Tangential Curves and Fatou's Theorem on Trees
1998
The paper shows that the distinction between tangential sequences
and tangential curves, which is known to be relevant for the boundary behaviour of harmonic functions in
Euclidean half spaces, is also meaningful and relevant in the discrete setting of the potential theory
on a tree associated to a very regular, nearest neighbour random walk. In particular, the paper first defines the
class of tangential curves on a tree, and then proves that, for any family of tangential curves which have the
same order of tangency and are associated (and converging) to points in the boundary, there is a bounded
harmonic function which, for (almost) every point in the boundary, has positive oscillation along the
corresponding curve. This result does not hold if, in place of tangential curves, tangential sequences are
admitted. The tree is not assumed to be homogeneous. In particular, the results
do not depend on the existence of isometries.
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