Higher order Poincaré inequalities on $\mathbb R$

Question

So, I know that for $f\in H^1(I) = W^{1,2}(I)$ one has $$ \|f-f_I\|_{L^2(I)}^2\,\le\,|I|^2\|f'\|_{L^2(I)}^2, $$ where $I$ is a bounded interval. I also know that for $f\in H^s(\mathbb R)$, $s\in (0,1)$, we have $$ \|f-f_I\|_{L^2(I)}^2\,\le\,|I|^{2s}\int_I\int_I\frac{|f(x)-f(y)|^2}{|x-y|^{1+2s}}\,dx\,dy. $$ But what about $f\in H^s(\mathbb R)$, when $s > 1$? Will we then have something like $$ \|f-f_I\|_{L^2(I)}^2\,\le\,|I|^{2s}\left(\|f\|_{H^s(I)}^2-\|f\|_{L^2(I)}^2\right)\,? $$ I already googled a lot, but could not find anything on this.

Answer 1

As Bananach noted, in order for the higher order of smoothness to yield higher decay of the norm, you need to subtract more than just a constant term (otherwise, the decay cannot be any faster than for a linear function). In general, this leads to estimates $\|f-P_{f,I}\|_{L^2(I)} \le \cdots $ where $P$ is a polynomial of some degree, depending on the smoothness of $f$. The key term is (higher order) Campanato space. Some references:

• Morrey-Campanato Spaces: an Overview by Humberto Rafeiro, Natasha Samko and Stefan Samko

• On the theory of homogeneous Lipschitz spaces and Campanato spaces by Harvey Greenwald (open access)