The following is part of Lemma $2.1$ found in Kyuya Masuda's paper "Weak Solutions of Navier Stokes Equations". Let $\rho$ be a $C^{\infty}$ function in $\mathbb{R}^{1}$ with support in $|t| \leq 1$, such that:
$\rho(t) = \rho(-t), \rho(t) \geq 0, \int_{-\infty}^{\infty} \rho(t) \text{d}t = 1.$
We set $\rho_{h} (t) = \frac{1}{h} \rho(t/h)$, for $h > 0$. Let $s, t$ be fixed real numbers such that: $0 \leq s < t < \infty$. Let $X$ be a Banach space. For $w$ in $L^{p}((s,t); X), 1 \leq p < \infty, w > 0$, we define the mollification of $w$:
$J_{h}[w] (\tau) = \int_{s}^{t} \rho_{h}(\tau - \sigma) w(\sigma) \text{d}\sigma$.
Then the following results hold:
(i). For each fixed $h$, $J_{h}$ is a bounded operator from $L^{p}((s,t); X)$ into $C^{1} ([s,t];X)$.
(ii.) For each $w$ fixed in $L^{p}((s,t) ; X)$, $J_{h}[w] \rightarrow w$, as $h \rightarrow 0$, in $L^{p}((s,t) ; X)$.
I have been able to show (i) holds, and I have been able to show that $J_{h}[w](\tau) \rightarrow w(\tau) \ \textit{pointwise}$. However, I do not know how I can show convergence in $L^{p}((s,t) ; X)$. Please offer some advice / hints as to how I can proceed. Let me know if I should show my work for the pointwise convergence proof. Thank you.