Unique continuation for Schrödinger operators in dimension three or less

We show that the differential inequality $\left|\Delta u\right|\le v\left|u\right|$ has the unique continuation property relative to the Sobolev space $H_{loc}^{2,1}\left(\Omega \right)$ , $\Omega \subset R^n$ , $n\le 3$ , if $v$ satisfies the condition $$(K_n^{\mathrm{loc}})\underset{r\to 0}{lim}\underset{x\in K}{sup}{\int }_{|x-y|<r}|x-y{|}^{2-n}v\left(y\right)dy=0$$ for all compact $K\subset \Omega$ , where if $n=2$ , we replace $|x-y{|}^{2-n}$ by $-log|x-y|$ . This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$ , in the case $n\le 3$ . The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,...$ and any $u\in H_c^{2,1}({\mathbf{R}}^3-\left\{0\right\}),$ $$\frac{\left|u\right(x\left)\right|}{|x{|}^N}\le C{\int }_{{\mathbf{R}}^3}|x-y{|}^{-1}\frac{\left|\Delta u\right(y\left)\right|}{|y{|}^N}dy\text{for}\text{a.e.}x\text{in}{\mathbf{R}}^3.$$