Understanding Sobolev spaces

Question

Given the Sobolev space $H^1((a,b);H^2(\mathbb{R}))$ and a function $g$ in that space. Consider now another function $f \in C_c^{\infty}((a,b) \times \mathbb{R}).$ Then for almost any $t \in (a,b)$ we have $g(t)f(t) \in H^2(\mathbb{R}).$ Now, the Laplacian on $\mathbb{R}$ maps $H^2$ into $L^2$, thus we can apply it $\Delta ( f(t)g(t)) \in L^2(\mathbb{R}).$

We can also first differentiate the product $\frac{d}{dt}(f(\cdot)g(\cdot)) \in L^2((a,b); H^2(\mathbb{R}))$ Then this is pointwise also a.e. in $H^2(\mathbb{R})$ and can then apply the Laplacian, i.e. $\Delta \left( \frac{d}{dt}(f(t)g(t))\right) \in L^2(\mathbb{R}).$

So far nothing but the definition.

But: Is it true that $t \mapsto \Delta(f(t)g(t)) \in H^1((a,b); L^2(\mathbb{R}))$ and is it then true that $\frac{d}{dt}\left(\Delta(f(t)g(t)) \right) = \Delta \left( \frac{d}{dt} (f(t)g(t))\right)$ for a.e. $t$?

Answer 1

Abstractly it is easier to prove the following two facts:

Proposition 1: If $f(t)g(t) \in H^1([a,b], H^2(\mathbb R))$.

Proof: Obvious. (Pick $g_n \in C^\infty ([a,b], H^2(\mathbb R))$ so that $g_n \to g$ in $H^1([a,b], H^2(\mathbb R))$ and use that $f$ is smooth and of compact support).

Proposition 2: Let $X$, $Y$ be Banach spaces, $f \in H^1([a,b] , X)$ and let $L :X\to Y$ be a bounded linear operator. Then $Lf : [a,b] \to Y$ is in $H^1([a,b], Y)$ and $\partial_t (Lf) = L \partial _t f$.

Proof: This is also obvious: let $f_n \in C^\infty([a,b], X)$ which approximates $f$. Then $L f_n \in C^\infty([a,b], Y)$ and $\partial_t (Lf_n) = L (\partial _tf_n)$ since $L$ is linear. Let $n\to \infty$, we conclude that $Lf \in H^1([a,b], Y)$ and that $\partial_t (Lf) = L \partial _t f$.

Now apply proposition 2 to $X = H^2$, $Y = L^2$ and $L = \Delta$.