The legacy of Jozef´ Marcinkiewicz: four hallmarks of genius Nikolay Kuznetsov In memoriam of an extraordinary analyst This article is a tribute to one of the most prominent Pol- ish mathematicians Jozef´ Marcinkiewicz who perished 80 years ago in the Katyn´ massacre. He was one of nearly 22000 Polish officers interned by the Red Army in Septem- ber 1939 and executed in April–May 1940 in the Katyn´ forest near Smolensk and at several locations elsewhere. One of these places was Kharkov (Ukraine), where more than 3800 Polish prisoners of war from the Starobelsk camp were executed. One of them was Marcinkiewicz; the plaque with his name (see photo at the bottom) is on the Memo- rial Wall at the Polish War Cemetery in Kharkov.∗ This industrial execution was authorized by Stalin’s secret order dated 5 March 1940, and organised by Beria, who headed the People’s Commissariat for Internal Affairs (the interior ministry of the Soviet Union) known as NKVD. Turning to the personality and mathematical achieve- ments of Marcinkiewicz, it is appropriate to cite the article [24] of his superviser Antoni Zygmund (it is published in the Collected Papers [13] of Marcinkiewicz; see p. 1): Considering what he did during his short life its proceedings, L. Maligranda published the detailed article and what he might have done in normal circum- [9] about Marcinkiewicz’s life and mathematical results; 16 stances one may view his early death as a great pages of this paper are devoted to his biography, where one blow to Polish Mathematics, and probably its finds the following about his education and scientific career. heaviest individual loss during the second world Education. Klemens Marcinkiewicz, Jozef’s´ father, was war. a well-to-do farmer to afford private lessons for him at home (the reason was Jozef’s´ poor health), before sending him From the Marcinkiewicz biography [9] to elementary school and then to gymnasium in Białystok. On the occasion of the centenary of Marcinkiewicz’s birth, a After graduating from it in 1930, Jozef´ enrolled in the De- conference was held on 28 June–2 July 2010, in Poznan.´ In partment of Mathematics and Natural Science of the Stefan Batory University (USB) in Wilno (then in Poland, now Vil- ∗ https://www.tracesofwar.com/sights/10355/Polish-War-Cemetery- Kharkiv.htm nius in Lithuania). From the beginning of his university studies, Jozef´ de- monstrated exceptional mathematical talent which attracted attention of his professors, in particular, of A. Zygmund. His lectures on orthogonal series, requiring some erudi- arXiv:1910.03480v2 [math.HO] 24 Jan 2020 tion, in particular, knowledge of the Lebesgue integral, Marcinkiewicz attended being just a second year student; this was the point, where their collaboration began. The first paper of Marcinkiewicz (see [13], p. 35) had been pub- lished when he was still an undergraduate student. It pro- vides a half-page proof of Kolmogorov’s theorem (1924) guaranteeing the convergence almost everywhere for partial sums of lacunary Fourier series. Marcinkiewicz completed his MSc and PhD theses (both supervised by Zygmund) in 1933 and 1935 respectively; to obtain PhD degree he also passed a rather stiff examination. The second dissertation was the fourth of almost five dozens his publications; it con- cerns interpolation by means of trigonometric polynomials and contains interesting results (see [24], p. 17, for a dis- 1 cussion), but a long publication history awaited this work. June 1939. On his way to Paris, he delivered a lecture there Its part was published in the Studia Mathematica the next and this, probably, was related to this impending appoint- year after the thesis defence (these two papers in French are ment; also, this was the reason to decline an offer of profes- reproduced in [13], pp. 171–185 and 186–199). The full, sorship in the USA during his stay in Paris. original text in Polish appeared in the Wiadomosci´ Matem- Marcinkiewicz still was in England, when the general atyczne (the Mathematical News) in 1939; finally, its En- mobilisation was announced in Poland in the second half glish translation was included in [13], pp. 45–70. of August 1939; the outbreak of war became imminent. His Scientific career. During the two years between de- colleagues advised him to stay in England, but his ill-fated fending his MSc and PhD theses, Marcinkiewicz did the decision was to go back to Poland. He regarded himself one year of mandatory military service and then was Zyg- a patriot of his homeland, which is easily explainable by mund’s assistant at USB. The academic year 1935/1936, the fact that he was just eight years old (very sensitive age Marcinkiewicz spent as an assistant at the Jan Kazimierz in forming a personality) when the independence of Poland University in Lwow.´ Despite 12 hours of teaching weekly, was restored. he was an active participant of mathematical discussions at the famous Scottish Cafe´ (see [3], ch. 10, where this unique Contribution of Marcinkiewicz to mathematics form of doing mathematics is described), and his contribu- tion to the Scottish Book compiled in this cafe´ was substan- Marcinkiewicz was a prolific author as demonstrates a list tial taking into account that his stay in Lwow´ lasted only of his almost five dozen papers written just in seven years nine months. One finds the history of this book in [14], (1933–1939); see Collected Papers [13], pp. 31–33. He was ch. I, whereas problems and their solutions, where applica- open to collaboration; indeed, more than one third of his ble, are presented in ch. II. Marcinkiewicz posed his own papers (19, to be exact) were written with five coauthors, of problem; it concerns the uniqueness of solution for the in- which the lion’s share belongs to his superviser Zygmund. tegral equation Marcinkiewicz is known, primarily, as an outstanding analyst, whose best results deal with various aspects of Z 1 y(t) f (x −t)dt = 0; x 2 [0;1]; real analysis; in particular, theory of series (trigonomet- 0 ric and others), inequalities and approximation theory. He also published several papers concerning complex and func- he conjectured that if f (0) , 0 and f is continuous, then this tional analysis and probability theory. In the extensive pa- equation has only the trivial solution y ≡ 0 (see problem no. per [9] dedicated to the centenary of Marcinkiewicz’s birth, 124, [14], pp. 211 and 212). He also solved three prob- one finds a detailed survey of all his results. lems; his negative answers to problems 83 and 106 posed by H. Auerbach and S. Banach, respectively, involve inge- This survey begins with the description of five topics con- nious counterexamples. His positive solution of problem cerning functional analysis ([9], pp. 153–175). No doubt, 131 (it was formulated by Zygmund in a lecture given in the first two of them—the Marcinkiewicz interpolation the- Lwow´ in the early 1930s) was published in 1938; see [13], orem and Marcinkiewicz spaces—are hallmarks of genius. pp. 413–417. An indirect evidence of ingenuity of the idea behind these During the next two academic years, Marcinkiewicz was results is that the note [11], in which they first appeared, is a senior assistant at USB and after completing his habili- the most cited work of Marcinkiewicz. tation in June 1937 became the youngest docent at USB. Another important point about his work is that he skill- The same year, he was awarded the Jozef´ Piłsudski Scien- fully applied methods of real analysis to questions border- tific Prize (the highest Polish distinction for achievements ing with complex analysis. A brilliant example of this mas- in science at that time). His last academic year 1938/1939, tery—one more hallmark of genius—is the Marcinkiewicz Marcinkiewicz was on leave from USB; a scholarship from function m introduced as an analogue of the Littlewood– the Polish Fund for National Culture yielded him opportu- Paley function g. It is worth mentioning that the short paper nity to travel. He spent October 1938–March 1939 in Paris [10], in which m first appeared, contains other fruitful ideas and moved to the University College London for April–Au- developed by many mathematicians subsequently. gust 1939, visiting also Cambridge and Oxford. One more hallmark of genius one finds in the paper This period was very successful for Marcinkiewicz; he [11] entitled “Sur les multiplicateurs des series´ de Fourier”. published several brief notes in the Comptes rendus de There are many generalisations of its results because of l’Academie´ des Sciences Paris. One of these, namely [12], their important applications. This work was the last of eight became widely cited because the celebrated theorem con- papers that Marcinkiewicz published in the Studia Mathe- cerning interpolation of operators was announced in it. matica; the first three he submitted during his stay in Lwow´ Now, this theorem is referred to as the Marcinkiewicz or and they appeared in 1936. Marcinkiewicz–Zygmund interpolation theorem (see be- Below, the above mentioned results of Marcinkiewicz are low). Moreover, an important notion was introduced in outlined in their historical context together with some fur- p the same note; the so-called weak-L spaces, known as ther developments. One can find a detailed presentation of Marcinkiewicz spaces now, are essential for the general all these results in the excellent textbook [18] based on lec- form of this theorem. tures of the eminent analyst Elias Stein, who made a con- Meanwhile, Marcinkiewicz was appointed to the position siderable contribution to further development of ideas pro- of Extraordinary Professor at the University of Poznan´ in posed by Marcinkiewicz. 2 Marcinkiewicz interpolation theorem and f ; here, mesf:::g denotes the Lebesgue measure of the and Marcinkiewicz spaces corresponding set and Z 1=p There are two pillars of the interpolation theory: the clas- h p i k f kp = j f (x)j dx sical Riesz–Thorin and Marcinkiewicz theorems.