  One key feature of isogeometric analysis is that it allows smooth shape
functions. Indeed, when isogeometric spaces are constructed from $p$-degree
splines (and extensions, such as NURBS), they enjoy up to $C^{p-1}$ continuity
within each patch. However, global continuity beyond $C^0$ on so-called
multi-patch geometries poses some significant difficulties. In this work, we
consider planar multi-patch domains that have a parametrization which is only
$C^0$ at the patch interface. On such domains we study the $h$-refinement of
$C^1$-continuous isogeometric spaces. These spaces in general do not have
optimal approximation properties. The reason is that the $C^1$-continuity
condition easily over-constrains the solution which is, in the worst cases,
fully locked to linears at the patch interface. However, recent studies by Kapl
et al. have given numerical evidence that optimal convergence occurs for
bilinear two-patch geometries and cubic (or higher degree) $C^1$ splines. This
is the starting point of our study. We introduce the class of analysis-suitable
$G^1$ geometry parametrizations, which includes piecewise bilinear
parametrizations. We then analyze the structure of $C^1$ isogeometric spaces
over analysis-suitable $G^1$ parametrizations and, by theoretical results and
numerical testing, discuss their approximation properties. We also consider
examples of geometry parametrizations that are not analysis-suitable, showing
that in this case optimal convergence of $C^1$ isogeometric spaces is
prevented.
