Norm of differential operator between Sobolev spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:44:35Z http://mathoverflow.net/feeds/question/119199 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119199/norm-of-differential-operator-between-sobolev-spaces Norm of differential operator between Sobolev spaces Sylvester-H 2013-01-17T17:31:06Z 2013-01-20T08:38:37Z <p>It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms) $W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, where $p\in [1,+\infty]$, and $U$ is an open subset of $\mathbb{R}^n$.</p> <p>My question is : do we know exactly the value of the norm of such (bounded) operator (in this generality, or with conditions on $U$ or the other parameters). (At least this norm is less than one, it is equal to one ?). </p> http://mathoverflow.net/questions/119199/norm-of-differential-operator-between-sobolev-spaces/119389#119389 Answer by Daniel Spector for Norm of differential operator between Sobolev spaces Daniel Spector 2013-01-20T08:38:37Z 2013-01-20T08:38:37Z <p>Indeed, the norm is one. </p> <p>To see this, fix a cutoff function $\phi \in C^\infty_c(U)$ (which we only need if $U$ is unbounded, to make sure the constructed functions are integrable) and define </p> <p>$f_n(x):=\phi(x) \frac{sin(nx_1)}{n^m}$. </p> <p>Then $f_n \to 0$ strongly in $W^{m-1,p}(U)$ and </p> <p>$||\nabla^m f_n||_{L^p}= ||\phi(x) sin(nx_1)||_{L^p} + o(\frac{1}{n})$.</p> <p>Therefore, $||\partial^\alpha f||_{W^{m-|\alpha|,p}} = ||\phi(x) sin(nx_1)||_{L^p} + o(\frac{1}{n})$,</p> <p>and so in the computation of</p> <p>$\sup \frac{||\partial^\alpha f||}{||f||}$,</p> <p>plugging in $f_n$ we have the following lower bound</p> <p>$\sup \frac{||\partial^\alpha f||}{||f||} \geq \frac{||\phi(x) sin(nx_1)||_{L^p}+o(\frac{1}{n})}{||\phi(x) sin(nx_1)||_{L^p}+o(\frac{1}{n})}$.</p> <p>Combining this with the upper bound mentioned in the question, we obtain that the norm is in fact one.</p>