UBC Probability Seminar: Naotaka Kajino

For each $p\in(1,\infty)$, a p-energy form $(\mathcal{E}_p,\mathcal{F}_p)$, a natural $L^p$-analog of the standard Dirichlet form for $p=2$, was constructed on the (two-dimensional standard) Sierpinski gasket $K$ by Herman--Peirone--Strichartz [Potential Anal. 20 (2004), 125--148]. As in the case of $p=2$, it satisfies the self-similarity (scale invariance)

$$\mathcal{E}_p(u)=\sum_{j=1}^3\rho_p \mathcal{E}_p(u\circ F_j),u\in\mathcal{F}_p$$

where $\{F_j\}^3_{j=1}$ are the contraction maps on $\mathbb{ℝ}^2$ defining $K$ through the equation $K=\bigcup^3_{j=1}F_j(K)$ and $\rho_p\in(1,\infty)$ is a scaling factor determined uniquely by $(K,\{F_i\}^3_{i=1})$ and $p$. While the construction of $(\mathcal{E}_p,\mathcal{F}_p)$ has been extended to general $p$.-c.f.\ self-similar sets by Cao--Gu--Qiu (2022), to Sierpinski carpets by Shimizu (2024) and Murugan--Shimizu (2024+) and to a large class of infinitely ramified self-similar fractals by Kigami (2023), very little has been understood concerning properties of important analytic objects associated with $(\mathcal{E}_p,\mathcal{F}_p)$ such as $p$-harmonic functions and p-energy measures, even in the (arguably simplest) case of the Sierpinski gasket.

This talk is aimed at presenting the result of the speaker's on-going joint work with Ryosuke Shimizu (Waseda University) that, for a class of p.-c.f. self-similar sets with very good geometric symmetry, the $p$-energy measure $\mu^p_{\langle u\rangle}$ of any $u\in\mathcal{F}_p$ and the $q$-energy measure $\mu^μ\langle v\rangle$ of any $v\in\mathcal{F}_q$ are mutually singular for any $p,q\in(1,\infty)$ with $p\neq q$. The keys to the proof are (1) new explicit descriptions of the global and local behavior of p-harmonic functions in terms of $\rho_p$, and (2) the highly non-trivial fact that $ρ^{1/(p−1)}_p$ is strictly increasing in $p\in(1,\infty)$, whose proof relies heavily on (1).