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 : 

Naotaka Kajino, Logperiodic asymptotic expansion of the spectral
partition function for selfsimilar sets,
Communications in Mathematical Physics, 2014, in press.

Naotaka Kajino, Nonregularly varying and nonperiodic
oscillation of the ondiagonal heat kernels on selfsimilar fractals,
in: Fractal Geometry and Dynamical Systems in Pure and
Applied Mathematics II: Fractals in Applied Mathematics,
Contemporary Mathematics, vol. 601, 2013, pp. 165194.

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. 91133.

Naotaka Kajino, Ondiagonal oscillation of the heat kernels
on postcritically finite selfsimilar fractals,
Probability Theory and Related Fields 156 (2013), 5174.

Naotaka Kajino, Time changes of local Dirichlet spaces by energy
measures of harmonic functions,
Forum Mathematicum 24 (2012), 339363.

Naotaka Kajino, Heat kernel asymptotics for the measurable
Riemannian structure on the Sierpinski gasket,
Potential Analysis 36 (2012), 67115.

Naotaka Kajino, Spectral asymptotics for Laplacians on selfsimilar sets,
Journal of Functional Analysis 258 (2010), 13101360.


