I am interested in the optimization problem known as "analysis regularization":

$$ {\rm argmin}_{x \in \mathbb{R}^{p}}\frac{1}{2}\|y - Ax\|_2^2 + \lambda \|D^T x\|_1,$$ where $y \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times p}$, and $D \in \mathbb{R}^{p \times d}$. In this paper, they state that the set of (global) minimizers to the above is nonempty and compact if and only if $$ {\rm kern}(A) \cap {\rm kern}(D^T) = 0.$$

How would one be able to prove this? They claim it to be a classic result, but I am not able to show it myself after a few attempts. I am unfamiliar with this area, so I may be overlooking a simpler argument.

Any references or pointers to relevant material would be helpful.