The paper deals with estimates for the solution of \(div(A(x)\nabla u(x))=0\) (A(x) a Lipschitz, positive definite matrix), and of the Schrödinger equation \(-\Delta u(x)+V(x)u(x)=0.\) The conclusions for the first equation are that \(| u|\) and \(| \nabla u|\) are an \(A_ p\) and an \(A_ q\) ``weight of Muckenhoupt'', respectively, over a ball contained in the domain \(\Omega\) on which that equation is considered, for appropriate p and q; that, if u vanishes to infinite order at \(x_ 0\in \Omega\), then \(u\equiv 0\) in \(\Omega\), and \(| \nabla u|\) cannot vanish to infinite order at \(x_ 0\in \Omega\) unless \(u\equiv\) constant in \(\Omega\). (Partial explanation: a ``Muckenhoupt weight'' w (w(x)\(\geq 0)\) satisfies a technical condition characterizing the measures w(x)dx for which the Hardy-Littlewood maximal operator is continuous from \(L^ p(wdx)\) into itself; a function u is said to ``vanish to infinite order'' at \(x_ 0\) if, for \(R>0\) sufficiently small, \(\int_{| x-x_ 0| 2\). Significant features are: for the equation in divergence form, u is not assumed to have a fixed sign; for the Schrödinger equation, in case \(m=2\), V is not in \(L^{n/2}_{loc}({\mathbb{R}}^ n)\).