  We study the problem of computing the $p\rightarrow q$ norm of a matrix $A
\in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\,
R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the
spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$,
$q=1$), and has been widely studied in various regimes. When $p \geq q$, the
problem exhibits a dichotomy: constant factor approximation algorithms are
known if $2 \in [q,p]$, and the problem is hard to approximate within almost
polynomial factors when $2 \notin [q,p]$.
  The regime when $p < q$, known as \emph{hypercontractive norms}, is
particularly significant for various applications but much less well
understood. The case with $p = 2$ and $q > 2$ was studied by [Barak et al,
STOC'12] who gave sub-exponential algorithms for a promise version of the
problem (which captures small-set expansion) and also proved hardness of
approximation results based on the Exponential Time Hypothesis. However, no
NP-hardness of approximation is known for these problems for any $p < q$.
  We study the hardness of approximating matrix norms in both the above cases
and prove the following results:
  - We show that for any $1< p < q < \infty$ with $2 \notin [p,q]$,
$\|A\|_{p\rightarrow q}$ is hard to approximate within
$2^{O(\log^{1-\epsilon}\!n)}$ assuming $NP \not\subseteq
BPTIME(2^{\log^{O(1)}\!n})$. This suggests that, similar to the case of $p \geq
q$, the hypercontractive setting may be qualitatively different when $2$ does
not lie between $p$ and $q$.
  - For all $p \geq q$ with $2 \in [q,p]$, we show $\|A\|_{p\rightarrow q}$ is
hard to approximate within any factor than $1/(\gamma_{p^*} \cdot \gamma_q)$,
where for any $r$, $\gamma_r$ denotes the $r^{th}$ norm of a gaussian, and
$p^*$ is the dual norm of $p$.
