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The Legacy of J´Ozef Marcinkiewicz: Four Hallmarks of Genius

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The Legacy of J´Ozef Marcinkiewicz: Four Hallmarks of Genius The Legacy of J´ozefMarcinkiewicz: Four Hallmarks of Genius In Memoriam of an Extraordinary Analyst Nikolay Kuznetsov This article is a tribute to one of the most prominent Pol- Considering what he did during his short life and ish mathematicians, J´ozefMarcinkiewicz, who perished what he might have done in normal circumstances eighty years ago in the Katy ´nmassacre. He was one of one may view his early death as a great blow to nearly 22,000 Polish officers interned by the Red Army Polish Mathematics, and probably its heaviest in- in September 1939 and executed in April–May 1940 in dividual loss during the second world war. the Katy ´nforest near Smolensk and at several locations elsewhere. One of these places was Kharkov (Ukraine), From the Marcinkiewicz Biography [9] where more than 3,800 Polish prisoners of war from On the occasion of the centenary of Marcinkiewicz’s birth, the Starobelsk camp were executed. One of them was a conference was held on 28 June–2 July 2010 in Pozna ´n. Marcinkiewicz; the plaque with his name (see Figure 1) In its proceedings, L. Maligranda published the detailed is on the Memorial Wall at the Polish War Cemetery article [9] about Marcinkiewicz’s life and mathematical re- in Kharkov.1 This industrial execution was authorized by sults; sixteen pages of this paper are devoted to his biog- Stalin’s secret order dated 5 March 1940 and organized by raphy, where one finds the following about his education Beria, who headed the People’s Commissariat for Internal and scientific career. Affairs (the interior ministry of the Soviet Union) known Education. Klemens Marcinkiewicz, J´ozef’s father, was a as NKVD. farmer well-to-do enough to afford private lessons for him Turning to the personality and mathematical achieve- at home (the reason was J´ozef’s poor health) before send- ments of Marcinkiewicz, it is appropriate to cite the article ing him to elementary school and then to gymnasium in [24] of his supervisor Antoni Zygmund (it is published in Białystok. After graduating in 1930, J´ozefenrolled in the the Collected Papers [13] of Marcinkiewicz; see p. 1): Department of Mathematics and Natural Science of the Stefan Batory University (USB) in Wilno (then in Poland, Nikolay Kuznetsov is a principal research scientist in the Laboratory for Math- now Vilnius in Lithuania). ematical Modelling of Wave Phenomena at the Institute for Problems in Me- From the beginning of his university studies, J´ozefde- chanical Engineering, Russian Academy of Sciences, St. Petersburg. His email monstrated exceptional mathematical talent that attracted address is [email protected]. the attention of his professors, in particular, of A. Zyg- 1https://www.tracesofwar.com/sights/10355/Polish-War mund. Being just a second-year student, Marcinkiewicz -Cemetery-Kharkiv.htm. attended his lectures on orthogonal series, requiring some Communicated by Notices Associate Editor Daniela De Silva. erudition, in particular, knowledge of the Lebesgue in- For permission to reprint this article, please contact: tegral; this was the point where their collaboration be- [email protected] . gan. The first paper of Marcinkiewicz (see [13, p. 35]) DOI: https://doi.org/10.1090/noti2084 690 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 to the Scottish Book compiled in this caf´ewas substantial, taking into account that his stay in Lw´ow lasted only nine months. One finds the history of this book in[14, ch. I], whereas problems and their solutions, where applicable, are presented in ch. II. Marcinkiewicz posed his own prob- lem; it concerns the uniqueness of the solution for the in- tegral equation 1 ∫ 푦(푡)푓(푥 − 푡) d푡 = 0, 푥 ∈ [0, 1]. 0 He conjectured that if 푓(0) ≠ 0 and 푓 is continuous, then this equation has only the trivial solution 푦 ≡ 0 (see prob- lem no. 124 in [14, pp. 211and 212]). He also solved three problems; his negative answers to problems 83 and 106 posed by H. Auerbach and S. Banach, respectively, involve ingenious counterexamples. His positive solution of prob- lem 131 (it was formulated by Zygmund in a lecture given in Lw´ow in the early 1930s) was published in 1938; see [13, pp. 413–417]. During the next two academic years, Marcinkiewicz was a senior assistant at USB and after completing his habili- tation in June 1937 became the youngest docent at USB. Figure 1. Plaque for Marcinkiewicz on the Memorial Wall at The same year, he was awarded the J´ozefPiłsudski Scien- the Polish War Cemetery in Kharkov. tific Prize (the highest Polish distinction for achievements in science at that time). His last academic year 1938–1939, was published when he was still an undergraduate student. Marcinkiewicz was on leave from USB; a scholarship from It provides a half-page proof of Kolmogorov’s theorem the Polish Fund for National Culture afforded him op- (1924) guaranteeing the convergence almost everywhere portunity to travel. He spent October 1938–March 1939 for partial sums of lacunary Fourier series. Marcinkiewicz in Paris and moved to the University College London for completed his MSc and PhD theses (both supervised by April–August 1939, also visiting Cambridge and Oxford. Zygmund) in 1933 and 1935, respectively; to obtain his This period was very successful for Marcinkiewicz; he PhD degree he also passed a rather stiff examination. published several brief notes in the Comptes rendus de The second dissertation was the fourth of his almost five l’Acad´emiedes Sciences Paris. One of these, namely [12], dozen publications; it concerns interpolation by means became widely cited because the celebrated theorem con- of trigonometric polynomials and contains interesting re- cerning interpolation of operators was announced in it. sults (see [24, p. 17] for a discussion), but a long publica- Now this theorem is referred to as the Marcinkiewicz or tion history awaited this work. Part of it was published in Marcinkiewicz–Zygmund interpolation theorem (see be- the Studia Mathematica the next year after the thesis defense low). Moreover, an important notion was introduced in (these two papers in French are reproduced in [13, pp. 171– the same note: the so-called weak-퐿푝 spaces, known as 185 and 186–199]). The full, original text in Polish ap- Marcinkiewicz spaces now, are essential for the general peared in the Wiadomo´sci Matematyczne (the Mathemati- form of this theorem. cal News) in 1939. Finally, its English translation was in- Meanwhile, Marcinkiewicz was appointed to the posi- cluded in [13, pp. 45–70]. tion of Extraordinary Professor at the University of Pozna ´n Scientific career. During the two years between defending in June 1939. On his way to Paris, he delivered a lecture his MSc and PhD theses, Marcinkiewicz did the one year of there and this, probably, was related to this impending ap- mandatory military service and then was Zygmund’s assis- pointment. Also, this was the reason to decline an offer of tant at USB. The academic year 1935–1936 Marcinkiewicz professorship in the USA during his stay in Paris. spent as an assistant at the Jan Kazimierz University in Marcinkiewicz still was in England when the general Lw´ow. Despite twelve hours of teaching weekly, he was mobilization was announced in Poland in the second half an active participant in mathematical discussions at the fa- of August 1939; the outbreak of war became imminent. mous Scottish Caf´e(see [3, ch. 10], where this unique form His colleagues advised him to stay in England, but his of doing mathematics is described), and his contribution ill-fated decision was to go back to Poland. He regarded MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 691 of genius. One indication of the ingenuity of the idea be- hind these results is that the note [11], in which they first appeared, is the most cited work of Marcinkiewicz. Another important point about his work is that he skill- fully applied methods of real analysis to questions border- ing with complex analysis. A brilliant example of this mas- tery—one more hallmark of genius—is the Marcinkiewicz function 휇 introduced as an analogue of the Littlewood– Paley function 푔. It is worth mentioning that the short pa- per [10], in which 휇 first appeared, contains other fruitful ideas developed by many mathematicians subsequently. One more hallmark of genius one finds in the paper [11] entitled “Sur les multiplicateurs des s´eriesde Fourier.” There are many generalizations of its results because of their important applications. This work was the last of eight papers that Marcinkiewicz published in the Studia Mathematica; the first three he submitted during his stay in Lw´ow, and they appeared in 1936. Below, the above-mentioned results of Marcinkiewicz are outlined in their historical context together with some further developments. One can find a detailed presenta- tion of all these results in the excellent textbook [18] based on lectures of the eminent analyst Elias Stein, who made a considerable contribution to further development of ideas proposed by Marcinkiewicz. Figure 2. J ´ozef Marcinkiewicz. Marcinkiewicz Interpolation Theorem and Marcinkiewicz Spaces himself as a patriot of his homeland, which is easily ex- There are two pillars of the interpolation theory: the clas- plainable by the fact that he was just eight years old (very sical Riesz–Thorin and Marcinkiewicz theorems. Each of sensitive age in forming a personality) when the indepen- these serves as the basis for two essentially different ap- dence of Poland was restored. proaches to interpolation of operators known as the com- Contribution of Marcinkiewicz to Mathematics plex and real methods. The term “interpolation of opera- tors” was, presumably, coined by Marcinkiewicz in 1939, Marcinkiewicz was a prolific author, as demonstrated by because Riesz and Thorin, who published their results in the almost five dozen papers he wrote in just seven years 1926 and 1938, respectively, referred to their assertions as (1933–1939); see Collected Papers [13, pp.
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