Biographical Sketch of Professor Eiichi Nakai

Nakai's initial mathematical accomplishment centered on characterizing pointwise multipliers for functions with bounded mean oscillation (BMO).For the case in which BMO was defined on the torus T n , Stegenga and Janson had independently solved this characterization.However, in 1983, while pursuing a master's degree under the guidance of his supervisor Kôzô Yabuta, Nakai successfully provided a solution for the case in which BMO was defined on Euclidean space R n .This work was published in 1985.To obtain this characterization, he devised a function space defined by mean oscillation controlled by variable growth conditions.Nakai's subsequent research consistently revolved around function spaces L p,ϕ (R n ), generalized Campanato spaces, based on mean oscillation controlled by variable growth conditions.In 2005, his work on pointwise multipliers for BMO(R n ) was used by Lerner to study the class P(R n ) of functions p(•) for which the Hardy-Littlewood maximal operator is bounded on Lebesgue spaces L p(•) (R n ) with variable exponents.This work positively resolved a conjecture by Deining, which had suggested the existence of discontinuous functions within P(R n ).In 2012, Nakai and Yoshihiro Sawano discovered that the dual space of the Hardy space H p(•) (R n ) with variable exponents can be expressed as L p,ϕ (R n ).Furthermore, in 2019, Nakai and Tsuyoshi Yoneda utilized the function space L p,ϕ (R n ) to study the Navier-Stokes equations.Nakai also introduced generalized Morrey spaces with variable growth conditions and investigated the boundedness of integral operators on them.These results were presented in a paper in 1993, which has been cited in more than 250 papers to date.In 2000, Nakai studied a generalization of the Hardy-Littlewood-Sobolev theorem, which concerned the boundedness of fractional integral operatorsHe extended this theorem to Orlicz spaces by introducing generalized fractional integral operators I ρ .He also extended the theorem to generalized Morrey-Campanato spaces in 2002.Moreover, in 2004, he introduced Orlicz-Morrey spaces and studied the boundedness of I ρ on them.