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Let $E=\{u\in H^{\frac{1}{2}}(\mathbb{R}^{2});\frac{\partial u}{\partial y}\in L^2(\mathbb{R}^{2}) \}$.

  1. Show that $E$ is a Hilbert space with following norm $$ (u,v)_{E}=(u,v)_{H^{\frac{1}{2}}}+\left(\frac{\partial u}{\partial y},\frac{\partial v}{\partial y} \right)_{L^2},\qquad u,v\in E$$
  2. Show that $C_{0}^{\infty}(\mathbb{R}^{2})$ is dense in $E$.
  3. show that if $u\in C_{0}^{\infty}(\mathbb{R}^{2})$, then $\forall y\in \mathbb{R}$ $$ u(x,y)=\int_{-\infty}^{y}\frac{\partial u}{\partial t}(x,t)dt $$ Then prove $\forall u\in E$,$ u(\cdot,0)\in L^2(\mathbb{R})$.

Here is my approch: Since Sobolev space $H^{\frac{1}{2}}$ is a hilbert space, if we have $(u_{m}-u_{n},u_{m}-u_{n})_{H^{1/2}}\to 0$ as $m,n\to \infty$, then we have a $u_{0}\in H^{1/2}$ which imply $ u_{n}\to u_{0}$ in $H^{\frac{1}{2}}$ norm. that is $$ \|(1+|y|^2)^{\frac{1}{4}}(\hat{u_{m}}-\hat{u_{n}})\|_{2}\to 0 $$ Now if $$ (u_{m}-u_{n},u_{m}-u_{n})_{E}\to 0\qquad m,n\to \infty$$ then $(u_{m}-u_{n},u_{m}-u_{n})_{H^{1/2}}\to 0$ as $m,n\to \infty$. we need to prove that $\left\|\frac{\partial u_{n}}{\partial y}-\frac{\partial u_{0}}{\partial y} \right\|_{2}\to 0$, I use the fact $$ \left\|\frac{\partial u_{n}}{\partial y}-\frac{\partial u_{0}}{\partial y} \right\|_{2}=\| 2\pi iy(\hat{u_{n}}-\hat{u_{0}})\|_{2} $$. Then I don't know how to deal with it. Can some one give a complete proof about these problems? Thanks a lot.

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