For each $p \in (1,\infty)$, a \emph{$p$-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, 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 $\rho_{p}^{1/(p-1)}$ is strictly increasing in $p \in (1,\infty)$,
whose proof relies heavily on (1).