Professor Hiroaki Masaoka

Area and Subject Taught Complex Analysis
Research Theme(s) Ideal Boundaries of harmonic spaces
Academic Degrees Doctor of Science (Mathematics), Kyoto University
Keywords for Research Field Harmonic Function, n-sheeted Unlimited Covering Surface,Fine Topology, Complex Structure
Office Phone Number Not Public
e-mail E-mail

Research Overview

Let D be the open unit disk excluding the origin and HP(D) a family of functions that are positive harmonic on D, and zero on the unit circle. It is known in this case (Bocher's Theorem) that the smallest linear space composed of the harmonic functions on D containing HP(D) is a one-dimensional linear space. If D is taken to be an n-sheeted unlimited covering surface of the open unit disk excluding the origin, then I consider, in the same way for such a D, the smallest linear space H composed of harmonic functions on D and containing HP(D). Letting k be a natural number satisfying 0<k<n+1, I can show the existence of D such that the dimension of H becomes k. It is also shown that the dimension of H is characterized by the fine topology on D (the weakest topology that makes superharmonic functions on D continuous). At present I am examining how the dimension of H changes when the complex structure of D is changed.

Notable Publications and Works in the Last Three Years

  1. T. Kilpelainen, P. Koskela, and H. Masaoka, “Examples of harmonic Hardy-Orlicz spaces on the plane with finitely many punctures” , RIMS Kokyuroku Bessatsu, 43 (2013), 47-59.
  2. T. Kilpelainen, P. Koskela, and H. Masaoka, “Harmonic Hardy-Orlicz spaces”, Ann. Acad. Sci. Fenn. Math., 38 (2013), 309-325.
  3. T. Kilpelainen, P. Koskela, and H. Masaoka, “On doubling measures”, Acta Human. et Sci. Univ. Sangio Kyotiensis, 42 (2013), 53-61(Japanese).
  4. H. Masaoka and M. Nakai, “Square means versus Dirichlet integrals for harmonic functions on Riemann surfaces”, Tohoku Mathematical Journal, 64 (2012), 233-259.
  5. H. Masaoka, “The classes of bounded harmonic functions and harmonic functions with finite Dirichlet integrals on hyperbolic Riemann surfaces”, Kodai Mathematical Journal, 33 (2010), 233-239.