Higher order Laplacians on p.c.f. fractals with three boundary points and dihedral symmetry

Abstract

We study higher order tangents and higher order Laplacians on fully symmetric p.c.f. self-similar sets with three boundary points. Firstly, we prove that for any function f defined near a vertex x, the higher order weak tangent of f at x, if exists, is the uniform limit of local multiharmonic functions that agree with f near x in some sense. Secondly, we prove that the higher order Laplacian on a fractal can be expressible as a renormalized uniform limit of higher order graph Laplacians. Some results can be extended to general p.c.f. self-similar sets. In the Appendix, we provide a recursive algorithm for the exact calculations of the boundary values of the monomials on D3 symmetric fractals, which is shorter and more direct than in the previous work on the Sierpiński gasket. 1. Introduction. Laplacians on post critically finite (p.c.f.) self-similar sets are defined as renormalized limits of Laplacians on the graphs that approximate the fractals [K1, K2]. An important feature of Kigami's theory is its intrinsic approach. The analytical properties derived only depend on the fractal itself, not its embedding in Euclidean space. There are many works exploring properties of these fractal Laplacians, which are natural analogs of the usual Laplacian; see [BK, BST, DSV, KL, KSS, MT, K3-K6, S1-S3, T1] and the references therein. Recently, there have been several works in connection with the differential calculus on p.c.f. self-similar sets that involve derivatives, tangents, energy measures, multiharmonic functions, and higher order Laplacians, analogous to the theory of analysis on manifolds; see [BSSY, CQ, DRS, NSTY, PT1, PT2, SU, P1, S3, T2] and the references therein. The study of "weak gradients" or "weak tangents" on p.c.f. self-similar sets, which is closely related to the theory of energy measures, goes back to the work of Kusuoka [Ku], under the assumption that the harmonic struc