how to understand this operator

Question

In Sobolev space $H^1(\mathbb R^6)$, how do we understand $$\langle f(x,y),(-\Delta_x) f(x,y) \rangle $$ where $f\in H^1(\mathbb R^6)$ and $x\in \mathbb R^3$, $y\in \mathbb R^3$.

Answer 1

$\Delta_x$ maps $H^1(\mathbb{R}^6)$ inside $H^0(\mathbb{R}^6) = L^2(\mathbb{R}^6)$. That will tell you how to justify your computation.