## UBC Probability Seminar: Naotaka Kajino

- Date: 02/14/2024
- Time: 15:15

University of British Columbia

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

For each p∈(1,∞), a \emph{p-energy form} (p,p),

a natural Lp-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)

p(u)=∑j=13ρpp(u∘Fj),u∈p,

where {Fj}3j=1 are the contraction maps on ℝ2 defining

K through the equation K=⋃3j=1Fj(K) and ρp∈(1,∞)

is a scaling factor determined uniquely by (K,{Fi}3i=1) and p.

While the construction of (p,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

(p,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 μp⟨u⟩

of any u∈p and the q-energy measure μq⟨v⟩

of any v∈q are mutually singular for any p,q∈(1,∞) with p≠q.

The keys to the proof are (1) new explicit descriptions of the global and local behavior

of p-harmonic functions in terms of ρp, and (2) the highly non-trivial fact

that ρ1/(p−1)p is strictly increasing in p∈(1,∞),

whose proof relies heavily on (1).

**Location:** ESB 4127

**Time:** 3:15pm Pacific