埃利亚斯·施泰因Elias M. Stein

埃利亚斯·施泰因的父母埃尔坎·施泰因和哈娜·戈德曼是犹太人，这意味着随着纳粹势力席卷欧洲，这个家庭陷入了严重困境。1940年5月德国军队入侵比利时，当时施泰因九岁。次年，施泰因一家逃离纳粹控制的国家，乘船前往美国。他们在纽约定居，施泰因就读于斯图文森高中。这所学校以17世纪荷兰驻纽约殖民总督彼得·斯图文森命名，享有极佳的声誉。他于1949年从高中毕业，同年进入芝加哥大学学习数学。1951年获得学士学位后，施泰因留在芝加哥攻读硕士学位。1953年，芝加哥大学授予他硕士学位，随后他在Antoni Zygmund的指导下攻读博士学位。1955年，施泰因凭借学位论文Linear Operators on $L^{p}$ Spaces获得博士学位。

施泰因在1955-56年间获得NSF博士后奖学金，并于1956年被任命为麻省理工学院数学讲师。他担任此职位两年，期间发表了一系列论文：Interpolation of linear operators (1956)、Functions of exponential type (1957)、Interpolation in polynomial classes and Markoff's inequality (1957)、Note on singular integrals (1957)、(与G Weiss合作)On the interpolation of analytic families of operators action on $H^{p}$ spaces (1957)、(与E H Ostrow合作)A generalization of lemmas of Marcinkiewicz and Fine with applications to singular integrals (1957)、A maximal function with applications to Fourier series (1958)、(与G Weiss合作)Fractional integrals on n-dimensional Euclidean space (1958)、(与G Weiss合作)Interpolation of operators with change of measures (1958)、Localization and summability of multiple Fourier series (1958)和On the functions of Littlewood-Paley, Luzin, Marcinkiewicz (1958)。

1958年，施泰因回到芝加哥大学，被任命为助理教授。次年，即1959年3月21日，他与Elly Intrator结婚；他们有一个儿子Jeremy（现为金融经济学教授）和一个女儿Karen。1961年，施泰因在芝加哥晋升为副教授。他获得了1962-63年度的NSF高级博士后奖学金和1961-63年度的Alfred P Sloan基金会奖学金，并于1962-63年在普林斯顿高等爱德华·斯图迪度过。1963年，他离开芝加哥的职位，被任命为普林斯顿大学数学教授。他的余下职业生涯都在普林斯顿度过，其中1968年至1971年以及1985年至1987年担任数学系主任，1984-85年由Guggenheim奖学金资助在高等斯图迪度过。他现在是普林斯顿的亚伯拉罕·阿德里安·艾伯特 Baldwin Dodd数学教授。

现在让我们给出一些指示，说明施泰因对调和分析[4]做出的显著贡献：-

施泰因塑造了数学分析领域，并改变了数学家在该领域几乎每个子领域中处理问题的方式。他是最早欣赏偏微分方程、经典约瑟夫·傅里叶分析、多复变量和表示论之间相互作用的人之一。他是第一个从这种相互作用中感知到每个领域的基本见解的人。施泰因是世界领先的调和分析权威。施泰因和同事引入了高维解析函数的推广，称为$H^{p}$-空间。这一理论导致了调和分析与概率论之间的重要联系，并促进了众多问题的解决。在他的研究中，他还展示了使用平方函数控制误差项的力量，这是一种他发明的技术，现在在调和分析中是基本的。

为了对他的工作有所了解，我们来看他写的一些书，以及他因领先贡献而获得的众多奖项中的一些引文。他在1970年出版了两本书，其中一本是Topics in harmonic analysis related to the Littlewood-Paley theory。以下是R E Edwards的一篇评论的摘录：-

已知Fourier series和积分的理论的相当部分可以推广到各类拓扑群。本书关注的是这类推广中较为精细和微妙的一种。在这种情况下，起点被恰当地描述为利特尔伍德-雷蒙德·佩利理论，而且推广甚至超出了群的情形。……作者以适中的篇幅和细节呈现了一个复杂主题的富有启发性和激励性的图景，值得祝贺，尤其是因为它还附有对未解决问题的评论和一些相当明确的进一步研究建议。

同样在1970年，施泰因出版了Singular integrals and differentiability properties of functions。这本书基于他1966-67年在巴黎奥赛讲授的一门课程Intégrales singulières et fonctions différentiables de plusieurs variables Ⓣ（奇异积分与多变量可微函数）。这本书由普林斯顿大学出版社出版，35多年后，该出版社这样描述这项工作：-

奇异积分是分析——数学三大主要分支之一——中最有趣和最重要的研究对象之一。它们涉及实数、复数及其函数。在这本书中，普林斯顿大学教授施泰因，一位领先的数学创新者也是一位有天赋的阐述者，写出了被称为过去三十五年中最有影响力的数学文本。它作为文本成功的一个原因是其几乎传奇般的呈现方式：施泰因将以前只有专家才能理解的深奥材料，变得连刚入学的研究生也能理解。读者反映，当你读这本书时，你不仅看到过去的伟人做了令人兴奋的工作，而且你还感到受到启发，相信自己能够掌握这个主题并为之做出贡献。当施泰因的书首次出版时，奇异积分只有少数专家知道。然而，随着时间的推移，这本书激励了整整一代研究人员将其方法应用于许多学科中的广泛问题，包括工程、生物学和金融。

1984年，施泰因 因这本书获得了 美国数学会 的 Leroy P 凯瑟琳·斯蒂尔 奖。然而，这并不是 施泰因 获得的唯一 Leroy P 斯蒂尔 奖，因为2002年他获得了 Leroy P 斯蒂尔 终身成就奖。引文如下 [1]：-

在跨越近半个世纪的科学职业生涯中，Eli 施泰因对分析的不同分支做出了基础性贡献。在调和分析中，他的插值定理是一个无处不在的工具。他关于约瑟夫·傅里叶变换与曲率之间关系的结果揭示了一个深刻且未被怀疑的性质，并具有深远的影响。他在Hardy空间上的工作改变了这个主题。他也对索菲斯·李群的表示论做出了重要贡献。他在多复变量上的工作同样引人注目。他为∂-问题给出的显式近似解使得证明强伪凸域中解的尖锐正则性结果成为可能。在这方面，他还获得了次椭圆估计，这些估计锐化并量化了拉尔斯·霍尔曼德尔关于二阶算子的亚椭圆性定理。除了通过自己的研究和优秀专著做出的贡献外，施泰因还与许多学生合作并影响了他们，这些学生后来自己也做出了深刻的贡献。

在回复中施泰因说[1]：-

一个多世纪以来，约瑟夫·傅里叶分析、复函数理论、偏微分方程、实分析，以及来自几何学和解析数论等其他学科的思想之间，一直存在着重要而富有成果的互动。这种情况已变得越来越明显，而所涉及的努力和发展，如果说有什么变化的话，那就是在过去二三十年中加速了。到了这个阶段，我们可以确信，我们远未到达这项事业的终点，许多激动人心和美妙的定理仍等待着我们去发现。

许多其他重要奖项也被授予施泰因。他获得了冯·洪堡奖（1989-90），1993年瑞典科学院颁发的肖克奖，1999年的沃尔夫奖，2002年6月13日由乔治·布什总统在白宫仪式上授予他的美国国家科学奖章，以及2005年的斯特凡·伯格曼奖。我们引用最后提到的奖项的颁奖词中的一小段[3]：-

斯特凡·伯格曼奖授予施泰因 M 施泰因，以表彰他在实分析、复分析和调和分析方面的工作。施泰因通过他的研究、他的阐述工作以及他对研究生的培养做出了决定性的贡献。……施泰因将复分析、偏微分方程、幂零索菲斯·李群上的分析和欧几里得调和分析融合在一起，深深影响了无数数学家。他的思想和技术将在未来许多年继续影响数学。

2001年，普林斯顿大学授予施泰因杰出教学校长奖。

让我们回到施泰因出版的杰出著作上。1971年，Analytic continuation of group representations问世，它基于施泰因1967年11月在耶鲁大学所做的一系列詹姆斯·惠特莫尔讲座。同样在1971年，施泰因与吉多·韦斯合作出版了Introduction to Fourier analysis on Euclidean spaces。Edwin Hewitt将这本书描述为“构思严谨，执行出色”，并以这样的话结束他的评论：-

这是一本出色的书，人们希望它能推动具体分析的研究并激发进一步的进展。

施泰因的其他著作包括：Boundary behavior of holomorphic functions of several complex variables（1992）；与P C Greiner合著的Estimates for the ∂-Neumann problem（1977）；与Alexander Nagel合著的Lectures on pseudodifferential operators: regularity theorems and applications to nonelliptic problems（1979）；与G B Folland合著的Hardy spaces on homogeneous groups（1982）；基于他1991-92年所作的三次Milliman讲座的Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals（1993）；以及与Rami Shakarchi合著的Fourier analysis: An introduction（2003）。评述最后提到的这部著作时，Steven George Krantz写道：-

施泰因无疑是现代约瑟夫·傅里叶分析最伟大的代表和发展者之一。R Shakarchi是查尔斯·费夫曼的近期学生，因此显然在这一学科中训练有素。我们确实很幸运，分析的核心部分之一有两位如此杰出的代表人物抽出时间撰写了这样一部教学著作。因为这不是奇异积分、拟微分算子或仿微分算子或波前集的入门书。相反，它是对约瑟夫·傅里叶分析非常经典主题的基础性介绍。……我期待阅读和评述这一系列中接下来的三本书（由同一作者撰写）。这一文集令人兴奋的特点之一是，它在分析的不同部分（实分析、复分析、约瑟夫·傅里叶分析和概率论）之间建立了许多不明显的联系。它对学生和导师都将具有教益。这第一卷是一个极好的开端，有望在未来许多年中成为经典。

这部三卷本系列Complex analysis的第二卷于2003年出版，第三卷Real analysis: Measure theory, integration, and Hilbert spaces于2005年出版。这些著作也获得了热烈的好评，表明它们都是以非凡的清晰和用心写成的杰出作品。

除了我们上面提到的奖项和荣誉外，施泰因还当选为国家科学院（1974）和美国艺术与科学院（1982）的成员。他获得了北京大学（1988）和芝加哥大学（1992）的荣誉学位。

施泰因的两位博士生，查尔斯·费夫曼和陶哲轩，获得了菲尔兹奖。查尔斯·费夫曼这样评价他的学位论文导师[2]：-

施泰因的工作常常结合两个非凡的品质：对数学若干分支的理解——其中每个分支通常只有专家才了解——以及发现它们之间联系的惊人能力。在施泰因告诉你他的解答之前，所涉及的问题看起来毫无希望。看起来似乎没有任何联系。然后，以完全正确的观点和完全正确的寥寥数语，人们看到了令人难以置信的洞见，将所有事物联系在一起，并使那些看起来完全不可能的事情变得显而易见。他一次又一次地做出了这样的事情。

施泰因在普林斯顿的同事Joseph Kohn说[2]：-

施泰因是世界上一流的调和分析专家之一，他对这一领域以及多复变函数论和偏微分方程等相关领域做出了卓越贡献。他有许多学生和合作者；他对一代数学家产生了深远影响。

Elias Stein's parents, Elkan Stein and Chana Goldman, were Jewish which meant that the family were in severe difficulties as the Nazi influence spread across Europe. Elias was nine years old when the German armies invaded Belgium in May 1940. In the following year the Stein family escaped from the Nazi controlled country and sailed to the United States. There they settled in New York and Elias attended Stuyvesant High School. This school, named after Peter Stuyvesant who was the 17th century Dutch colonial governor of New York, had an excellent reputation. He graduated from the High School in 1949 and in the same year entered the University of Chicago to study mathematics. After the award of his bachelor's degree in 1951, Stein remained at Chicago studying for his Master's degree. He was awarded an M.A. by Chicago in 1953 and undertook research for his doctorate with Antoni Zygmund as his thesis advisor. Stein was awarded a Ph.D. in 1955 for his thesis Linear Operators on $L^{p}$ Spaces.

Stein held an NSF Postdoctoral Fellowship during 1955-56 and was appointed as an Instructor in Mathematics at the Massachusetts Institute of Technology in 1956. He held this position for two years during which time a whole series of his papers appeared in print: Interpolation of linear operators (1956), Functions of exponential type (1957), Interpolation in polynomial classes and Markoff's inequality (1957), Note on singular integrals (1957), (with G Weiss) On the interpolation of analytic families of operators action on $H^{p}$ spaces (1957), (with E H Ostrow) A generalization of lemmas of Marcinkiewicz and Fine with applications to singular integrals (1957), A maximal function with applications to Fourier series (1958), (with G Weiss) Fractional integrals on n-dimensional Euclidean space (1958), (with G Weiss) Interpolation of operators with change of measures (1958), Localization and summability of multiple Fourier series (1958), and On the functions of Littlewood-Paley, Luzin, Marcinkiewicz (1958).

In 1958 Stein returned to the University of Chicago where he was appointed as an Assistant Professor. In the following year, on 21 March 1959, he married Elly Intrator; they had a son Jeremy (who is now a professor of financial economics) and a daughter Karen. In 1961 Stein was promoted to Associate Professor at Chicago. Awarded an NSF Senior Postdoctoral Fellowship for 1962-63 and an Alfred P Sloan Foundation Fellowship for 1961-63, he spent 1962-63 at the Institute for Advanced Study at Princeton. He left his position in Chicago in 1963 when he was appointed as Professor of Mathematics at Princeton University. He has spent the rest of his career at Princeton where he was chairman of the Mathematics Department from 1968 to 1971 and again from 1985 to 1987, having spent 1984-85 at the Institute for Advanced Study supported by a Guggenheim Fellowship. He is now the Albert Baldwin Dodd Professor of Mathematics at Princeton.

Let us now give some indication of the remarkable contributions that Stein has made to harmonic analysis [4]:-

Elias Stein has shaped the field of mathematical analysis and has changed the way mathematicians approach problems in nearly every subarea of the field. He was among the first to appreciate the interplay among partial differential equations, classical Fourier analysis, several complex variables and representation theory. He was the first to perceive the fundamental insights in each field arising from that interplay. Stein is the world's leading authority in harmonic analysis. Stein and colleagues introduced a generalization of analytic functions in higher dimensions known as $H^{p}$-spaces. This theory led to important connections between harmonic analysis and probability theory, and facilitated the solution of numerous problems. In his studies, he also showed the power of using square functions to control error terms, a technique that he invented and that is now fundamental in harmonic analysis.

To gain some understanding of his work we look at some of the books he has written and the citations from some of the many awards he has been given for his leading contributions. He published two books in 1970, one being Topics in harmonic analysis related to the Littlewood-Paley theory. Here are extracts from a review by R E Edwards:-

Considerable portions of the theory of Fourier series and integrals are known to extend to various categories of topological groups. This book is concerned with one of the more elaborate and delicate extensions of this sort. In this case the starting point is aptly described as Littlewood-Paley theory, and the extension goes even beyond the group case. ... The author is to be congratulated on presenting in moderate space and detail an illuminating and stimulating picture of a complex topic, the more so since it is garnished with comments on open problems and some quite explicit suggestions for further research.

Also in 1970 Stein published Singular integrals and differentiability properties of functions. This book was based on a course Intégrales singulières et fonctions différentiables de plusieurs variables Ⓣ which he had given at Orsay, Paris, in 1966-67. The book was published by Princeton University Press who, over 35 years later, described the work as follows:-

Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance.

In 1984 Stein was awarded the American Mathematical Society's Leroy P Steele Prize for this book. This, however, was not the only Leroy P Steele Prize won by Stein for in 2002 he received the Leroy P Steele Prize for Lifetime Achievement. The citation reads [1]:-

During a scientific career that spans nearly half a century, Eli Stein has made fundamental contributions to different branches of analysis. In harmonic analysis, his Interpolation Theorem is a ubiquitous tool. His result about the relation between the Fourier transform and curvature revealed a deep and unsuspected property and has far reaching consequences. His work on Hardy spaces has transformed the subject. He has made important contributions to the representation theory of Lie groups as well. His work on several complex variables is equally striking. His explicit approximate solutions for the ∂-problems made it possible to prove sharp regularity results for solutions in strongly pseudoconvex domains. In this connection he also obtained subelliptic estimates which sharpened and quantified Hörmander's hypoellipticity theorem for second order operators. Besides his contributions through his own research and excellent monographs, Stein has worked with and influenced many students, who have gone on to make profound contributions of their own.

In reply Stein said [1]:-

For more than a century there has been a significant and fruitful interaction between Fourier analysis, complex function theory, partial differential equations, real analysis, as well as ideas from other disciplines such as geometry and analytic number theory, etc. That this is the case has become increasingly clear, and the efforts and developments involved have, if anything, accelerated in the last twenty or thirty years. Having reached this stage, we can be confident that we are far from the end of this enterprise and that many exciting and wonderful theorems still await our discovery.

Many other major prizes have been awarded to Stein. He received the von Humboldt Award (1989-90), the Schock Prize from the Swedish Academy of Sciences in 1993, the Wolf Prize in 1999, the United States National Medal of Science which was presented to him by President George Bush at a White House ceremony on 13 June 2002, and the Stefan Bergman Prize in 2005. We give a short quote from the citation for the last mentioned award [3]:-

The Bergman prize is awarded to Elias M Stein in recognition of his work in real, complex, and harmonic analysis. Stein has made decisive contributions through his research, his expository efforts, and his training of graduate students. ... Stein's fusion of complex analysis, partial differential equations, analysis on nilpotent Lie groups, and Euclidean harmonic analysis has deeply influenced countless mathematicians. His ideas and techniques will continue to impact mathematics for years to come.

In 2001, Princeton University awarded Stein its President's Award for Distinguished Teaching.

Let us return to looking at the outstanding books Stein has published. In 1971 Analytic continuation of group representations appeared based on a series of James Whittemore lectures that Stein had given at Yale University in November 1967. Also in 1971, in collaboration with Guido Weiss, Stein published Introduction to Fourier analysis on Euclidean spaces. Edwin Hewitt describes the book as "soundly conceived and brilliantly executed", ending his review with the words:-

This is a splendid book, destined, one hopes, to further the study of concrete analysis and to inspire further advances.

Other books by Stein include: Boundary behavior of holomorphic functions of several complex variables (1992); (with P C Greiner) Estimates for the ∂-Neumann problem (1977); (with Alexander Nagel) Lectures on pseudodifferential operators: regularity theorems and applications to nonelliptic problems (1979); (with G B Folland) Hardy spaces on homogeneous groups (1982); Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals (1993) based on the three Milliman Lectures he gave in 1991-92; and (with Rami Shakarchi) Fourier analysis: An introduction (2003). Reviewing this last mention text, Steven George Krantz writes:-

E M Stein is certainly one of the great avatars and developers of Fourier analysis in modern times. R Shakarchi is a recent student of Charles Fefferman, so obviously is very well trained in the discipline. We are fortunate indeed that two such prominent exponents of one of the central parts of analysis have taken the time to write an instructional book of this kind. For this is not an entree to singular integrals, nor to pseudodifferential operators or paradifferential operators or wave front sets. It is instead a basic introduction to very classical topics of Fourier analysis. ... I look forward to reading and reviewing the next three books in this series (by the same authors). One of the exciting features of this collection is that it establishes many non-obvious connections among different parts of analysis (real analysis, complex analysis, Fourier analysis, and probability). It will be instructive for student and mentor alike. This first volume is a terrific beginning, and promises to stand as a classic for many years to come.

The second in this three volume series Complex analysis was published in 2003 and the third volume Real analysis: Measure theory, integration, and Hilbert spaces in 2005. These texts also received rave reviews indicating they are all outstanding works written with remarkable clarity and care.

In addition to the prizes and awards we have mentioned above, Stein has been honoured with election to membership of the National Academy of Sciences (1974) and the American Academy of Arts and Sciences (1982). He has received honorary degrees from Peking University (1988) and the University of Chicago (1992).

Two of Stein's doctoral students, Charles Fefferman and Terence Tao, have won Fields medals. Fefferman said this of his thesis advisor [2]:-

Stein's work often combines two remarkable qualities: an understanding of several branches of math, each of which normally is known only by specialists, and an astonishing ability to find connections between them. Before Stein tells you his solution, the problems involved look utterly hopeless. It looks as if there is no connection. Then, with exactly the right point of view and exactly the right few words, one sees incredible insights that link everything together and make obvious things that would have appeared to be totally impossible. He has done things like this over and over again.

Stein's Princeton colleague Joseph Kohn said [2]:-

Stein is one of the foremost experts in harmonic analysis in the world, and he has made stellar contributions to this field as well as related fields such as the theory of several complex variables and partial differential equations. He has had many students and collaborators; he has had a profound influence on a generation of mathematicians.