Exercise 32, § 3

Let $n\geqslant 1$ be an integer. Let Δ denote the Laplacian on $\mathscr{D}'(\mathbf{R}^n)$, defined by

Let $u$ be a locally integrable function on $\mathbf{R}^n$ such that $\Delta (u)$ is a locally integrable function.

Let $v\in \mathscr{L}^{\infty}(\mathbf{R}^n)$ be the function defined by

$\{0$ if $u(x) = 0$,

$\overline{u(x)}/|u(x)|$ if $u(x)\not = 0$.

a) For $\varepsilon >$ 0, set $u_{\varepsilon}= (|u|^2+\varepsilon^2)^{1/2}$. Prove that the distribution $-\Delta (u_{\varepsilon})-\mathscr{R}(u/u_{\varepsilon}\Delta (u))$ is positive. (First consider the case where $u\in \mathscr{C}^{\infty}(\mathbf{R}^n)$, then use convolution to deal with the general case.)

b) The distribution $-\Delta (|u|)-\mathscr{R}(v\Delta (u))$ is positive ("Kato's inequality").