Geometry and analysis of groups and manifolds

Green Functions in Metric Measure Spaces

speaker: Mario Bonk (University of California)

abstract: Let $X$ be an unbounded Ahlfors $Q$-regular $Q$-Loewner space with $Q> 1$. Then $X$ admits a measurable differential structure in the sense of Cheeger and one can define a notion of ``Cheeger harmonic" functions on $X$. In my talk I will discuss the statement that in this setting for every point $x_0\in X$ there exists a Green function $G_{x0}$ with pole at $x_0$. It is $Q$-harmonic in $X\setminus\{x_0\}$ and has logarithmic blow-up near $x_0$ and near $\infty$ (with different signs). With suitable normalization, this Green function $G_{x0}$ is unique. While for $Q\ge 2$ a proof can be given along the lines of earlier work by Balogh--Holopainen--Tyson for Carnot groups, the case $1
timetable:
Fri 30 Jun, 11:30 - 12:30, Aula Dini
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