The exterior derivative on a compact manifold with boundary always admits a Hodge decomposition on the $L^2$-space of differential forms (Conner 1956). In contrast, for the delbar operator on a complex manifold we need the boundary to be sufficiently convex in order to obtain a Hodge decomposition (Kohn 1964). In this talk we will incorporate these two results into the framework of Lie algebroids. After defining the notions of Cauchy-Riemann structure and q-convexity in this context, we will state a theorem about Hodge decompositions for Lie algebroids that generalizes the cases of the exterior derivative and the delbar operator. As an application, we will discuss a neighbourhood theorem for a special class of Poisson submanifolds in generalized complex geometry. This last part is based on joint work with M.Bailey and G.Cavalcanti.