Let $p^*$ and $q^*$ be the conjugate exponents. Some (slightly laborious) algebra shows that the dual-norm is $\|A\|_{p^*,q^*}$Some (slightly laborious) algebra shows that the dual-norm is $\|A\|_{p^*,q^*}$. The conjugate function is the indicator function for the (unit) dual-norm ball.
added link to your preprint where the computation is preformed