I want to prove the following inequality
$$
\left\Vert \left( I-\partial _{x}^{2}\right) ^{-1}\partial
_{x}^{4}u\right\Vert _{L^{2}(0,1)}\leq C\left\Vert \partial
_{x}^{2}u\right\Vert
_{L^{2}(0,1)}
$$
for all $v \in (H_0^2 \cap {H^4})(0,1)$.
$\textbf{My attempt}$
let $v\in D(0,1)$ ($D(0,1)$ is the space of $C^{\infty }$ function with compact support in $(0,1))$
It is well known that $D(0,1)$ is dense in $L^{2}(0,1).$ we can extend easelly $v\in D(0,1)$ to $\widetilde{v}\in D(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )$with support in $(0,1).$
the symbole of the operator $\left( I-\partial _{x}^{2}\right) ^{-1}\partial _{x}^{4}$ in Fourier space is $\frac{\xi ^{4}}{\xi ^{2}+1}$ so we have \begin{eqnarray*} \left\Vert \frac{\xi ^{4}}{\xi ^{2}+1}\widehat{\widetilde{v}}(\xi )\right\Vert _{L^{2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )} &=&\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion }\left( \frac{\xi ^{4}}{\xi ^{2}+1}\right) ^{2}\widehat{\widetilde{v}}% ^{2}(\xi )d\xi \leq \int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion }\left( \xi ^{2}\right) ^{2}\widehat{\widetilde{v}}^{2}(\xi )d\xi \\ &\leq &\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion }\left( \xi ^{2}+1\right) ^{2}\widehat{\widetilde{v}}^{2}(\xi )d\xi =\left\Vert \widetilde{v}\right\Vert _{H^{2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )} \end{eqnarray*}
by using the Plancherel theorem we get $$ \left\Vert \left( I-\partial _{x}^{2}\right) ^{-1}\partial _{x}^{4}% \widetilde{v}\right\Vert _{L^{2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )}\leq \left\Vert \widetilde{v}\right\Vert _{H^{2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )} $$
but we have $supp(\widetilde{v})\subset (0,1)$ so we have
$$ \left\Vert \left( I-\partial _{x}^{2}\right) ^{-1}\partial _{x}^{4}v\right\Vert _{L^{2}(0,1) )}\leq \left\Vert v\right\Vert _{H_{0}^{2}(0,1) %EndExpansion )} $$
for all $v$ in $D(0,1).$
Now we use the density argument to prove the desired result.
My question is: is my attempt is write ? Is there any problems in it ? Thanks