Here are some of my thoughts on the question. Fix $s\in(0,\frac{1}{2})$" />. Then $C:=\sup\sb {r\geq 0}\frac{(1+r^{s})^{2}}{1+r}](http://latex.mathoverflow.net/png?%24C%3A%3D%5Csup%5F%7Br%5Cgeq%200%7D%5Cfrac%7B%281%2Br%5E%7Bs%7D%29%5E%7B2%7D%7D%7B1%2Br%7D). Notice then that , but I'm not sure. In fact this might be another question... I'm not very familiar with fractional Sobolev spaces - much less on subsets of . If it were true (it should be true for  an integer - See Evans page 282), then your result would be that ![$||f||\sb {H^{s}(D)}\lesssim\sb {D}C\sb {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.

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.

Here are some of my thoughts on the question. Fix $s\in(0,\frac{1}{2})$" />. Then $C:=\sup\sb {r\geq 0}\frac{(1+r^{s})^{2}}{1+r}](http://latex.mathoverflow.net/png?%24C%3A%3D%5Csup%5F%7Br%5Cgeq%200%7D%5Cfrac%7B%281%2Br%5E%7Bs%7D%29%5E%7B2%7D%7D%7B1%2Br%7D). Notice then that , but I'm not sure. In fact this might be another question... I'm not very familiar with fractional Sobolev spaces - much less on subsets of . If it were true (it should be true for  an integer - See Evans page 282), then your result would be that ![$||f||\sb {H^{s}(D)}\lesssim\sb {D}C\sb {n}\lambda$" /> 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.

Here are some of my thoughts on the question. Fix $s\in(0,\frac{1}{2})$" />. Then $C:=\sup\sb {r\geq 0}\frac{(1+r^{s})^{2}}{1+r}](http://latex.mathoverflow.net/png?%24C%3A%3D%5Csup%5F%7Br%5Cgeq%200%7D%5Cfrac%7B%281%2Br%5E%7Bs%7D%29%5E%7B2%7D%7D%7B1%2Br%7D). Notice then that , but I'm not sure. In fact this might be another question... I'm not very familiar with fractional Sobolev spaces - much less on subsets of . If it were true (it should be true for  an integer - See Evans page 282), then your result would be that ![$||f||\sb {H^{s}(D)}\lesssim\sb {D}C\sb {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.