Further study of modulation spaces as Banach algebras　　　　　　　　　　　　　　　
Hans G. Feichtinger, Masaharu Kobayashi, Enji Sato
Annales Univ. Sci. Budapest., Sect. Comp., 56, 151, 166, Sep. 2024, [Peer-reviewed], [Invited], [Corresponding author]
English, Scientific journal

A note on operating functions of modulation spaces
Masaharu Kobayashi, Enji Sato
Journal of Pseudo-Differential Operators and Applications, 13, 4, Springer Science and Business Media LLC, Dec. 2022, [Peer-reviewed], [Corresponding author]
English, Scientific journal

Operating Functions on $$A^q_s({ { \mathbf {T } } })$$
Masaharu Kobayashi, Enji Sato
Journal of Fourier Analysis and Applications, 28, 3, Springer Science and Business Media LLC, Jun. 2022, [Peer-reviewed]
English, Scientific journal

Representation of higher-order dispersive operators via short-time Fourier transform and its application
Keiichi Kato, Masaharu Kobayashi, Shingo Ito, Tadashi Takahashi
Tohoku Mathematical Journal, 73, 1, 105, 118, Mar. 2021, [Peer-reviewed]
English, Scientific journal, 13576533

Operating functions on modulation and Wiener amalgam spaces　　　　　　　　　　　　　　　
Masaharu Kobayashi, Enji Sato
Nagoya Math. J., 230, 72, 82, 2018, [Peer-reviewed]
English, Scientific journal

REMARK ON CHARACTERIZATION OF WAVE FRONT SET BY WAVE PACKET TRANSFORM
Keiichi Kato, Masaharu Kobayashi, Shingo Ito
OSAKA JOURNAL OF MATHEMATICS, 54, 2, 209, 228, OSAKA JOURNAL OF MATHEMATICS, Apr. 2017, [Peer-reviewed]
English, Scientific journal, In this paper, we give characterizations of usual wave front set and wave front set in H-s in terms of wave packet transform without any restriction on basic wave packet, which give complete answers of the question raised by G.B. Folland.

Estimates for Schrödinger operators on modulation spaces　　　　　　　　　　　　　　　
K.Kato, M.Kobayashi, S.Ito
RIMS Kôkyûroku Bessatsu, B60, 129, 143, 2016, [Peer-reviewed]
English, Research institution

Inclusion relations between L-P-Sobolev and Wiener amalgam spaces
Jayson Cunanan, Masaharu Kobayashi, Mitsuru Sugimoto
JOURNAL OF FUNCTIONAL ANALYSIS, 268, 1, 239, 254, ACADEMIC PRESS INC ELSEVIER SCIENCE, Jan. 2015, [Peer-reviewed]
English, Scientific journal, We determined optimal inclusion relations between L-P-Sobolev and Wiener amalgam spaces. For applications, we discuss mapping properties of unimodular Fourier multipliers e(i vertical bar D vertical bar alpha) between L-P-Sobolev and Wiener amalgam spaces and derive some Littlewood-Paley type inequalities. (C) 2014 Elsevier Inc. All rights reserved.

Wave front set defined by wave packet transform and its application
Keiichi Kato, Masaharu Kobayashi, Shingo Ito
NONLINEAR DYNAMICS IN PARTIAL DIFFERENTIAL EQUATIONS, 64, 417, 425, MATH SOC JAPAN, 2015, [Peer-reviewed]
English, International conference proceedings, We introduce the wave front set WFsp,q by using the wave packet transform. This is another characterization of the Fourier Lebesgue type wave front set WFFLqs. We apply this to the propagation of singularities for the wave equation.

Estimates on modulation spaces for Schrodinger evolution operators with quadratic and sub-quadratic potentials
Keiichi Kato, Masaharu Kobayashi, Shingo Ito
JOURNAL OF FUNCTIONAL ANALYSIS, 266, 2, 733, 753, ACADEMIC PRESS INC ELSEVIER SCIENCE, Jan. 2014, [Peer-reviewed]
English, Scientific journal, In this paper we give new estimates for the solution to the Schrodinger equation with quadratic and sub-quadratic potentials in the framework of modulation spaces. (C) 2013 Published by Elsevier Inc.

Characterization of Wave Front Sets in Fourier-Lebesgue Spaces and Its Application
Keiichi Kato, Masaharu Kobayashi, Shingo Ito
FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, 56, 1, 1, 17, KOBE UNIV, DEPT MATHEMATICS, Apr. 2013, [Peer-reviewed]
English, Scientific journal, In this paper, we characterize the Fourier-Lebesgue type wave front set by using the wave packet transform. We apply this to the propagation of singularities for the first order hyperbolic partial differential equations with constant coefficient.

REPRESENTATION OF SCHRODINGER OPERATOR OF A FREE PARTICLE VIA SHORT-TIME FOURIER TRANSFORM AND ITS APPLICATIONS
Keiichi Kato, Masaharu Kobayashi, Shingo Ito
TOHOKU MATHEMATICAL JOURNAL, 64, 2, 223, 231, TOHOKU UNIVERSITY, Jun. 2012, [Peer-reviewed]
English, Scientific journal, We propose a new representation of the Schrodinger operator of a free particle by using the short-time Fourier transform and give its applications.

SCHATTEN p-CLASS PROPERTY OF PSEUDODIFFERENTIAL OPERATORS WITH SYMBOLS IN MODULATION SPACES
Masaharu Kobayashi, Akihiko Miyachi
NAGOYA MATHEMATICAL JOURNAL, 205, 119, 148, DUKE UNIV PRESS, Mar. 2012, [Peer-reviewed]
English, Scientific journal, It is proved that the pseudodifferential operators sigma(t) (X, D) belong to the Schatten p-class C-p, 0 < p <= 2, the symbol sigma(x; omega) is in certain modulation spaees on R-x(d) x R-omega(d).

Application of wave packet transform to Schrödinger equations
K.Kato, S. Ito, M.Kobayashi
RIMS Kôkyûroku Bessatsu, B33, 29, 39, Kyoto University, 2012, [Peer-reviewed]
English, Research institution

Remarks on Wiener amalgam space type estimates for Schrödinger equation
K.Kato, M.Kobayashi, S. Ito
RIMS Kôkyûroku Bessatsu, B33, 41, 48, Kyoto University, 2012, [Peer-reviewed]
English, Research institution

The inclusion relation between Sobolev and modulation spaces
Masaharu Kobayashi, Mitsuru Sugimoto
JOURNAL OF FUNCTIONAL ANALYSIS, 260, 11, 3189, 3208, ACADEMIC PRESS INC ELSEVIER SCIENCE, Jun. 2011, [Peer-reviewed]
English, Scientific journal, The inclusion relations between the L(p)-Sobolev spaces and the modulation spaces is determined explicitly. As an application, mapping properties of unimodular Fourier multiplier e(i|D|alpha) between L(p)-Sobolev spaces and modulation spaces are discussed. (C) 2011 Elsevier Inc. All rights reserved.

Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator　　　　　　　　　　　　　　　
K.Kato, M.Kobayashi, S. Ito
SUT J. Math., 47, 2, 175, 183, 2011, [Peer-reviewed]
English, Scientific journal

MOLECULAR DECOMPOSITION OF THE MODULATION SPACES
Masaharu Kobayashi, Yoshihiro Sawano
OSAKA JOURNAL OF MATHEMATICS, 47, 4, 1029, 1053, OSAKA JOURNAL OF MATHEMATICS, Dec. 2010, [Peer-reviewed]
English, Scientific journal, The aim of this paper is to develop a theory of decomposition in the weighted modulation spaces M(p,q)(s,W) with 0 < p, q <= infinity, s is an element of R and W is an element of A(infinity), where W belongs to the class of A(infinity) defined by Muckenhoupt. For this purpose we shall define molecules for the modulation spaces. As an application we give a simple proof of the boundedness of the pseudo-differential operators with symbols in M(infinity,min(1,p,q))(0). We shall deal with dual spaces as well.

Modulation spaces and their applications　　　　　　　　　　　　　　　
Masaharu Kobayashi
RIMS Kôkyûroku Bessatsu, B22, 131, 135, 2010, [Peer-reviewed]
English, Research institution

On the L-2-boundedness of pseudo-differential operators and their commutators with symbols in alpha-modulation spaces
Masaharu Kobayashi, Mitsuru Sugimoto, Naohito Tomita
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 350, 1, 157, 169, ACADEMIC PRESS INC ELSEVIER SCIENCE, Feb. 2009, [Peer-reviewed]
English, Scientific journal, The results of [J.Sjostrand U. Sjostrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994) 185-192] and Sugimoto [M. Sugimoto, L-p-boundedness of pseudo-differential operators satisfying Besov estimates, [J. Math. Soc. Japan 40 (1988) 105-122] oil a mapping property of pseudo-differential operators are two different kinds of extensions of the pioneering work by Calderon and Vaillancourt [A.P. Calderon, R. Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971) 374-378]. The objective of this paper is to show that these two results, which appeared to be independent ones, can be proved based on the same principle. For the purpose, we use the alpha-modulation spaces, a parameterized family of function spaces, which include Besov spaces and Modulation spaces as special cases. As an application, we also discuss the L-2- boundedness of the commutator [sigma(X. D),a]. where a(x) is a Lipschitz function and sigma belongs to an alpha-modulation space. (c) 2008 Elsevier Inc. All rights reserved.

Embedding relations between local Hardy and modulation spaces
Masaharu Kobayashi, Akihiko Miyachi, Naohito Tomita
STUDIA MATHEMATICA, 192, 1, 79, 96, POLISH ACAD SCIENCES INST MATHEMATICS, 2009, [Peer-reviewed]
English, Scientific journal, A sharp embedding relation between local Hardy spaces and modulation spaces is given.

Trace ideals for pseudo-differential operators and their commutators with symbols in alpha-modulation spaces
Masaharu Kobayashi, Mitsuru Sugimoto, Naohito Tomita
JOURNAL D ANALYSE MATHEMATIQUE, 107, 141, 160, SPRINGER, Jan. 2009, [Peer-reviewed]
English, Scientific journal, That symbols in the modulation space M (1,1) generate pseudo-differential operators of the trace class was first stated by Feichtinger and proved by Grochenig in [13]. In this paper, we show that the same is true if we replace M (1,1) by the more general alpha-modulation spaces, which include modulation spaces (alpha = 0) and Besov spaces (alpha = 1) as special cases. The result with alpha = 0 corresponds to that of Grochenig, and the one with alpha = 1 is a new result which states the trace property of the operators with symbols in the Besov space. As an application, we discuss the trace property of the commutator [alpha (X, D), a], where; a(chi) is a Lipschitz function and sigma(chi, xi) belongs to an alpha-modulation space.

Dual of modulation spaces
Masaharu Kobayashi
JOURNAL OF FUNCTION SPACES AND APPLICATIONS, 5, 1, 1, 8, SCIENTIFIC HORIZON, 2007, [Peer-reviewed]
English, Scientific journal, We have constructed the modulation spaces M-p q (R (d)) in [2] for general 0 < p, q <= infinity , which coincide with the ususal modulation spaces when 1< p, q <= infinity, and studied their basic properties. The aim of this paper is the study of the dual of M-p,M-q (R (d)) for O<p,q<infinity. When 1 <= p,q<infinity, the fact that M-p',M-q' (R-d) is the dual of M-p,M-q(R-d) is already known, where 1/p + 1/p' = 1/q + 1/q' = 1. (See Feichtinger [1].) So in this paper we are concerned with the dual, in particular when p < 1 or q < 1. Motivated by the fact that the modulation spaces have similar properties to that of the Besov spaces (Proposition 2.2), we employ)J. Triebel's method [3] to study the dual. But gained results are similar to the sequence spaces V rather than the Besov spaces B-p,B-q (s)(R-d).

Multipliers on modulation spaces　　　　　　　　　　　　　　　
Masaharu Kobayashi
SUT J. Math., 42, 2, 305, 312, 2006, [Peer-reviewed]
English, Scientific journal

ルベーグ積分 要点と演習　　　　　　　　　　　　　　　
相川 弘明, 小林政晴
共立出版, Sep. 2018, 9784320113411, 244, Japanese, Scholarly book, [Joint work]

Gabor解析における諸問題の解決
科学研究費助成事業 基盤研究(C)
01 Apr. 2022 - 31 Mar. 2025
小林 政晴
日本学術振興会, 基盤研究(C), 北海道大学, 22K03328

調和解析における実関数論の方法とその応用
科学研究費助成事業
01 Apr. 2020 - 31 Mar. 2025
宮地 晶彦, 古谷 康雄, 田中 仁, 冨田 直人, 筒井 容平, 澤野 嘉宏, 小林 政晴, 中井 英一
多重線形の擬微分作用素でシンボルの導関数が決まった関数で抑えられるクラスの作用素に対して、新しいシンボルのクラスを導入し、これまで知られていたLebesgue空間での有界性を含む精密な結果を示した。この研究においては、アマルガム空間と呼ばれる関数空間とBrascamp-Lieb型不等式を利用することが重要な鍵となった。また3重線形Hilbert変換について、双線形の場合を単純に一般化した有界性は成り立たないことを示した。
双線形の分数階積分作用素に対する重み付き評価について新しい不等式を得た。その不等式には2進立方体の直積に対するFefferman-Phong型不等式やCarleson型埋め込み不等式が密接に関係していることを示した。この研究にはスパース作用素が有効に利用された。関数のメディアンと最大関数に対する一般論を整備した。
Morrey空間に関して、変動指数型Morrey空間の相対コンパクト集合の特徴付け、複素補間空間の性質、各点乗子となる関数の特徴付けについて、標準的な設定の下でこれまでに知られていた結果を、一般的な設定の下へ拡張した。また、分数階積分作用素や特異積分作用素のMorrey空間における評価も一般化した。
非圧縮粘Navier-Stokes方程式をBesov空間で考察し、定常解の存在と安定性を示した。同じく非圧縮粘Navier-Stokes方程式を臨界のルベーグ空間においても考察し、弱解が強解になるための条件を得た。消散型偏微分方程式に対して、短時間Fourier変換を用いた解の表示を示し、それを用いてStrichartz型評価などを示した。
日本学術振興会, 基盤研究(B), 東京女子大学, 20H01815

モジュレーション空間とHRT予想の研究
Apr. 2019 - Mar. 2022
小林政晴
文部科学省：科学研究費補助金(基盤研究(C)), Principal investigator, Competitive research funding

Harmonic Analysis by the methods of real analysis
Grants-in-Aid for Scientific Research
01 Apr. 2016 - 31 Mar. 2020
Miyachi Akihiko
For multilinear pseudo-differential operators of ordinal symbol class, we identified sharp differentiability conditions that assure boundedness of the operators in Lebesgue and Hardy spaces. For multilinear pseudo-differential operators of exotic class, we introduced new class of symbols related to general weight functions and obtained sharp estimates for the multilinear pseudo-differential operators in amalgam spaces. For multilinear fractional integral operators, we obtained new inequalities that involve the class of summable functions and found sharp conditions for weight functions of power form. We found a new method to estimate strong maximal functions. We investigated properties of several function spaces including Morrey spaces and their variants, and applied them to study the solutions to partial differential equations.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Tokyo Woman's Christian University, 16H03943

モジュレーション空間とその偏微分方程式への応用　　　　　　　　　　　　　　　
科学研究費補助金(若手研究(B) )
Apr. 2016 - Mar. 2019
小林政晴
文部科学省, Principal investigator, Competitive research funding

Study of the operators on some function spaces in harmonic analysis
Grants-in-Aid for Scientific Research
01 Apr. 2014 - 31 Mar. 2017
SATO ENJI, KOBAYASHI Masaharu
Study of the operators in function spaces by harmonic analysis is very effective for partial differentiable equations. Moreover, it is important that an operator in some function spaces is bounded. Main subjects in our research are study of Fourier multiplier operators, study of fractional integral operators in Morrey spaces, and study of modulation spaces which are related to partial differential equations. First, we gave a simple proof of the restriction theorem of Fourier multipliers, and generalized the result of the fractional integral operators in Morrey spaces. Also we developed the result in modulation spaces by the study of operating functions.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Yamagata University, 26400129

Research on Schroedinger equations by wave packet transform
Grants-in-Aid for Scientific Research
01 Apr. 2013 - 31 Mar. 2016
Kato Keiichi, Ito Shingo, Kobayashi Masaharu
By using the representation of solutions to Schroedinger equations in terms of wave packet transform given by the representative of this research project and the co-workers, the representative and the co-workers has studied properties of solutions to Schroedinger equations. More precisely, we have characterize singularities to Schroedinger equations with time dependent sub-quadratic potentials and for purturbed harmonic oscillator in terms of information of
initial data.
We have studied existence and completeness of wave operators for Schroedinger equations with time dependent potentials. We have shown existence and completeness of wave operator for Schroedinger equations with time dependent shortrange potentials.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Tokyo University of Science, 25400183

Harmonic analysis by real variable methods and its applications
Grants-in-Aid for Scientific Research
01 Apr. 2011 - 31 Mar. 2015
MIYACHI Akihiko, OKADA Masami, FURUYA Yasuo, KIKUCHI Masato, TANAKA Hitoshi, TOMITA Naohito, SAWANO Yoshihiro, NAKAI Eiichi, TSUTSUI Yohei, SATO Shuichi, KOBAYASHI Masaharu, TACHIZAWA Kazuya
Using product type Sobolev norm, we determined the critical differentiability orders in the Hormander-Mihlin type conditions for bilinear Fourier multiplier operators. We generalized the Calderon-Vaillancourt theorem for linear pseudo-differential operators to the case of bilinear pseudo-differential operators. We obtained several new estimates for various operators of harmonic analysis in various function spaces.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Tokyo Woman's Christian University, 23340034

偏微分方程式に対するモジュレーション空間からのアプローチ　　　　　　　　　　　　　　　
科学研究費補助金(若手研究(B) )
Apr. 2012 - Mar. 2015
小林政晴
文部科学省, Principal investigator, Competitive research funding

Harmonic analysis by real variable methods and its applications
Grants-in-Aid for Scientific Research
2006 - 2009
MIYACHI Akihiko, KANJIN Yuichi, KOZONO Hideo, SATO Shuichi, SATO Enji, FURUYA Yasuo, TACHIZAWA Kazuya, SHINOHARA Masahiko, OAKU Toshinori, OKADA Masami, SUGIMOTO Mitsuru, TOMITA Naohito, KOBAYASHI Masaharu, SAWANO Yoshihiro, NAKAI Eiichi, KANJIN Yuichi, SATO Enji
We introduced a function space on a domain of the Euclidean space and established its fundamental properties. The function space has several properties similar to the Hardy space on the whole Euclidean space introduced by Fefferman and Stein. In particular, we showed that the change of variables defined through diffeomorphisms, with certain properties, of the basic domains transforms the function space into another function space of the same kind. We used the function space to study classical orthogonal series. We investigated several other function spaces used in the field of time-frequency analysis and obtained several results concerning the operators acting in those spaces.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Tokyo Woman's Christian University, 18340043