Recent studies in analysis on fractals revealed that the
value called $p$-walk dimension is deeply connected to the
$(1,p)$-Sobolev spaces on fractals, where the exponent $p$ is between
$1$ and $\infty$. Some singularity phenomenon on fractals are expected
due to the anomalous behavior of the $p$-walk dimension. (It is known
that the $2$-energy measures and the reference measure are singular on
many fractals.) However, even for the Sierpinski gasket, it is not
easy to prove such a new singularity result so that many problems
remain open. In this talk, I will provide a class of self-similar
sets, self-similar Laakso-type spaces (or edge-iterated graph
systems), for which $p$-energy forms and $p$-energy measures (of a
nice potential function) can be understood in a transparent way.
Moreover, I will explain the first example on which the new phenomena,
which we call singularity of Sobolev spaces meaning that Sobolev
spaces for distinct exponents intersect only at constant functions,
can be verified.
This is based on joint work with Riku Anttila (University of
Jyväskylä) and Sylvester Eriksson-Bique (University of Jyväskylä).