  Isogeometric analysis is a powerful paradigm which exploits the high
smoothness of splines for the numerical solution of high order partial
differential equations. However, the tensor-product structure of standard
multivariate B-spline models is not well suited for the representation of
complex geometries, and to maintain high continuity on general domains special
constructions on multi-patch geometries must be used. In this paper we focus on
adaptive isogeometric methods with hierarchical splines, and extend the
construction of $C^1$ isogeometric spline spaces on multi-patch planar domains
to the hierarchical setting. We introduce a new abstract framework for the
definition of hierarchical splines, which replaces the hypothesis of local
linear independence for the basis of each level by a weaker assumption. We also
develop a refinement algorithm that guarantees that the assumption is fulfilled
by $C^1$ splines on certain suitably graded hierarchical multi-patch mesh
configurations, and prove that it has linear complexity. The performance of the
adaptive method is tested by solving the Poisson and the biharmonic problems.
