Let D be a domain in R^n, and let f be an eigenfunction of the Laplacian with Dirichlet boundary condition with eigenvalue $\lambda$. Assume that f has L^2 norm 1. I want to know if I can say anything about the Sobolev s-norm of f (interms of s \lambda and D) ?

In particular, I want to know if it is true that |f|_s is like \lambda^{\frac{s}{2}}.

Same question for the Neumann and dbar-Neumann boundary conditions.