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 operators I α from L p (R n ) to L q (R n ).He 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.After 2008, he started to research general-
Watari, Chao and Ombe proved the boundedness of the fractional integral Iα, which is a martingale transform, from Lp to Lq for 1 < p < q < ∞, 0 < α < 1 and −1/p+α = −1/q. This boundedness was also extended to martingale Orlicz and Morrey spaces by Arai et al. and Nakai and Sadasue, respectively. In this paper, we extend these boundedness to martingale Orlicz-Morrey spaces by using the generalized fractional integral Iγ as a martingale transform.
In this note, we give a remark on the dual spaces of bi-parabolic Bergman spaces, where the bi-parabolic Bergman spaces are spaces of all integrable solutions of bi-parabolic equation (∂t + (−∆)α)2u = 0 for 0 < α ≤ 1 on the upper half space.
Humio Ichimura's research area is algebraic number theory.He addressed various classical problems on • class number and class group
In the previous paper [2], we have classified simple regular polyhedral BP-complexes, which are polyhedral complexes satisfying certain natural conditions (B) and (P) on their vertex structures. A regular polyhedral complex X is defined as satisfying the condition (B) if the diameter of the vertex structure of X is equal to its injectivity radius. On the other hand, we have shown in [3] that 2-skeletons of higher dimensional regular polytopes do not satisfy the condition (B). So it seems very likely that the diameter of the vertex structure and its injective radius are not equal.
We investigate the foliation defined by the kernel of an exact presymplectic form dα of rank 2n on a (2n + r)-dimensional closed manifold M. For r = 2, we prove that the foliation has at least two leaves which are homeomorphic to a 2-dimensional torus, if M admits a locally free T2-action which preserves dα and satisfies that the function α(Z2) is constant, where Z1, Z2 are the infinitesimal generators of the T2-action. We also give its generalization for r ≥1.
In this paper we prove the boundedness of the generalized fractional integral operator Iρ on generalized Campanato spaces with variable growth condition, which is a generalization and improvement of previous results, and then, we establish the boundedness of Iρ on their bi-preduals. We also prove the boundedness of Iρ on their preduals by the duality.
In this paper, we prove that any non-constant real rational function appears as a time transformation of a caloric morphism, mapping which preserves caloric functions, between semi-eucledean spaces.
For a fixed integer $n \geq 1$, let $p=2n\ell +1$ be a prime number with an odd prime number $\ell $ and let $F=F_{p,\ell }$ be the real abelian field of conductor $p$ and degree $\ell $. We prove that for each fixed $n$, there exist only finitely many pa
We consider a p-Laplace equation ∆pV + h(V ) = 0, with an arbitrary C-nonlinearity h, in a bounded domain and supplemented with the Neumann boundary condition. We prove a necessary condition for zeros of h = h(V ) to be touched by non-constant solutions to this problem.
Let Ω be a bounded domain of ℝ^N whose boundary is a ℂ^2 compact manifolds. In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we adopted powers of the distance function δ(x) to the boundary ∂Ω.
Let 1 < p < 1 and let Ω be a bounded domain of R N ( N (cid:21) 1). In this paper, we consider a class of second order quasilinear elliptic operators A in Ω including the p -Laplace operator ∆ p . First we establish various type of Kato’s inequalities for A when A u is a Radon measure. Then we prove the inverse maximum principle and describe the strong maximum principle. For this purpose it is crucial to introduce a notion of admissible class for the operator A and use it effectively. y
In semi-euclidean spaces, conformal mappings are consists of similarities, inversions, and Bateman mapping [14]. In this note, we shall discuss problems whether there exist caloric morphisms with Bateman space mapping for radial semi-euclidean metrics. It is based on the similar arguments as were used in [8], [9], and [10].
We raise a problem on coextensions and find partial answers.
A subsemigroup of a torsion-free abelian group is called grading monoid. This is a note on star operations in ideal theory of grading monoids. Explicitly, we study stability, ascents-descents, and Kronecker function rings of semistar operations on grading monoids.
Let p be a prime number of the form p = 2 ℓ +1 with some odd prime number ℓ . For such a prime number p , it is shown that the relative class number h (cid:0) p of the p th cyclotomic (cid:12)eld Q ( (cid:16) p ) is odd when 2 remains prime in Q ( (cid:16) ℓ ) + by Estes [3], Stevenhagen [11] and Mets(cid:127)ankyl(cid:127)a [8] using a Bernoulli number associated to Q ( (cid:16) p ). In this note, we give an alternative proof of the assertion using a cyclotomic unit of Q ( (cid:16) p ) + .
In [8] Matsuda has investigated stability of a semistar operation. In this paper we extend the notion of stability of a semistar operation to the presemistar operation case and we shall study stability properties of presemistar operations.
Let R be an integral domain with quotient field K, let h (resp., g, f) be the non-zero R-submodules of K (resp., the non-zero fractional ideals of R, the finitely generated non-zero fractional ideals of R), and let {x, y} be a subset of the set {f, g, h} of symbols. For a semistar operation ★ on R, if (EE1)★ = (EE2)★ implies E1★ = E2★ for every E ∈ x and every E1, E2 ∈ y, then ★ is called xy-cancellative. Let ★ be a gg-cancellative semistar operation on R which is an extension of a star operation on R. In this paper, we show that ★ need not be gh-cancellative.
