# Conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~\text{otherwise}\end{cases}$$ where $$\frac{1}{p}+\frac{1}{q}=1$$ for $$p\geq 1$$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $$\|\mathbf{A}\|_{p,q}$$ defined for matrix $$\mathbf{A}$$?

$$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$$ where $$\mathbf{a}_i$$ is the $$i^{\text{th}}$$ column of matrix $$\mathbf{A}$$.

Let $$p^*$$ and $$q^*$$ be the conjugate exponents. 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.
• This has also been shown in much more generality in the context of (weighted) Besov (sequence) spaces (characterised by wavelets) - which are a vast generalisation matrix spaces endowed with $\| \cdot \|_{p, q}$ - in the paper On the smoothness and convexity of Besov spaces by Kazimierski. Mar 8 at 21:58