Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces

Abstract

We construct self-similar $p$-energy forms ${\mathcal{E}}_p$ on a rich class of \emph{Laakso-type fractal spaces} and study the properties of the associated Sobolev spaces ${\mathcal{F}}_p$. The main result of the paper is the discovery of a new analytic phenomenon, which we refer to as \emph{singularity of Sobolev spaces}. This means that the associated Sobolev spaces ${\mathcal{F}}_{p_1}$ and ${\mathcal{F}}_{p_2}$ for distinct $p_1,p_2\in (1,\infty )$ intersect only at constant functions. We show that the Laakso diamond space of Lang--Plaut is one such example, and explain why this does not contradict the inverse limit construction of Cheeger--Kleiner which proves Laakso Diamond to support Poincaré inequality of Heinonen--Koskela.

Revision: Results about Lipschitz functions have been included. Certain parts in the previous version have been removed to improve readability, but the main results are the same. References have been updated. 69 pages, comments are welcome!