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Much of this paper will be concerned with the proof of the following Theorem 1. Suppose d ≥ 3, r = max {d, (3d - 4)/2}. If V ∈ L_loc ^r(R^d), then the differential inequality |Δu| ≤ V |∇u| has the strong unique continuation property in the following sense: If u belongs to the Sobolev space W_loc ^2,p and if |Δu| ≤ V |∇u| and ...

Let $L_{\alpha }^q(R^D),\alpha \>0,1\<q\<\infty$, denote the space of Bessel potentials $f=G_{\alpha }*g$, $g\in L^q$, with norm $\parallel f{\parallel }_{\alpha ,q}=\parallel g{\parallel }_q$. For $\alpha$ integer $L_{\alpha }^q$ can be identified with the Sobolev space $H^{\alpha ,q}$. One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space $H^{1;2}$, and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the positive cone...