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Let $n\in \mathbb{N}_{0}$. I am interested in the quantity

$\inf\{\|\psi\|_{W^{1,n}(\mathbb{R})}\mid \psi\in W^{1,n}(\mathbb{R}), 0\leq \psi \leq 1, \psi\equiv 1 \text{ on }[-1/2,1/2], \text{ supp}(\psi)\subseteq[-1,1]\}$.

Here $1$ is the integration parameter and $n$ the smoothness parameter. I can obtain estimates for this quantity in a fairly simple manner using for example convolutions of suitably supported functions, by using the Denjoy-Carleman theorem, or by just trying natural choices of $\psi$. However, this does not tell me how close I am to the exact value of this quantity, for example asymptotically as a function of $n$ (or even better, explicitly). I would guess that this quantity has been studied before. Does someone know the answer to my question or where I might find it? Thanks in advance.

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    $\begingroup$ My guess would be that the "optimizer" is obtained by taking $\psi^{(n-1)}$ piecewise constant (quotes because $\psi^{(n)}$ is a combination of $\delta$'s and thus not in $W^{1,n}$ in a literal sense, but of course we can just compute $\|\psi\|$ anyway). $\endgroup$
    – Christian Remling
    Commented Jun 18, 2017 at 19:17

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