AIKAWA, Hiroaki

Research Field
Department of Mathematics
Research Interest
Potential theory
Research Activities

I have been studying potential theory. The main theme of potential theory is the investigation of fundamental functions such as (super)harmonic functions, subharmonic functions, plurisubharmonic functions. The importance of the deep analysis of these functions has been recognized as they play crucial role in analysis, geometry, probability, applied mathematics and so on. On the other hand, methods, notions and aims of real and complex analysis, functional analysis and probabilistic analysis have enriched potential theory. Generally speaking, the interior properties of solutions to partial differential equations have been exploited even though the equations are complicated; if the domain is sufficiently smooth, the boundary behavior of the solutions can be studied to some extent. Nevertheless the boundary behavior of harmonic functions has many open problems if the domain is not smooth. My research interest is the boundary behavior of these functions in a wide sense. More specifically, I have been studying the Fatou theorem, minimal fine topology, the integrability of super harmonic functions, the additivity of capacity, the norm estimate of the Green operator, Martin boundary and the boundary Harnack principle.

1. H. Aikawa, Modulus of continuity of the Dirichlet solutions, Bull. Lond. Math. Soc., 42, 5 (2010), 857-867

2. H. Aikawa, Equivalence between the boundary Harnack principle and the Carleson estimate, Math. Scand., 103, 1 (2008), 61-76