Naotaka KAJINO

Naotaka KAJINO

Department of Mathematics,
Graduate School of Science, Kobe University
Division : Applied Mathematics
Associate Professor
Building B, Room 426

Personal Website

Research Field : Analysis on fractals and on metric measure spaces

Research Summary : "Fractal" is a generic name for figures or sets whose geometric properties are totally different from smooth spaces such as the Euclidean space and Riemannian manifolds. Since B. B. Mandelbrot pointed out their frequent appearances and importance in nature, fractals have been widely studied in various fields of natural sciences. Although the usual notion of differentiation does not make any sense on fractals, certain mathematically idealized, technically tractable fractals are known to admit canonical ``Laplacians", which are rigorously defined as a kind of differential operators. It has then turned out that a rich theory of analysis can be developed through studies of the eigenvalues of those Laplacians and their associated heat and wave equations.
The principal theme of my research is to figure out how the geometric properties of fractals are reflected in analytical phenomena, mainly through the analysis of Laplacian eigenvalues and heat kernel asymptotics on fractals. I am also interested in analysis of singular differential operators defined through singular measures on the Euclidean space and Riemannian manifolds, in which case various phenomena typical of Laplacians on fractals are similarly observed.

Primary Publications :
  1. Naotaka Kajino, Log-periodic asymptotic expansion of the spectral partition function for self-similar sets, Communications in Mathematical Physics, 2014, in press.
  2. Naotaka Kajino, Non-regularly varying and non-periodic oscillation of the on-diagonal heat kernels on self-similar fractals, in: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics, Contemporary Mathematics, vol. 601, 2013, pp. 165-194.
  3. Naotaka Kajino, Analysis and geometry of the measurable Riemannian structure on the Sierpiński gasket, in: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics, Contemporary Mathematics, vol. 600, 2013, pp. 91-133.
  4. Naotaka Kajino, On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals, Probability Theory and Related Fields 156 (2013), 51-74.
  5. Naotaka Kajino, Time changes of local Dirichlet spaces by energy measures of harmonic functions, Forum Mathematicum 24 (2012), 339-363.
  6. Naotaka Kajino, Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, Potential Analysis 36 (2012), 67-115.
  7. Naotaka Kajino, Spectral asymptotics for Laplacians on self-similar sets, Journal of Functional Analysis 258 (2010), 1310-1360.