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Show that $\delta$, function of Dirac, defined than $\left<{\delta_0,\phi}\right> = \phi(0)$ belongs to $W^{-1,p}(]-1,1[)$ and $\delta_0 \notin L^p(-1,1)$ $\forall p\geq 1$.

How I will be able to start?

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If $u$ is a test function (smooth function with compact support), then $$|\delta_0(u)|=|u(0)|=\left|\int_{—1}^0u'(t)dt\right|\leq \lVert u'\rVert_p\leq \lVert u\rVert_{W^{1,p}(-1,1)}.$$ By density of test function, we can extend $\delta_0$ to the functions of $W^{1,p}(-1,1)$.

Assume that $\delta_0$ can be represented by $u\in L^p$. Take $\phi_n\in C^{\infty}_0(-1,1)$ such that $\phi_n=1$ on $(-1/2,1/2)$, $\phi_n$ is supported in $(-1/n,1/n)$ and $0\leq \phi_n\leq 1$. Then $$\int_{(-1,1)}u(t)\phi_n(t)dt=\delta_0(\phi_n)=1.$$ But by the dominated convergence theorem the LHS should converge to $0$ as $n\to +\infty$.

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I know that this question is well done, only a one comment according to my lecture $$W^{-1,p}(]-1,1[) = \left\{{u\in L^p(\Omega); D^{a}u\in L^p(\Omega) \forall |\alpha|\leq -1}\right\}$$ in ohter words. If $\delta_0 \in W^{-1,p}(]-1,1[)$ then $\delta_0\in L^p(]-1,1[)$. Then exist a condradictory? – juaninf Nov 6 '12 at 11:01
I need show that $\delta_0 \in W^{-1,p}(-1,1)$ but you wrote $\delta_0 \in W^{1,p}(-1,1)$ – juaninf Nov 8 '12 at 18:52
How I will be able to prove the linearity of $\delta_0$? – juaninf Nov 9 '12 at 1:40
It's the same as showing that the map $f\mapsto f(0)$ is linear, which is a consequence of the definition of the sum (and the multiplication by a scalar) of functions. – Davide Giraudo Nov 9 '12 at 9:15
in your map ... $f$ belong what space? – juaninf Nov 9 '12 at 13:07

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