On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from H3/2 into B3/22,∞; for elementwise polynomials these are bounded from H1/2 into B1/22,∞. As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.