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, JózefMarcinkiewicz, who perished what he might have done in normal circumstances eighty years ago in the Katy ńmassacre. 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 ńforest 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 ń. 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, Józef’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 Józef’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, Józefenrolled 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, Józefde- 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 caféwas substantial, taking into account that his stay in Lwów 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 problem; it concerns the uniqueness of the solution for the integral 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 problem 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 problem 131 (it was formulated by Zygmund in a lecture given in Lwów 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 JózefPiłsudski 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’Académiedes 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 ń 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 Lwów. 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 Café(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 paper [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 sériesde 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 Lwów, 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 presentation 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 ózef 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 operators” 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. 31–33]. He was “convexity theorems.” open to collaboration; indeed, more than one third of his It is worth emphasizing again that a characteristic fea- papers (nineteen, to be exact) were written with five coau- ture of Marcinkiewicz’s work was applying real methods thors, of which the lion’s share belongs to his supervisor to problems that other authors treated with the help of Zygmund. complex analysis. It was mentioned above that in his Marcinkiewicz is known, primarily, as an outstanding paper [10] published in 1938, Marcinkiewicz introduced analyst, whose best results deal with various aspects of real the function 휇 without using complex variables but so analysis, in particular, theory of series (trigonometric and that it is analogous to the Littlewood–Paley function 푔, others), inequalities, and approximation theory. He also whose definition involves these variables. In the same year, published several papers concerning complex and func- 1938, Thorin published his extension of the Riesz con- tional analysis and probability theory. In the extensive pa- vexity theorem, which exemplifies the approach based on per [9] dedicated to the centenary of Marcinkiewicz’s birth, complex variables. Possibly this stimulated Marcinkiewicz one finds a detailed survey of all his results. to seek an analogous result with proof relying on real anal- This survey begins with the description of five topics ysis. Anyway, Marcinkiewicz found his interpolation the- concerning functional analysis ([9, pp. 153–175]). No orem and announced it in [12]; concurrently, a letter was doubt, the first two of them—the Marcinkiewicz interpo- sent to Zygmund that contained the proof concerning a lation theorem and Marcinkiewicz spaces—are hallmarks

692 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 particular case. Ten years after World War II, Zygmund re- Marcinkiewicz spaces. Another crucial step, made by constructed the general proof and published it in 1956; Marcinkiewicz in [12], was the introduction of the weak for this reason the theorem is sometimes referred to as the 퐿푝 spaces playing the essential role in his general interpola- Marcinkiewicz–Zygmund interpolation theorem. tion theorem. They are now called the Marcinkiewicz spaces An excellent introduction to the interpolation theory and usually denoted 퐿푝,∞. one finds in the book [1] based on the works of Jaak Pee- To give an idea of these spaces, let us consider a measure tre (he passed away on 1 April 2019 at age eighty-three), space (푈, Σ, 푚) over real scalars with a nonnegative mea- whose contribution to this theory cannot be overestimated. sure 푚 (just to be specific). For a real-valued 푓, which is In collaboration with Jacques-Louis Lions, he introduced finite almost everywhere and 푚-measurable, we introduce the “real method interpolation spaces” (see their funda- its distribution function mental article [8]), which can be considered as “descendants” of the Marcinkiewicz interpolation theorem. 푚({푥 ∶ |푓(푥)| > 휆}), 휆 ∈ (0, ∞) An important fact of Peetre’s biography is that his life and put was severely changed during World War II (another re- 1/푝 minder about that terrible time). With his parents, Jaak |푓|푝,∞ = sup 휆[푚({푥 ∶ |푓(푥)| > 휆})] for 푝 ∈ [1, ∞). escaped from Estonia in September 1944 just two days be- 휆>0 fore his home town of Pärnu was destroyed in an air raid 푝,∞ 푝 푝,∞ Then 퐿 = {푓 ∶ |푓|푝,∞ < ∞}, and it is clear that 퐿 ⊂ 퐿 of the Red Army. He was only ten years old when his fam- for 푝 ∈ [1, ∞), because |푓|푝,∞ ≤ ‖푓‖푝 in view of Cheby- ily settled in Lund (Sweden), where he spent most of his shev’s inequality. The Marcinkiewicz space for 푝 = ∞ is life. But let us turn to mathematics again. 퐿∞ by definition. The Marcinkiewicz interpolation theorem for operators It occurs that |푓|푝,∞ is not a norm for 푝 ∈ [1, ∞), but a 푝 푛 in 퐿 (ℝ ). We begin with this simple result because it has quasi-norm because numerous applications, being valid for subadditive opera- 푝 푛 tors mapping the Lebesgue spaces 퐿 (ℝ ) with 푝 ≥ 1 into |푓 + 푔|푝,∞ ≤ 2(|푓|푝,∞ + |푔|푝,∞) themselves (see, e.g., [18, ch. 1, sect. 4]). We recall that an (see, e.g., [1, p. 7]). However, it is possible to endow 퐿푝,∞, operator 푇 ∶ 퐿푝 → 퐿푝 is subadditive if 푝 ∈ (1, ∞), with a norm ‖⋅‖푝,∞, converting it into a Banach |푇(푓1 + 푓2)(푥)| = |푇(푓1)(푥)| + |푇(푓2)(푥)| for every 푓1, 푓2 . space. Moreover, the inequality

Furthermore, 푇 is of weak type (푟, 푟) if the inequality −1 |푓|푝,∞ ≤ ‖푓‖푝,∞ ≤ 푝(푝 − 1) |푓|푝,∞ 푟 푟 훼 mes{푥 ∶ |푇(푓)(푥)| > 훼} ≤ 퐴푟‖푓‖푟 holds for all 푓 ∈ 퐿푝,∞. It is worth mentioning that 퐿푝,∞ holds for all 훼 > 0 and all 푓 ∈ 퐿푟 with 퐴 independent of 푟 belongs (as a limiting case) to the class of Lorentz spaces 훼 and 푓. Here, mes{… } denotes the Lebesgue measure of 퐿푝,푞, 푞 ∈ [1, ∞] (see, e.g., [1, sect. 1.6] and references cited the corresponding set, and in this book). 1/푝 푝,∞ 푝 Another generalization of 퐿 , known as the Marcinkie- ‖푓‖푝 = [ ∫ |푓(푥)| d푥] ℝ푛 wicz space 푀휑, is defined with the help of a nonnegative, is the norm in 퐿푝(ℝ푛). Now, we are in a position to formu- concave function 휑 ∈ 퐶[0, ∞). This Banach space con- late the following. sists of all (equivalence classes of) measurable functions for which the norm Theorem 1. Let 1 ≤ 푟1 < 푟2 < ∞, and let 푇 be a subadditive 푡 operator acting simultaneously in 퐿푟푖 (ℝ푛), 푖 = 1, 2. If it is of 1 ‖푓‖ = sup ∫ 푓∗(푠) d푠 (푟, 푟) 푖 = 1, 2 푝 ∈ (푟 , 푟 ) 휑 weak type 푖 푖 for , then for every 1 2 the 푡>0 휑(푡) 0 푝 푛 inequality ‖푇(푓)‖푝 ≤ 퐵‖푓‖푝 holds for all 푓 ∈ 퐿 (ℝ ) with 퐵 is finite. Here 푓∗ denotes the nonincreasing rearrangement depending only on 퐴푟 , 퐴푟 , 푟1, 푟2, and 푝. 1 2 of 푓, i.e., When 퐵 is independent of 푓 in the last inequality, the operator 푇 is of strong type (푝, 푝); it is clear that 푇 is also of 푓∗(푠) = inf{휆 ∶ 푚({푥 ∶ |푓(푥)| > 휆}) ≤ 푠} for 푠 ≥ 0, 휆>0 weak type (푝, 푝) in this case. In the letter to Zygmund mentioned above, Marcinkie- and so is nonnegative and right-continuous. Moreover, its ∗ wicz included a proof of this theorem for the case 푟1 = 1 distribution function 푚({푥 ∶ |푓 (푥)| > 휆}) coincides with 1−1/푝 and 푟2 = 2. Presumably, it was rather simple; indeed, even that of 푓. If 휑(푡) = 푡 , then the corresponding Marcin- 푝,∞ 1 when 푟2 < ∞ is arbitrary, the proof is less than two pages kiewicz space is 퐿 , whereas 휑(푡) ≡ 1 and 휑(푡) = 푡 give 퐿 long in [18, ch. 1, sect. 4]. and 퐿∞, respectively.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 693 The Marcinkiewicz interpolation theorem for bounded See Vol. II, ch. XVI, sects. 2 and 3, where, in particular, linear operators. This kind of continuous operator is usu- it is demonstrated that 퐹, originally defined on a dense ally considered as mapping one normed space to another set in 퐿푝, 푝 ∈ [1, 2], is extensible to the whole space as a ′ one, in which case the operator’s norm is an important bounded operator 퐹 ∶ 퐿푝 → 퐿푝 , 푝′ = 푝/(푝 − 1), and so the ′ characteristic. However, the latter can be readily general- integral converges in 퐿푝 . To prove this assertion and its 푝 푝,∞ ized for a mapping of 퐿 to 퐿 . Indeed, if |푇푓|푝,∞ ≤ 푛-dimensional analogue one can use Theorem 2. Indeed, 1 ∞ 퐶‖푓‖푝, then it is natural to introduce the norm (or quasi- 퐹 ∶ 퐿 → 퐿 is bounded (this is straightforward to see), norm) of 푇 as the infimum over all possible values of 퐶. and by Plancherel’s theorem 퐹 is bounded on 퐿2, and so Now we are in a position to formulate the following. this theorem is applicable. On the other hand, the Riesz– Thorin theorem, which has no restriction 푝 ≤ 푞, yields a Theorem 2. Let 푝0, 푞0, 푝1, 푞1 ∈ [1, ∞] satisfy the inequali- −1 푝 ≤ 푞 푝 ≤ 푞 푞 ≠ 푞 푝, 푞 ∈ [1, ∞] more complete result valid for the inverse transform 퐹 as ties 0 0, 1 1, and 0 1, and let be 푝′ 푝 such that 푝 ≤ 푞 and the equalities well. The latter operator acting from 퐿 to 퐿 is bounded; here 푝′ ∈ [2, ∞), and so 푝 = 푝′/(푝′ − 1) ∈ (1, 2]. 1 1 − 휃 1 1 1 − 휃 1 = + and = + (2) In studies of conjugate Fourier series, the singular 푝 푝 푝 푞 푞 푞 0 1 0 1 integral operator (the periodic Hilbert transform) hold for some 휃 ∈ (0, 1). If 푇 is a linear operator that maps 퐿푝0 푞0,∞ 푝1 1 푡 into 퐿 and its norm is 푁0 and simultaneously 푇 ∶ 퐿 → 퐻(푓)(푠) = lim ∫ 푓(푠 − 푡) cot d푡 푞1,∞ 푝 푞 2휋 휖→0 2 퐿 has 푁1 as its norm, then 푇 maps 퐿 into 퐿 and its norm 휖≤|푡|≤휋 푁 satisfies the estimate plays an important role. Indeed, by linearity it is sufficient 1−휃 휃 2 푁 ≤ 퐶푁0 푁1 , (1) to define 퐻 on a basis in 퐿 (−휋, 휋), and the relations with 퐶 depending on 푝0, 푞0, 푝1, 푞1, and 휃. 퐻(cos 푛푡) = sin 푛푠 for 푛 ≥ 0, 퐻(sin 푛푡) = − cos 푛푠 for 푛 ≥ 1 The convexity inequality (1) is a characteristic feature show that it expresses passing from a trigonometric series of the interpolation theory. The general form of this theo- to its conjugate. Moreover, these formulae show that 퐻 is rem (it is valid for quasi-additive operators, whose special bounded on 퐿2(−휋, 휋) and its norm is equal to one. case are subadditive ones described prior to Theorem 1) is In the mid-1920s, Marcel Riesz obtained his celebrated proved in [23, ch. XII, sect. 4]. In particular, it is shown result about this operator; first, he announced it in a brief that one can take note in the Comptes rendus de l’Acad´emiedes Sciences Paris, 푞 푞 1/푞 푝(1−휃)/푝0 푝휃/푝1 퐶 = 2 + 0 1 ; and three years later published his rather long proof that ( ) 1/푝 푝 |푞 − 푞0| |푞 − 푞1| 푝 퐻 is bounded on 퐿 (−휋, 휋) for 푝 ∈ (1, ∞); i.e., for every see [23, Vol. II, p. 114, formula (4.18)], where, unfortu- finite 푝 > 1 there exists 퐴푝 > 0 such that nately, the notation differs from that adopted here. Special ‖퐻(푓)‖푝 ≤ 퐴푝‖푓‖푝 for all 푓 ∈ 퐿푝(−휋, 휋). (2) cases of Theorem 2 and diagrams illustrating them can be found in [9, pp. 155–156]. It should be emphasized that However, (2) does not hold for 푝 = 1 and ∞; see [23, Vol. I, the restriction 푝 ≤ 푞 is essential; indeed, as early as 1964, ch. VII, sect. 2] for the corresponding examples and a proof R. A. Hunt [6] constructed an example demonstrating that of this inequality. Theorem 2 is not true without it. For a description of this There are several different proofs of this theorem; the example see, e.g., [1, pp. 16–17]. original proof of M. Riesz was reproduced in the first edi- It was Marcinkiewicz himself who proposed an exten- tion of Zygmund’s monograph [23], which appeared in sion of his interpolation theorem to other function spaces; 1935. In the second edition published in 1959, this proof namely, the so-called diagonal case (when 푝0 = 푞0 and was replaced by that of Calder´onobtained in 1950. Let us 푝1 = 푞1) of his theorem is formulated for Orlicz spaces outline another proof based on the Marcinkiewicz inter- in [12]. References to papers containing further results on polation theorem analogous to Theorem 1 but involving interpolation in these and other spaces (e.g., Lorentz and 퐿푝-spaces on (−휋, 휋) instead of the spaces on ℝ. 푀휑) can be found in [1, pp. 128–129] and [9, pp. 163– First we notice that it is sufficient to prove (2) only for 166]. 푝 ∈ (1, 2]. Indeed, assuming that this is established, then Applications of the interpolation theorems. (1) In his for 푓 ∈ 퐿푝 and 푔 ∈ 퐿푝′ we have monograph [23], Zygmund gave a detailed study of the 휋 one-dimensional Fourier transform ∫ [퐻(푓)(푠)] 푔(−푠) d푠 ≤ 퐴푝‖푓‖푝 ‖푔‖푝′ 1 −휋 ∫ 퐹(푓)(휉) = 푓(푥) exp{−푖 휉푥} d푥, 휉 ∈ ℝ. ′ √2휋 ℝ by the Hölder inequality (as above 푝 = 푝/(푝 − 1), and so

694 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 푝′ ≥ 2 when 푝 ≤ 2). Since for which the norm ∞ 1/푝 푝 d푡 ‖푎‖ = {∫ [푡−휃 퐾(푡, 푎)] } 휋 휋 휃,푝 0 푡 ∫ [퐻(푓)(푠)] 푔(−푠) d푠 = ∫ 푓(−푠) [퐻(푔)(푠)] d푠 , −휋 −휋 is finite. Here 퐾(푡, 푎) is defined on 퐴0 + 퐴1 for 푡 ∈ (0, ∞) by

−1 inf {‖푎0‖퐴0 +푡‖푎1‖퐴1 ∶ 푎0 ∈ 퐴0, 푎1 ∈ 퐴1 and 푎0 +푎1 = 푎}. the inequality ‖퐻(푔)‖푝′ ≤ 퐴푝 ‖푔‖푝′ is a consequence of 푎0,푎1 the assertion converse to the Hölder inequality. 퐾 푝 = ∞ 2 This -functional was introduced by Peetre. If , then It was mentioned above that 퐻 is bounded in 퐿 . Hence, −휃 the expression sup푡>0{푡 퐾(푡, 푎)} gives the norm ‖푎‖휃,∞ in order to apply Theorem 1 for 푝 ∈ (1, 2], it is sufficient when finite. to show that this operator is of weak type (1, 1), and this is Every 퐴휃,푝 is an intermediate space with respect to the an essential part of Calder´on’s proof; see [23, Vol. I, ch. IV, pair (퐴0, 퐴1), i.e., sect. 3]. Moreover, an improvement of the latter proof al- lowed S. K. Pichorides [16] to obtain the least value of the 퐴0 ∩ 퐴1 ⊂ 퐴휃,푝 ⊂ 퐴0 + 퐴1. constant 퐴 in (2). It occurs that 퐴 = tan 휋/(2푝) and 푝 푝 Moreover, if 퐴0 ⊂ 퐴1, then cot 휋/(2푝) is this value for 푝 ∈ (1, 2] and 푝 ≥ 2, respec- 퐴 ⊂ 퐴 ⊂ 퐴 ⊂ 퐴 , tively. 0 휃0,푝0 휃1,푝1 1

There are many other applications of interpolation the- provided either 휃0 > 휃1 or 휃0 = 휃1 and 푝0 ≤ 푝1. For any orems in analysis; see, e.g., [1, ch. 1], [23, ch. XII], and 푝, it is convenient to put 퐴0,푝 = 퐴0 and 퐴1,푝 = 퐴1. Now references cited in these books. we are in a position to explain what the interpolation of an Further development of interpolation theorems. Results operator is in terms of the family {퐴휃,푝} and another family constituting the interpolation space theory were obtained of spaces {퐵휃,푝} constructed by using some Banach spaces in the early 1960s and are classical now. This theory 퐵0 and 퐵1 in the same way as 퐴0 and 퐴1. was created in the works of Nachman Aronszajn, Alberto Let 푇 ∶ 퐴0 + 퐴1 → 퐵0 + 퐵1 be a linear operator such Calderón,Mischa Cotlar, Emilio Gagliardo, Selim Grig- that its norm as the operator mapping 퐴0 (퐴1) to 퐵0 (퐵1) orievich Krein, Jacques-Louis Lions, and Jaak Peetre, to is equal to 푀0 (푀1). Then the operator 푇 ∶ 퐴휃,푝 → list a few. We leave aside several versions of complex in- 퐵휃,푝 is also bounded, and its norm is less than or equal 1−휃 휃 terpolation spaces developed from the Riesz–Thorin the- to 푀0 푀1 . Along with the method based on the 퐾- orem (see, e.g., [1, ch. 4]) and concentrate on “espaces functional, there is an equivalent method (also developed de moyennes” introduced by Lions and Peetre in their by Peetre) involving the so-called 퐽-functional. Further de- celebrated article [8]. These “real method interpolation tails concerning this approach to interpolation theory can spaces,” usually denoted (퐴0, 퐴1)휃,푝, are often considered be found in [1, chs. 3 and 4]. as “descendants” of the Marcinkiewicz interpolation theorem. The Marcinkiewicz Function Prior to describing these spaces, it is worth mention- In the Annales de la SociétéPolonaise de Mathématique, vol- ing another germ of interpolation theory originating from ume 17 (1938), Marcinkiewicz published two short pa- Lwów. Problem 87 in the Scottish Book [14] posed by Ba- pers. Two remarkable integral operators were considered nach demonstrates his interest in nonlinear interpolation. in the first of these notes (see [10] and [13, pp. 444–451]); Presumably, it was formulated during Marcinkiewicz’s stay they and their numerous generalizations became indis- in Lwów. Indeed, he solved problems 83 and 106 in pensable tools in analysis. One of these operators is al- [14], which were posed before and after, respectively, Ba- ways called the “Marcinkiewicz integral”; see [23, ch. IV, nach’s problem on interpolation. A positive solution of sect. 2] for its definition and properties. In particular, it the latter problem (due to L. Maligranda) is presented in is used for investigation of the structure of a measurable [14, pp. 163–170]. set near an “almost arbitrary” point; see [18, sects. 2.3 and Let us turn to defining the family of spaces {(퐴0, 퐴1)휃,푝} 2.4], whereas further references to papers describing some involved in the real interpolation method; here 휃 ∈ (0, 1) of its generalizations can be found in the monographs [18] and 푝 ∈ [1, ∞]. In what follows, we write 퐴휃,푝 instead and [23]. The second operator is usually referred to as the of (퐴0, 퐴1)휃,푝 for the sake of brevity. Let 퐴0 and 퐴1 be “Marcinkiewicz function” (see, e.g., [9, pp. 192–194]), but two Banach spaces, both continuously embedded in some it also appears as the “Marcinkiewicz integral.” Presum- (larger) Hausdorff topological vector space. Then for a pair ably, the mess with names began as early as 1944, when (휃, 푝) the space 퐴휃,푝 with 푝 < ∞ consists of all 푎 ∈ 퐴0 + 퐴1 Zygmund published the extensive article [22], section 2 of

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 695 which was titled “On an Integral of Marcinkiewicz.” In Furthermore, Marcinkiewicz conjectured that for 푝 > 1 fact, this 14-page section is devoted to a detailed study of the inequalities the Marcinkiewicz function 휇, whose properties were just outlined by Marcinkiewicz himself in [10]. It is not clear 퐴푝‖푓‖푝 ≤ ‖휇‖푝 ≤ 퐵푝‖푓‖푝 (3) whether Zygmund had already received information about hold, where again 푓 must have the zero mean value in the Marcinkiewicz’s death when he decided to present in de- second inequality. Moreover, he foresaw that it would not tail the results from [10] (the discovery of mass graves in be easy to prove these inequalities; indeed, the proof given the Katy ´nforest was announced by the Nazi government by Zygmund in his article [22] is more than 11 pages long. in April 1943). The first step towards generalization of the Marcinkie- Zygmund begins his presentation with a definition of wicz function was made by Daniel Waterman; his paper the Littlewood–Paley function 푔(휃; 푓), which is a nonlin- [21] was published seven (!) years after presentation of the ear operator applied to an integrable, 2휋-periodic 푓. The work to the AMS. However, its abstract appeared in the purpose of introducing 푔(휃; 푓) was to provide a characteri- 푝 Proceedings of the International Congress of Mathematicians zation of the 퐿 -norm ‖푓‖푝 in terms of the Poisson integral held in 1954 in Amsterdam. Waterman considered the 휇- of 푓. After describing some properties of 푔(휃), Zygmund function notes: ∞ [퐹(휏 + 푡) + 퐹(휏 − 푡) − 2퐹(휏)]2 1/2 It is natural to look for functions analogous to 푔(휃) 휇(휏; 푓) = { ∫ d푡} , 푡3 but defined without entering the interior of the 0 unit circle. where 휏 ∈ (−∞, ∞) and 퐹 is a primitive of 푓 ∈ 퐿푝(−∞, ∞), After a reference to [10], Zygmund continues: 푝 > 1. His proof of inequalities (3) for 휇(휏; 푓) heavily re- Marcinkiewicz had the right idea of introducing lies on the M. Riesz theorem about conjugate functions on 1 the function ℝ (see [21, p. 130] for the formulation), and its proof involves the Marcinkiewicz interpolation theorem described 휇(휃) = 휇(휃; 푓) above. 휋 [퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃)]2 1/2 Another consequence of inequalities (3) for 휇(휏; 푓) is a 1,푝 = { ∫ 3 d푡} characterization of the Sobolev space 푊 (ℝ), 푝 ∈ (1, ∞). 0 푡 휋 Indeed, putting 퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃) 2 1/2 = { ∫ 푡[ ] d푡} ∞ 푡2 [푓(휏 + 푡) + 푓(휏 − 푡) − 2푓(휏)]2 1/2 0 M(휏; 푓) = { ∫ d푡} 푡3 where 퐹(휃) is the integral of 푓, 0 1,푝 ′ 휃 for 푓 ∈ 푊 (ℝ), we have that M(휏; 푓) = 휇(휏; 푓 ). Then (3) 퐹(휃) = 퐶 + ∫ 푓(푢) d푢 . can be written as 0 퐴 ‖푓′‖ ≤ ‖M(⋅; 푓)‖ ≤ 퐵 ‖푓′‖ , More generally, he considers the functions 푝 푝 푝 푝 푝

휋 which implies the following assertion. Let 푝 ∈ (1, ∞). |퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃)|푟 1/푟 휇 (휃) = { ∫ d푡} Then 푓 ∈ 푊 1,푝(ℝ) if and only if 푓 ∈ 퐿푝(ℝ) and M(⋅; 푓) ∈ 푟 푡푟+1 0 퐿푝(ℝ). 휋 푟 |퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃)| 1/푟 Stein extended these results to higher dimensions in the = { ∫ 푡푟−1 | | d푡} , | 2 | late 1950s and early 1960s (it is worth mentioning that 0 푡 휇 is referred to as the Marcinkiewicz integral in his paper so that 휇2(휃) = 휇(휃). He proves the following facts [17]). For this purpose he applied the real-variable tech- which are clearly analogues of the corresponding nique used in the generalization of the Hilbert transform properties of 푔(휃). ∞ 푓(푥 + 푡) − 푓(푥 − 푡) These facts are the estimates P.V. ∫ d푡 0 푡 ‖휇푞‖푞 ≤ 퐴푞‖푓‖푞 and ‖푓‖푝 ≤ 퐴푝‖휇푝‖푝 to higher dimensions. Indeed, this can be written as valid for 푞 ≥ 2 and 1 < 푝 ≤ 2, respectively, where 푓 ∞ has the zero mean value in the second inequality and the 퐹(푥 + 푡) + 퐹(푥 − 푡) − 2퐹(푥) assertion: For every 푝 ∈ (1, 2] there exists a continuous, 2휋- ∫ 2 d푡, 0 푡 periodic function 푓 such that 휇푝(휃; 푓) = ∞ for almost every 휃. which resembles the expression for 휇(휏; 푓), and so Stein, in

696 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 his own words, was Stein begins his investigation of properties of this func- guided by the techniques used by A. P. Calderón tion by proving that ‖휇(⋅; 푓)‖2 ≤ 퐴‖푓‖2, where 퐴 is and A. Zygmund [2] in their study of the 푛-di- independent of 푓, and his proof involving Plancherel’s mensional generalizations of the Hilbert trans- theorem is not elementary at all. Even less elementary form; connected with this are some earlier ideas is his proof that 휇(⋅; 푓) is of weak type (1, 1). Then of Marcinkiewicz. the Marcinkiewicz interpolation theorem (see Theorem 1 above) implies that ‖휇(⋅; 푓)‖ ≤ 퐴‖푓‖ for 푝 ∈ (1, 2] pro- The definition of singular integral given in[2], to which 푝 푝 vided 푓 ∈ 퐿푝(ℝ푛). For all 푝 ∈ (1, ∞) this inequality is Stein refers, involves a function Ω(푥) defined for 푥 ∈ ℝ푛 proved in [17] with assumptions (i)–(iii) changed to the and assumed (i) to be homogeneous of degree zero, i.e., following ones: Ω(푥′) is absolutely integrable on the unit to depend only on 푥′ = 푥/|푥|; (ii) to satisfy the Hölder sphere and is odd there, i.e., Ω(−푥′) = −Ω(푥′). A few years condition with exponent 훼 ∈ (0, 1]; and (iii) to have the later, A. Benedek, A. P. Calderón,and R. Panzone demon- zero mean value over the unit sphere in ℝ푛. Then strated that for a 퐶1-function Ω, condition (iii) implies the Ω(푦′) 푆(푓)(푥) = lim ∫ 푓(푥 − 푦) d푦 last inequality for all 푝 ∈ (1, ∞). 휖→0 푛 |푦|>휖 |푦| In another note, Stein obtained the following generalization of the one-dimensional result. exists almost everywhere provided 푓 ∈ 퐿푝(ℝ푛), 푝 ∈ [1, ∞). Let 푝 ∈ (2푛/(푛 + 2), ∞) and 푛 ≥ 2. Then 푓 belongs to the Furthermore, this singular integral operator is bounded in 1,푝 푛 푝 푛 푝 푛 Sobolev space 푊 (ℝ ) if and only if 푓 ∈ 퐿 (ℝ ) and 퐿 (ℝ ) for 푝 > 1; i.e., the inequality ‖푆(푓)‖푝 ≤ 퐴푝‖푓‖푝 2 1/2 holds with 퐴푝 independent of 푓. [푓(⋅ + 푦) + 푓(⋅ − 푦) − 2푓(⋅)] 푝 푛 { ∫ 푛+2 d푦} ∈ 퐿 (ℝ ). Moreover, in the section dealing with background facts, ℝ푛 |푦| Stein notes that 휇 is a nonlinear operator and writes (see For 푛 > 2 this does not cover 푝 ∈ (1, 2푛/(푛 + 2)] and so is [17, p. 433]): weaker than the assertion formulated above for 푛 = 1. An “interpolation” theorem of Marcinkiewicz is In the survey article [9, pp. 193–194], one finds a list of very useful in this connection. papers concerning the Marcinkiewicz function. In particu- In quoting the result of Marcinkiewicz, [.. .] we lar, further properties of 휇 were considered by A. Torchin- shall not aim at generality. For the sake of sim- sky and S. Wang [19] in 1990, whereas T. Walsh [20] pro- plicity we shall limit ourselves to the special case posed a modification of the definition (4), (5) in 1972. that is needed. After that the required form of the interpolation theorem Multipliers of Fourier Series and Integrals (see Theorem 1 above) is formulated and used later in the During his stay in Lwów, Marcinkiewicz collaborated with paper, thus adding one of the first items in the now long Stefan Kaczmarz and Juliusz Schauder,2 who had awak- list of its applications. Since the term interpolation was ened his interest in multipliers of orthogonal series. Studies novel, quotation marks are used by Stein in the quoted in this area of analysis were initiated by Hugo Steinhaus in piece. Indeed, Zygmund’s proof of the Marcinkiewicz the- the 1920s; in its general form, the problem of multipliers orem had appeared in 1956, just two years earlier than is as follows. Let 퐵1 be a Banach space with a Schauder ∞ Stein’s article. basis {푔푛}푛=1. The (linear) operator 푇 is called a multiplier ∞ Stein begins his generalization of the Marcinkiewicz when there is a sequence {푚푛}푛=1 of scalars of this space function 휇(휏; 푓) with the case when 푓 ∈ 퐿푝(ℝ푛), 푝 ∈ [1, 2]. and 푇 acts as follows: Realizing the analogy described above, he puts ∞ ∞ 퐵 ∋ 푓 = ∑ 푐 푔 → 푇푓 ∼ ∑ 푚 푐 푔 . Ω(푦′) 1 푛 푛 푛 푛 푛 퐹 (푥) = ∫ 푓(푥 − 푦) d푦 , 푥 ∈ ℝ푛, (4) 푛=1 푛=1 푡 |푦|푛−1 |푦|≤푡 Here ∼ means that the second sum assigned as 푇푓 can be- where Ω satisfies conditions (i)–(iii), and notes that if 푛 = long to the same space 퐵1 or be an element of another Ba- 1 and Ω(푦) = sign 푦, then nach space 퐵2; this depends on properties of the sequence. 푥 Multipliers of Fourier series are of paramount interest, and

퐹푡(푥) = 퐹(푥+푡)+퐹(푥−푡)−2퐹(푥) with 퐹(푥) =∫ 푓(푠) d푠. this was the topic of the remarkable paper [11] published 0 by Marcinkiewicz in 1939. Therefore, it is natural to define the 푛-dimensional Marcin- kiewicz function as follows: 2Both perished in World War II. Being in the reserve, Kaczmarz was drafted and killed during the first week of war; the circumstances of his death are un- ∞ 2 1/2 [퐹푡(푥)] clear. Schauder was in hiding in occupied Lw´ow, and the Gestapo killed him in 휇(푥; 푓) = { ∫ 3 d푡} . (5) 0 푡 1943 while he was trying to escape arrest.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 697 Not long before Marcinkiewicz’s visit to Lw´ow started, The trigonometric form was used by Marcinkiewicz for Kaczmarz investigated some properties of multipliers in double Fourier series as well, and his sufficient condi- 푝 the function spaces (mainly 퐿 (0, 1) and 퐶[0, 1]) under tions on bounded real multipliers {휆푚푛} look rather awk- ∞ rather general assumptions about the system {푔푛}푛=1. Fur- ward. Now, the restrictions on {휆푚푛} ⊂ ℂ are usually ex- ther results about multiplier operators were obtained in pressed in a rather condensed form by using the so-called the joint paper [7] of Kaczmarz and Marcinkiewicz. It dyadic intervals; see, e.g., [18, sect. 5.1]. Applying these was submitted to the Studia Mathematica in June 1937; conditions to multipliers acting on the expansion i.e., their collaboration lasted for another year after ∞

Marcinkiewicz left Lw´ow. This paper has the same title as ∑ 푐푚푛 exp 푖{푚푥 + 푛푦} 푝 that of Kaczmarz and concerns the case when 퐿 (0, 1) with 푚,푛=−∞ 푝 ≠ ∞ is mapped to 퐿푞(0, 1), 푞 ∈ [1, ∞]; it occurs that the of 푓 ∈ 퐿푝((0, 2휋)2), 푝 ∈ (1, ∞), one obtains an updated case 푞 = ∞ is the simplest one. In this paper, it is assumed formulation of the multiplier theorem; see, e.g., [9, p. 201]. that every function 푔 is bounded, whereas the sequence 푛 A simple corollary derived by Marcinkiewicz from this {푔 }∞ is closed in 퐿1(0, 1). In each of four theorems that 푛 푛=1 theorem is as follows (see [11, p. 86]). The fractions differ by the ranges of 푝 and 푞 involved, certain conditions are imposed on {푚 }∞ , and these conditions are neces- 푚2 푛2 |푚푛| 푛 푛=1 , , (7) sary and sufficient for the sequence to define a multiplier 푚2 + 푛2 푚2 + 푛2 푚2 + 푛2 푝 푞 operator 푇 ∶ 퐿 → 퐿 . provide examples of multipliers in 퐿푝 for double Fourier After returning to Wilno, Marcinkiewicz kept on his series. The reason to include these examples was to an- studies of multipliers initiated in Lw´ow, and in May 1938, swer a question posed by Schauder, and this is specially he submitted (again to the Studia Mathematica) the semi- mentioned in a footnote. Moreover, after remarking that nal paper [11], in which the main results are presented in a his Theorem 2 admits an extension to multiple Fourier se- curious way. Namely, Theorems 1 and 2, concerning mul- ries, Marcinkiewicz added a straightforward generalization tipliers of Fourier series and double Fourier series, are for- of formulae (7) to higher dimensions, again referring to mulated in the reverse order. Presumably, the reason for Schauder’s question. This is evidence that the question was this is the importance of multiple Fourier series for appli- an important stimulus for Marcinkiewicz in his work. cations and generalizations. Let us formulate Theorem 1 A natural way to generalize Marcinkiewicz’s theorems is in a slightly updated form. to consider multipliers of Fourier integrals. Study of these 푝 Let 푓 ∈ 퐿 (0, 2휋), 푝 ∈ (1, ∞), be a real-valued function operators was initiated by Mikhlin in 1956; see note [15], and let its Fourier series be in which the first result of that kind was announced. Sev- ∞ eral years later, Mikhlin’s theorem was improved by Lars 푎0/2 + ∑ 퐴푛(푥), 푤ℎ푒푟푒 퐴푛(푥) = 푎푛 cos 푛푥 + 푏푛 sin 푛푥. Hörmander [5], and since then it has been widely used for 푛=1 various purposes. To formulate this theorem we need the ∞ If a bounded sequence {휆푛}푛=1 ⊂ ℝ is such that 푛-dimensional Fourier transform

2푘+1 퐹(푓)(휉) = (2휋)−푛/2 ∫ 푓(푥) exp{−푖 휉 ⋅ 푥} d푥, 휉 ∈ ℝ푛, 푛 ∑ |휆푛 − 휆푛+1| ≤ 푀 for all 푘 = 0, 1, 2, … , (6) ℝ 푘 푛=2 defined for 푓 ∈ 퐿2(ℝ푛) ∩ 퐿푝(ℝ푛), 푝 ∈ (1, ∞). It is clear where 푀 is a constant independent of 푘, then the mapping 푓 ↦ that any bounded measurable function Λ on ℝ푛 defines ∞ 푝 the mapping ∑푛=1 휆푛 퐴푛 is a bounded operator in 퐿 (0, 2휋). It is well known that for 푝 = 2 this theorem is true with 푇 (푓)(푥) = 퐹−1[Λ(휉)퐹(푓)(휉)](푥) , 푥 ∈ ℝ푛, condition (6) omitted, but this is not mentioned in [11]. Λ ∞ 2 푛 푝 푛 The assumptions that 푓 is real-valued and {휆푛}푛=1 ⊂ ℝ such that 푇Λ(푓) ∈ 퐿 (ℝ ). If 푇Λ(푓) is also in 퐿 (ℝ ) and 푇Λ were not stated in [11] explicitly but used in the proof. is a bounded operator, i.e., This was noted by Solomon Grigorievich Mikhlin [15], ‖푇 (푓)‖ ≤ 퐵 ‖푓‖ for all 푓 ∈ 퐿푝(ℝ푛) (8) who extended this theorem to complex-valued multipliers Λ 푝 푝,푛 푝 and functions. Also, he used the exponential form of the with 퐵푝 independent of 푓, then Λ is called a multiplier Fourier expansion: for 퐿푝. 2 ∞ The description of all multipliers for 퐿 is known as well for 퐿1 and 퐿∞ (it is the same for these two spaces); see [18, 푓(푥) = ∑푐푛 exp 푖푛푥 . 푛=−∞ pp. 94–95]. However, the question about characterization of the whole class of multipliers for other values of 푝 is far

698 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 from resolved. The following assertion gives widely used [4] Loukas Grafakos and Lenka Slav´ıková, The Marcinkiewicz sufficient conditions. multiplier theorem revisited, Arch. Math. (Basel) 112 (2019), no. 2, 191–203, DOI 10.1007/s00013-018-1269-7. Theorem (Mikhlin, Hörmander). Let Λ be a function of the 푘 푛 MR3908837 퐶 -class in the complement of the origin of ℝ . Here 푘 is the [5] Lars Hörmander, Estimates for translation invariant oper- least integer greater than 푛/2. If there exists 퐵 > 0 such that ators in 퐿푝 spaces, Acta Math. 104 (1960), 93–140, DOI ℓ 10.1007/BF02547187. MR121655 ℓ | 휕 Λ(휉) | |휉| | | ≤ 퐵 , 1 ≤ 푗1 < 푗2 < ⋯ < 푗ℓ ≤ 푛 , [6] Richard A. Hunt, An extension of the Marcinkiewicz interpo- |휕휉 휕휉 ⋯ 휕휉 | 푗1 푗2 푗ℓ lation theorem to Lorentz spaces, Bull. Amer. Math. Soc. 70 for all 휉 ∈ ℝ푛, ℓ = 0, … , 푘, and all possible ℓ-tuples, then (1964), 803–807, DOI 10.1090/S0002-9904-1964-11242- inequality (8) holds; i.e., Λ is a multiplier for 퐿푝. 8. MR169037 [7] S. Kaczmarz and J. Marcinkiewicz, Sur les multiplicateurs In various versions of this theorem, different assump- des sériesorthogonales, Studia Math. 7 (1938), 73–81. Also tions are imposed on the differentiability of Λ. In par- [13, pp. 389–396]. ticular, Hörmander [5, pp. 120–121] replaced the point- [8] J.-L. Lions and J. Peetre, Sur une classe d’espaces wise inequality for weighted derivatives of Λ by a weaker d’interpolation (French), Inst. Hautes Etudes´ Sci. Publ. one involving certain integrals (see also [18, p. 96]). Re- Math. 19 (1964), 5–68. MR165343 cently, Loukas Grafakos and Lenka Slavíková[4] obtained [9] Lech Maligranda, Józef Marcinkiewicz (1910–1940)—on new sufficient conditions for Λ in the multiplier theorem, the centenary of his birth, Marcinkiewicz centenary vol- thus improving Hörmander’s result. Their conditions are ume, Banach Center Publ., vol. 95, Polish Acad. Sci. Inst. optimal in a certain sense explicitly described in [4]. Math., Warsaw, 2011, pp. 133–234, DOI 10.4064/bc95-0- 10. MR2918095 Corollary. Every function that is smooth everywhere except at [10] J. Marcinkiewicz, Sur quelques intégrales du type de the origin and is homogeneous of degree zero is a Fourier multi- Dini, Ann. Soc. Polon. Math. 17 (1938), 42–50. Also plier for 퐿푝. [13, pp. 444–451]. [11] J. Marcinkiewicz, Sur les multiplicateurs des séries de Its immediate consequence is the Schauder estimate Fourier, Studia Math. 8 (1939), no. 2, 78–91. Also ‖ 휕2푢 ‖ [13, pp. 501–512]. ‖ ‖ ≤ 퐶푝,푛‖Δ 푢‖푝 , 1 ≤ 푗1, 푗2 ≤ 푛 , [12] J. Marcinkiewicz, Sur l’interpolation d’opérations, C. R. ‖휕푥푗1 휕푥푗2 ‖ 푝 Acad. Sci. Paris 208 (1939), 1272–1273. Also [13, pp. 539– valid for 푢 belonging to the Schwartz space of rapidly de- 540]. caying infinitely differentiable functions. For this purpose [13] Józef Marcinkiewicz, Collected papers, edited by An- one has to use the equality toni Zygmund. With the collaboration of Stanislaw Lo- 2 jasiewicz, Julian Musielak, Kazimierz Urbanik, and An- 휕 푢 휉푗 휉푗 퐹 ( ) (휉) = 1 2 퐹(Δ 푢)(휉) , 1 ≤ 푗 , 푗 ≤ 푛 , toni Wiweger. Instytut Matematyczny Polskiej Akademii 2 1 2 휕푥푗1 휕푥푗2 |휉| Nauk, Pa ństwowe Wydawnictwo Naukowe, Warsaw, 1964. and the fact that the function 휉 휉 /|휉|2 is homogeneous MR0168434 푗1 푗2 [14] R. Daniel Mauldin (ed.), The Scottish Book: Mathematics of degree zero. from the Scottish Café, Birkhäuser, Boston, Mass., 1981. In- cluding selected papers presented at the Scottish Book Con- ACKNOWLEDGMENTS. The author thanks Irina ference held at North Texas State University, Denton, Tex., Egorova for the photo of Marcinkiewicz’s plaque and May 1979. MR666400 Alex Eremenko, whose comments helped to improve [15] S. G. Mihlin, On the multipliers of Fourier integrals the original manuscript. (Russian), Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701– 703. MR0080799 [16] S. K. Pichorides, On the best values of the constants in the References theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. [1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. 44 (1972), 165–179. (errata insert), DOI 10.4064/sm-44- An introduction, Grundlehren der Mathematischen Wis- 2-165-179. MR312140 senschaften, No. 223, Springer-Verlag, Berlin-New York, [17] E. M. Stein, On the functions of Littlewood-Paley, Lusin, and 1976. MR0482275 Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430– [2] A. P. Calderon and A. Zygmund, On the existence of cer- 466, DOI 10.2307/1993226. MR112932 tain singular integrals, Acta Math. 88 (1952), 85–139, DOI [18] Elias M. Stein, Singular integrals and differentiability 10.1007/BF02392130. MR52553 properties of functions, Princeton Mathematical Series, No. [3] Roman Duda, Pearls from a lost city: The Lvov school of math- 30, Princeton University Press, Princeton, N.J., 1970. ematics, History of Mathematics, vol. 40, American Mathe- MR0290095 matical Society, Providence, RI, 2014. Translated from the 2007 Polish original by Daniel Davies. MR3222779

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 699 [19] Alberto Torchinsky and Shi Lin Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), no. 1, 235–243, DOI 10.4064/cm-60-61-1-235-243. MR1096373 [20] T. Walsh, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203–217. (errata insert), DOI 10.4064/sm-44- 3-203-217. MR315368 Why navigate your [21] Daniel Waterman, On an integral of Marcinkiewicz, Trans. Amer. Math. Soc. 91 (1959), 129–138, DOI graduate school 10.2307/1993149. MR103953 [22] A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. experience alone? 55 (1944), 170–204, DOI 10.2307/1990189. MR9966 [23] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cam- bridge University Press, New York, 1959. MR0107776 [24] A. Zygmund, J´ozefMarcinkiewicz (Polish), Wiadom. Mat. (2) 4 (1960), 11–41. Also [13, pp. 1–33]. MR115885 “CROSSWORD! (or: Diversion as a vehicle “Solvitur for conversation on Ambulando” power and usage)”

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