UBC Probability Seminar: Naotaka Kajino

  • Date: 02/14/2024
  • Time: 15:15
Naotaka Kajino, Kyoto University

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)

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).

Other Information: 

Location: ESB 4127


Time: 3:15pm Pacific