Sobolev norms of eigenfunctions 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.
 A: There is an estimate of the form
|f|_s < C(D, \lambda, s)|f|_0
where |f|_0 = L^2 norm of f and |f|_s = Sobolev s-norm. There are different ways to get this estimate, depending on how sharp an estimate you need and the regularity of the boundary of D.
In particular, you can apply any a priori estimate for the Sobolev norm of the solution f to 
Laplacian(f) = h
in terms of h and bootstrap to get whatever you want.
A: Here are some of my thoughts on the question. Fix $s\in(0,\frac{1}{2})$. Then $C:=\sup_{r\geq 0}\frac{(1+r^{s})^{2}}{1+r}$. Notice then that $\int(1+|\xi|^{s})^{2}||\widehat{f}(\xi)|^{2}d\xi\leq C\int(1+|\xi|)^{2}|\widehat{f}(\xi)|^{2}d\xi\int |\widehat{f}(\xi)|^{2}d\xi$. Now, as mentioned by Deane, there may be some issues with the boundary $\partial D$. Suppose that $f\in H^{1}_{0}(D)\cap H^{2}(D)$ and $-\triangle f = \lambda f$ (We maybe be able to drop the second order regularity of $f$ if more regularity is assumed on the boundary for example). After integrating by parts and using perhaps using some sort of Poincaré inequality (need some sort of boundedness for the domain), one can see by integration by parts that $||f||_{H^{1}_{0}(D)}\sim\lambda$. I THINK that $||f||_{H^{s}(D)} \sim\big(\int(1+|\xi|^{s})^{2}|\widehat{f}(\xi)|^{2}d\xi\big)^{\frac{1}{2}}$, but I'm not sure. In fact this might be another question... I'm not very familiar with fractional Sobolev spaces - much less on subsets of $\mathbb{R}^{n}$. If it were true (it should be true for $s$ an integer - See Evans page 282), then your result would be that $||f||_{H^{s}(D)}\lesssim_{D}C_{n}\lambda$ (modulo $D$ because of the Poincaré inequality - which would require some sort of boundedness of one coordinate). This was my first idea. I'm sure there are better ideas/results. I hope this helps.
A: Let me make the question more precise -- f is normalized to have L^2 norm 1.
Then f is in $H^1_0(D)$ and the Sobolev 1-norm of f is \lambda.
We want the norm of f in H^s(D).
I'll be glad if anybody can help. 
