Let $V\subset H$ be densely, compactly and continuously embedded, which, among others, means that $$\|x\|_H \leq \|x\|_V\quad (*)$$ for all $x\in V $.
I have the following stability estimates for a solution $v\in L^2 ((0,\infty); V )$ and a right hand side $f\in ((0,\infty); H)$:
- For any time $T > 0$, it holds that $$\|v\|_{L^2 ((0,T); V)} \leq C_T \|f\|_{L^2 ((0,T); H)}$$ with a constant $C_T$ depending on $T$
- In the weaker norm: $$\|v\|_{L^2 ((0,\infty); H)} \leq C_\infty \|f\|_{L^2 ((0,\infty); H)}$$
Does this somehow imply that also $$\|v\|_{L^2 ((0,\infty); V)} \leq \tilde C_\infty \|f\|_{L^2 ((0,\infty); H)}\quad(?)$$
edit: with constants $C_T, C_\infty, \tilde C_\infty$ independent of $f$.
I am convinced that such an estimate should hold in general since, by 2.), $v(t)\to 0$ as $t\to \infty$ and, at least in my case with $H=L^2(\Omega)$ and $V=W^{1,2}(\Omega)$, it holds that $$\|x\|_H = 0 \text{ and }x\in V, \text{ then }\|x\|_V=0.$$
Take $v_n(t, x) = f_n(t, x) = \sum_{k = 1}^n \theta_k(t) \lambda_k^{-1/2} \varphi_k(x)$ composed of eigenpairs of a Laplacian (with summable $\lambda_k^{-1}$) and the indicator function $\theta_k$ of the temporal interval $[k, k + 1]$. I think the sequence $(v_n, f_n)$ satisfies 1. & 2. uniformly but not "Ineq. (?)".