### Harnack inequalities for Schrodinger operators

Let $\mu$ be a signed Radon measure on a domain $X$ in $\mathbb R^d$, $d\ge1$, with Green function $G_X$, assume that $\mu$ is potentially bounded, i.e., that the potential $G_X^{1_B|\mu|}$ is bounded for every ball $B$ in $X$, and define $d_{\mu^{\pm}}(x)=\limsup_{y\to x} G_X^{1_B\mu^{\pm}}(y)-G_X^{1_B\mu^{\pm}}(x)$, $x\in B,\overline B\subset X$ ($\mu$ is a local Kato measure if and only if $d_{\mu^+}=d_{\mu^-}=0$). The question, if positive $\mu$-harmonic functions, i.e., positive finely continuous solutions of $\Delta h - h\mu=0$, satisfy Harnack inequalities, is completely solved: If $U$~ is a domain in~ $X$ admitting a positive $\mu$-harmonic function which is locally bounded and not identically zero, then Harnack inequalities hold for positive $\mu$-harmonic functions on $U$ and every $\mu$-harmonic function on ~$U$ is locally bounded. In particular, Harnack inequalities always hold as long as $d_{\mu^-}\le\gamma <1$ (they may already fail for $d_{\mu^-}\le 1$, but it is it possible that they hold non-trivially in spite of big values of $d_{\mu^-}$). The results are presented in a general setting covering uniformly elliptic operators and sums of squares of smooth vector fields.

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