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

Abstract

We construct self-similar $p$-energy forms as normalized limits of discretized $p$-energies on a rich class of Laakso-type fractal spaces. Collectively, we refer to them as IGS-fractals, where IGS stands for (edge-)iterated graph systems. We propose this framework as a rich source of "toy models" that can be consulted for tackling challenging questions that are not well understood on most other fractal spaces. Supporting this, our framework uncovers a novel analytic phenomenon, which we term as singularity of Sobolev spaces. This means that the associated Sobolev spaces $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ for distinct $p_1,p_2 \in (1,\infty)$ intersect only at constant functions. We provide the first example of a self-similar fractal on which this singularity phenomenon occurs for all pairs of distinct exponents. In particular, we show that the Laakso diamond space is one such example.