  This paper discusses the incorporation of local sparsity information, e.g. in
each pixel of an image, via minimization of the $\ell^{1,\infty}$-norm. We
discuss the basic properties of this norm when used as a regularization
functional and associated optimization problems, for which we derive equivalent
reformulations either more amenable to theory or to numerical computation.
Further focus of the analysis is put on the locally 1-sparse case, which is
well motivated by some biomedical imaging applications. Our computational
approaches are based on alternating direction methods of multipliers (ADMM) and
appropriate splittings with augmented Lagrangians. Those are tested for a model
scenario related to dynamic positron emission tomography (PET), which is a
functional imaging technique in nuclear medicine. The results of this paper
provide insight into the potential impact of regularization with the
$\ell^{1,\infty}$-norm for local sparsity in appropriate settings. However, it
also indicates several shortcomings, possibly related to the non-tightness of
the functional as a relaxation of the $\ell^{0,\infty}$-norm.
