What does "$\Delta\varphi \in L^{2}(\mathbb{R}^{d})$ in the sense of distributions" means?

Question

I'm reading Reed & Simon's book and, at some point, the authors define the following object. If $\Delta$ denotes the Laplacian operator on $\mathbb{R}^{d}$, they set the domain $D_{\text{\max}}$ of $-\Delta$ as: $$D_{\text{max}} := \{\varphi \in L^{2}(\mathbb{R}^{d}): \hspace{0.1cm} \mbox{$\Delta\varphi \in L^{2}(\mathbb{R}^{d})$ in the sense of distributions}\}$$

Question: What does "$\Delta\varphi \in L^{2}(\mathbb{R}^{d})$ in the sense of distributions" means?

Answer 1

For any $L^2$ function $\varphi$, one can define the distribution derivatives as a functional $C^\infty_c(\mathbb R^d)\to \mathbb R$ by

$$ \partial_{x_i} \varphi (f) : = -\int_{\mathbb R^d} \varphi f_{x_i}. $$

Then $\Delta \varphi$ is just the functional

\begin{align} (\Delta \varphi) (f) &= \left( \varphi_{x_1x_1} + \cdots + \varphi_{x_dx_d}\right) (f) \\ &= \int_{\mathbb R^d} \varphi \left( f_{x_1x_1} + \cdots + f_{x_dx_d}\right) \\ &= \int_{\mathbb R^d} \varphi \Delta f. \end{align}

So $\Delta\varphi$ as defined is just a functional. We say that $\Delta \varphi \in L^2(\mathbb R^d)$ in the sense of distribution, if this functional is given by integration of a $L^2$ function: there is $g\in L^2(\mathbb R^d)$ so that

$$ (\Delta \varphi) (f) = \int _{\mathbb R^d} \varphi \Delta f = \int_{\mathbb R^d } g f, \ \ \ \forall f\in C^\infty_c(\mathbb R^d).$$