Strong convergence of average function

Question

Let $f\in L^{2}(0,T;L^{2}(\Omega))$ where $T$ is a finite number and $\Omega\subset \mathbb{R}^{3}$ is a bounded domain. I'm consider the average function with respect to time that is \begin{equation*} f_{\kappa}(t)=\frac{1}{\kappa}\int_{t-\kappa}^{t}f(\tau) d\tau, \end{equation*} after extending $f$ to zero outside $(0,T)$. One can see $f_{\kappa}\to f$ a.e. in $(0,T)$ due to Lebesgue differentiation theorem. Then I think that by Hardy-Littlewood maximal inequality (I am not sure whether it is still valid for e.g. vector valued functions, because I think I can view this function $f$ as map from $(0,T)$ to $L^{2}(\Omega)$. Can someone give me a reference?) \begin{align} \lVert f_{\kappa}\rVert_{L^{2}(0,T;L^{2}(\Omega))}\leq C\lVert f \rVert_{L^{2}(0,T;L^{2}(\Omega))} \end{align} I can first deduce that $f_{\kappa}\in L^{2}(0,T;L^{2}(\Omega))$ and moreover weak convergence in $L^{2}(0,T;L^{2}(\Omega))$. But I am thinking whether I can say some strong convergenve result about $f_{\kappa}$ as $\kappa\to0$.
If not, can we deduce some convergence result for the $L^{2}(\Omega)$ norm for a.e. $t\in(0,T)$ e.g. \begin{align} \lVert f_{\kappa}(t)\rVert_{L^{2}(\Omega)} \to \lVert f(t)\rVert_{L^{2}(\Omega)} \end{align}
for a.e. $t\in(0,T)$.
Update: Now I have a new idea. By the vector-valued version of Lebesgue differentiation theorem, $f_{\kappa}(t)\to f(t)$ for a.e. $t\in(0,T)$. So \begin{align} f_{\kappa}(t)\to f(t)\mbox{ in }L^{2}(\Omega) \end{align}
for a.e. $t\in(0,T)$. Does this make any sense?

Answer 1

Suppose $f$ is of the form $\sum_{k = 1}^n c_k \varphi_k(t)$ where $c_k \in L^2(\Omega)$ and $\varphi_k \in C^\infty_c(0, T)$. Then $$ \begin{align*} \Vert f(t) - f_\kappa(t) \Vert_{L^2(\Omega)} &= \left\Vert \frac{1}{\kappa} \int_{t-\kappa}^t (f(t) - f(s)) \ \mathrm{d}s \right\Vert_{L^2(\Omega)}\\ & \le \sum_{k=1}^n \Vert c_k\Vert_{L^2(\Omega)} \frac{1}{\kappa} \int_{t - \kappa} ^ t |\varphi_k(t) - \varphi_k(s)|\ \mathrm{d}s, \end{align*} $$ which tends to $0$ as $\kappa \to 0$, since $\varphi_k$ are smooth. By the dominated convergence theorem, $\int_0^T \Vert f(t) - f_\kappa(t) \Vert^2_{L^2(\Omega)} \mathrm{d} t \to 0$. Since functions of the preceding form are dense in $L^2(0, T; L^2(\Omega))$, we may select, given general $f \in L^2(0, T; L^2(\Omega))$ and $\epsilon > 0$, a function $g$ of this form for which $\Vert f - g\Vert_{L^2(0, T; L^2(\Omega))} < \epsilon$. Then (omitting $L^2(0, T; L^2(\Omega))$ from the norms), $$ \Vert f_\kappa - f \Vert \le \Vert f_\kappa - g_\kappa \Vert + \Vert g_\kappa - g \Vert + \Vert g - f \Vert \le 3\epsilon $$ for small $\kappa$.