Abstract of Wolfhard Hansen's talk on February 4, 2000
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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1999/2000 Analysis Seminar