Let $f\in L^2((0,T); H^2) $ with $ \partial_t f \in L^{2}((0,T);H^{-2}) $ and let $ \eta_{\varepsilon} $ a standard mollifier sequence in $ (t,x) $, then there exists a constant $ C $ independent of $ \varepsilon $ such that
$$ \| \partial_t f_{\varepsilon} \|_{L^2((0,T);H^{-2})} = \|\partial_{t}(\eta_{\varepsilon} *f)\|_{L^2((0,T);H^{-2})} \leq C\|\partial_t f\|_{L^2((0,T);H^{-2})}?$$