Speaker: Naotaka Kajino

Title: On singularity of p-energy measures among distinct values of p for some p.-c.f. self-similar sets

Abstract: 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) Sierpi\'{n}ski 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)

\[\mathcal{E}_{p}(u)=\sum_{j=1}^{3}\rho_{p}\mathcal{E}_{p}(u\circ F_{j}), \qquad u \in \mathcal{F}_{p},\]

where $\{F_{j}\}_{j=1}^{3}$ are the contraction maps on $\mathbb{R}^{2}$ defining $K$ through the equation $K=\bigcup_{j=1}^{3}F_{j}(K)$ and $\rho_{p} \in (1,\infty)$ is a scaling factor determined uniquely by $(K,\{F_{i}\}_{i=1}^{3})$ 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 Sierpi\'{n}ski 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 Sierpi\'{n}ski gasket.

This talk is aimed at presenting the result of the speaker's on-going joint work with Ryosuke Shimizu (Waseda University) that, \emph{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 $\mu^{q}_{\langle v\rangle}$ of any $v \in \mathcal{F}_{q}$ are mutually singular for any $p,q \in (1,\infty)$ with $p\not=q$}. 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 \emph{$\rho_{p}^{1/(p-1)}$ is strictly increasing in $p \in (1,\infty)$}, whose proof relies heavily on (1).