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?
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?
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 $n\to +\infty$.