volberg for cyclic groups. We study these coefficient inequalities in a broader framework of finite systems of functions that encompasses and extends the Boolean and cyclic settings. This framework also admits a natural $\ell_p$ formulation, leading to Hardy--Littlewood-type inequalities. For orthonormal systems, we obtain polynomial bounds for the Bohnenblust--Hille constants and, for sufficiently large $p$, for the constants in these Hardy--Littlewood-type inequalities. In the Boolean case, the polynomial power is $5/2$, with a cubic logarithmic factor.