So, I was going over my notes on differential equations and I stumbled upon a certain observation that left me scratching my head.
The observation is about the function $u\ast\eta_{\epsilon}$, where $u\in C^{\infty}(\mathbb{R}^{n})$, $D^{\alpha}u\in L^{2}(\mathbb{R}^{n})$ for all multi-indices $\alpha$ and $\eta_{\epsilon}=\epsilon^{-n}\eta(\epsilon^{-1}x)$. Here $\eta$ is the standard mollifier on $\mathbb{R}^{n}$.
The observation claims that Jensen's inequality implies that $u\ast\eta_{\epsilon}\in L^{2}(\mathbb{R}^{n})$, but I was stupid enough not to outline why this is the case.
I took this course several years ago and for the life of me I can't figure out how to apply the inequality to obtain the desired observation. I tried a couple of things but they didn't lead me anywhere.
I'm sure this must be somewhat trivial (otherwise I would have elaborated on my notes) and if I wasn't in a hurry I would keep trying to figure it out on my own, but given the circumstances I would appreciate your help.
I hope I understand your question correctly. Let me repeat the important things.
You suppose $C(\mathbb{R})\cap L^2(\mathbb{R})$ and $\eta$ is a mollifier with $\int_{\mathbb{R}} \eta(y)\;dy = 1$. Now, Jensen's inequality states that $$ \varphi\left(\int_\Omega g\,d\mu\right) \leq \int_\Omega \varphi\circ g \;d\mu $$ for all convex functions $\varphi$ and $\mu$-integrable functions $g$, where $\mu(\Omega) = 1$. In particular, since $$ \int_\mathbb{R} \eta_\epsilon(y)\;dy = 1, $$ we get \begin{align*} \int_{\Bbb R} (u\ast\eta_\epsilon)(x)^2 \;dx &= \int_{\Bbb R} \left(\int_{\mathbb{R}} u(x-y)\eta_\epsilon(y)\;dy\right)^2\;dx\\ &\leq \int_{\Bbb R}\left( \int_{\Bbb R} u(x-y)^2\eta_\epsilon(y)\;dy\right)\;dx\\ &= \int_{\Bbb R}\left( \int_{\Bbb R} u(x-y)^2\;dx\right)\eta_\epsilon(y)\;dy\\ &= \int_{\Bbb R}\left( \int_{\Bbb R} u(x)^2\;dx\right)\eta_\epsilon(y)\;dy\\ &= \Vert u\Vert_{L^2}^2\cdot \underbrace{\int_{\Bbb R} \eta_\epsilon(y)\;dy }_{=1} = \Vert u\Vert_{L^2}^2. \end{align*} In particular, $u\ast\eta_\epsilon$ is $L^2$.
As you can see, we don't even need the regularity you have. We could even weaken it further to $u\in L^1_{loc}(\Bbb R)\cap L^2(\Bbb R)$.
Let me know if this answers your question or if I used assumptions I was not allowed to use.