What is the Lp norm of the $N$-dimensional Hadamard matrix $H = ((-1)^{i \cdot j})_{i,j}$ for $p > 2$? I know that $\|H\|_1 = N$, $\|H\|_2 = \sqrt{N}$, $\|H\|_\infty = N$ but I can't figure out what it is for other values of $p$. Can we at least give a good upper-bound on it?

Here I consider the induced norm: $\|H\|_p = \max_{x : \|x\|_p = 1} \|Hx\|_p$

(a previous version of this question incorrectly said that $\|H\|_\infty = 1$)

incorrect statement:since the Hadamard matrix is real-symmetric, by duality we have $\Vert H \Vert_p = \Vert H \Vert_q$ when $p$ and $q$ are conjugate indices. Could you please update the question? $\endgroup$1more comment