一般社団法人 日本数学会 Application Server

# アブストラクト事後公開 — 2018年度年会(於:東京大学)

## 実函数論分科会

 特別講演 Geometric techniques in Banach space theory: Challenges to Tingley’s problem 田中亮太朗 (九大数理) Mathematical developments are always based on problems. Challenges to difficulties generate new ideas. In this talk, we focus on a 30 years old open problem in Banach space theory, so-called Tingley’s problem, and present new geometric techniques (and results) derived from our challenges. PDF 特別講演 多孔質媒質中の二重拡散対流現象を記述する方程式系の可解性について 内田　俊 (早大理工) We consider some equations describing double-diffusive convection phenomena of incompressible viscous fluid in a porous medium. Roughly speaking, this system consists of the Stokes equation and two advection-diffusion equations. Although the fluid equation in our model is linearized, the others still possess convection terms as non-monotone perturbations, which make it difficult to deal with this system. Main topic of this talk is to show the global solvability of this double-diffusive convection system. In particular, we focus on the existence of time periodic solutions to the system in the whole space domain for large data, i.e., without any smallness conditions for given external forces. In previous results for periodic problems of parabolic type equations with non-monotone perturbation terms (e.g., incompressible Navier–Stokes equations and Boussinesq system), it seems that either of the smallness of given data or the boundedness of space domain is essential. However, in spite of the presence of non-monotone terms, the solvability of our problem in the whole space is shown for large external forces via the convergence of solutions to approximate equations in bounded domains. PDF 1. Henstock–Kurzweil主値積分について 川﨑敏治 (日大工／玉川大工) There are the wide Denjoy integral, the approximately continuous Perron integral, the approximately continuous Henstock integral, the approximately continuous Denjoy integral and the distributional denjoy integral as the wider integrals than Denjoy–Perron–Henstock–Kurzweil integral. These integrals are defined by replacing derivative with approximately derivative and distributuinal derivative. In this talk, we extend an integral by the Cauchy’s principal value and show the obtained results. PDF 2. 準劣加法的単調測度に関する弱 $L_p$ 空間 $L^{p,\infty}$ 本田あおい (九工大情報工)・​岡崎悦明 (ファジィシステム研) The weak $L_p$ space $L^{p,\infty}(\mu)$ is introduced for the quasi-subadditive monotone measure $\mu$. If $\mu$ is continuous from below, them $L^{p,\infty}(\mu)$ is a quasi-Banach space. As an application it is shown that there exists a real number $\alpha \in (0,1]$ such that the power transformatin $\mu^{\alpha}$ is uniformly quasi-subadditive. Furthermore there exists a subadditive monotone measure $\lambda$ satisfying $\lambda \le \mu^{\alpha} \le 2\lambda$. PDF 3. 非加法的測度論における強形のEgorovの定理の成立条件 室伏俊明 (東工大情報理工)・​榎本直樹 (東工大情報理工) The consequent of the strong form of the Egorov theorem in non-additive measure theory is that strong almost everywhere convergence implies strong almost uniform convergence. This paper shows that the conjunction of the uniform subadditive continuity and the order continuity of the non-additive measure is a sufficient condition for the consequent of the strong form of the Egorov theorem, and that the monotone continuity is a necessary condition. PDF 4. 完備CAT(1)空間における凸関数に対する近接点法 高阪史明 (東海大理) Using the recently introduced resolvent of a convex function in a complete CAT(1) space, we obtain existence and convergence theorems for the proximal point algorithm in such a space. PDF 5. 作用素分割法の収束について 松下慎也 (秋田県立大) Let $H$ be a real Hilbert space and let $f:H\rightarrow (-\infty,\infty]$ and $g:H\rightarrow (-\infty,\infty]$ be proper, lower semicontinuous and convex functions. This talk considers a problem of finding the resolvent $J_{\partial (f+g)}$ of the subdifferential $\partial (f+g)$. It is assumed that both the resolvents $J_{\partial f}$ and $J_{\partial g}$ of $\partial f$ and $\partial g$ can be easily computed. This enables us to consider the case in which a solution to the problem cannot be computed easily. PDF 6. 強擬非拡大写像について 青山耕治 (千葉大社会科学) In this talk, we introduce and study a quasi-nonexpansive mapping, a strictly quasi-nonexpansive mapping, and a strongly quasi-nonexpansive mapping in an abstract space. In particular, we give some basic properties of such mappings. PDF 7. Weak and strong convergence theorems for a sequence of nonlinear operators 厚芝幸子 (山梨大教育) In this talk, we study the relations among $k$-acute points, attractive points and fixed points. Further, we apply these to rearrange proofs of some known convergence theorems and to prove new convergence theorems for nonlinear mappings in Hilbert spaces. Using the ideas of attractive points, acute points and fixed points, we also prove convergence theorems for nonlinear mappings in Banach spaces. PDF 8. $\nu$-generalized metric space の2つの位相 鈴木智成 (九工大工) We will talk about two topologies on $\nu$-generalized metric spaces. PDF 9. π/2回転不変ノルムによる幾何学的定数 冨澤佑季乃 (新潟工大工) In this talk, we study the von Neumann–Jordan constant of $\pi/2$-rotation invariant norms on $R^2$. We know that any $\pi/2$-rotation invariant normed space is isometrically isomorphic to some Day–James space. Since the von Neumann–Jordan constant is invariant under isometrically isomorphic, for characterization of the constant, it is enough to consider the Day–James space. From this fact, we can give some estimations of the constant. PDF 10. 総和法と$\mathbb{R}^{\times}$上の調和解析 国定亮一 (早大教育) We introduce a certain class of summability methods which are defined by the convolution operation in the group algebra $L^1(\mathbb{R}^{\times})$ and study $b$-strongness and $b$-equivalence between them. In particular, this class contains an integral version of Cesàro summability method and we give a necessary and sufficient condition for a summability method in the class to equivalent to this one. PDF 11. The dual inequality of the boundedness fot the Hardy–Littlewood maximal operator and the fractional integrals 飯田毅士 (福島工高専) In this talk, we consider the dual inequality for the Hardy–Littlewood maximal operator $M$ and the fractional integral operator $I_{\alpha}$. Since the fractional integral operator $I_{\alpha}$ has the property $\int I_{\alpha} f g dx=\int f I_{\alpha}g dx$ for $f\geq 0$ and $g\geq 0$, the weight norm inequality $I_{\alpha}: L^{p}(v^{p})\to L^{q}(u^{q})$ is equivalent to $I_{\alpha}:L^{q'}(u^{-q'})\to L^{p'}(v^{-p'})$, where $\frac{1}{p}+\frac{1}{p'}=\frac{1}{q}+\frac{1}{q'}=1$ (\$1