  While inverse estimates in the context of radial basis function approximation
on boundary-free domains have been known for at least ten years, such theorems
for the more important and difficult setting of bounded domains have been
notably absent. This article develops inverse estimates for finite dimensional
spaces arising in radial basis function approximation and meshless methods. The
inverse estimates we consider control Sobolev norms of linear combinations of a
localized basis by the $L_p$ norm over a bounded domain. The localized basis is
generated by forming local Lagrange functions for certain types of RBFs (namely
Mat\'ern and surface spline RBFs). In this way it extends the boundary-free
construction of Fuselier, Hangelbroek, Narcowich, Ward and Wright.
