The Legacy of Jozef Marcinkiewicz: Four Hallmarks of Genius

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 achievements 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´ demonstrated 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 published 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 concerns 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 contribution 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 ∈ [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 µ 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 µ 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 proof Extraordinary Professor at the University of Poznan´ in posed by Marcinkiewicz.

2 Marcinkiewicz interpolation theorem and f ; here, mes{...} 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 = | f (x)| dx sical Riesz–Thorin and Marcinkiewicz theorems. Each of Rn these serves as the basis for two essentially different ap- proaches to interpolation of operators known as the com- is the norm in Lp(Rn). Now, we are in a position to formu- plex and real methods. The term ‘interpolation of opera- late the following. tors’ was, presumably, coined by Marcinkiewicz in 1939, Theorem 1. Let 1 6 r1 < r2 < ∞, and let T be a sub-additive because Riesz and Thorin, who published their results in operator acting simultaneously in Lri (Rn), i = 1,2. If it is of 1926 and 1938, respectively, referred to their assertions as weak type (ri,ri) for i = 1,2, then for every p ∈ (r1,r2) the p n ‘convexity theorems’. inequality kT( f )kp 6 Bk f kp holds for all f ∈ L (R ) with

It is worth emphasising again that a characteristic fea- B depending only on Ar1 , Ar2 , r1, r2 and p. ture of Marcinkiewicz’s work was applying real methods to When B is independent of f in the last inequality, the problems that other authors treated with the help of com- operator T is of strong type (p, p); it is clear that T is also plex analysis. It was mentioned above that in his paper [10] of weak type (p, p) in this case. published in 1938, Marcinkiewicz introduced the function In the letter to Zygmund mentioned above, Marcinkie- µ without using complex variables, but so that it is analo- wicz included a proof of this theorem for the case r1 = 1 gous to the Littlewood–Paley function g, whose definition and r2 = 2. Presumably, it was rather simple; indeed, even involves these variables. In the same year 1938, Thorin when r2 < ∞ is arbitrary, the proof is less than two pages published his extension of the Riesz convexity theorem, long in [18], ch. 1, sect. 4. which exemplifies the approach based on complex vari- Marcinkiewicz spaces. Another crucial step, made by ables. Possibly, this stimulated Marcinkiewicz to seek an Marcinkiewicz in [12], was introduction of the weak Lp analogous result with proof relying on real analysis. Any- spaces playing the essential role in his general interpolation way, Marcinkiewicz found his interpolation theorem and theorem. Now, they are called the Marcinkiewicz spaces announced in [12]; concurrently, a letter was sent to Zyg- and usually denoted Lp,∞. mund which contained the proof concerning a particular To give an idea of these spaces, let us consider a measure case. Ten years after World War II, Zygmund reconstructed space (U,Σ,m) over real scalars with a non-negative mea- the general proof and published it in 1956; for this reason sure m (just to be specific). For a real-valued f , which is the theorem is sometimes referred to as the Marcinkiewicz– finite almost everywhere and m-measurable, we introduce Zygmund interpolation theorem. its distribution function An excellent introduction to the interpolation theory one finds in the book [1] based on the works of Jaak Peetre m({x : | f (x)| > λ}), λ ∈ (0,∞), (he passed away on 1 April 2019 aged 83), whose contri- and put bution to this theory cannot be overestimated. In collab- 1/p | f |p,∞ = sup λ[m({x : | f (x)| > λ})] for p ∈ [1,∞). oration with Jacques-Louis Lions, he introduced the ‘real λ>0 method interpolation spaces’ (see their fundamental arti- p,∞ p p,∞ Then L = { f : | f |p,∞ < ∞}, and it is clear that L ⊂ L cle [8]), which can be considered as ‘descendants’ of the for p ∈ [1,∞), because | f |p,∞ 6 k f kp in view of Cheby- Marcinkiewicz interpolation theorem. shev’s inequality; the Marcinkiewicz space for p = ∞ is L∞ An important fact of Peetre’s biography is that his life by definition. was severely changed during World War II (another re- It occurs that | f |p,∞ is not a norm for p ∈ [1,∞), but a minder about that terrible time). With his parents, Jaak es- quasi-norm because caped from Estonia in September 1944 just two days before his home town Parnu¨ was destroyed in an air raid of the Red | f + g|p,∞ 6 2(| f |p,∞ + |g|p,∞) Army. He was only ten years old when his family settled in (see, e.g., [1], p. 7). However, it is possible to endow Lp,∞, Lund (Sweden), where he spent most of his life. But let us p ∈ (1,∞), with a norm k · kp,∞ converting it into a Banach turn to mathematics again. space; moreover, the inequality The Marcinkiewicz interpolation theorem for opera- −1 | f |p,∞ 6 k f kp,∞ 6 p(p − 1) | f |p,∞ tors in Lp(Rn). We begin with this simple result because p,∞ p,∞ it has numerous applications, being valid for sub-additive holds for all f ∈ L . It is worth mentioning that L be- p,q operators mapping the Lebesgue spaces Lp(Rn) with p > 1 longs (as a limiting case) to the class of Lorentz spaces L , into themselves (see, e.g., [18], ch. 1, sect. 4). We recall q ∈ [1,∞] (see, e.g., [1], sect. 1.6, and references cited in this that an operator T : Lp → Lp is sub-additive if book). Another generalisation of Lp,∞, known as the Marcinkie- |T( f1 + f2)(x)| = |T( f1)(x)| + |T( f2)(x)| for every f1, f2 . wicz space Mϕ , is defined with the help of a non-negative, concave function ϕ ∈ C[0,∞). This Banach space consists Furthermore, T is of weak type (r,r) if the inequality of all (equivalence classes of) measurable functions for which the following norm αrmes{x : |T( f )(x)| > α} 6 A k f kr r r Z t 1 ∗ r k f kϕ = sup f (s)ds holds for all α > 0 and all f ∈ L with Ar independent of α t>0 ϕ(t) 0

3 is finite. Here f ∗ denotes the non-increasing rearrangement one-dimensional Fourier transform: of f , i.e., 1 Z F( f )(ξ) = √ f (x) exp{−iξx}dx, ξ ∈ R. f ∗(s) = inf {λ : m({x : | f (x)| > λ}) 6 s} for s > 0, 2π R λ>0 See vol. II, ch. XVI, sections 2 and 3, where, in particu- and so is non-negative and right-continuous; moreover, its lar, it is demonstrated that F, originally defined on a dense distribution function m({x : | f ∗(x)| > λ}) coincides with set in Lp, p ∈ [1,2], is extensible to the whole space as a 0 that of f . If ϕ(t) = t1−1/p, then the corresponding Marcin- bounded operator F : Lp → Lp , p0 = p/(p − 1), and so the 0 kiewicz space is Lp,∞, whereas ϕ(t) ≡ 1 and ϕ(t) = t give integral converges in Lp . To prove this assertion and its L1 and L∞, respectively. n-dimensional analogue one can use Theorem 2; indeed, The Marcinkiewicz interpolation theorem for boun- F : L1 → L∞ is bounded (this is straightforward to see), ded linear operators. This kind of continuous operators is and by Plancherel’s theorem F is bounded on L2, and so usually considered as mapping one normed space to another this theorem is applicable. On the other hand, the Riesz– one, in which case the operator’s norm is an important char- Thorin theorem, which has no restriction p 6 q, yields a acteristic. However, the latter can be readily generalised for more complete result valid for the inverse transform F−1 as p p,∞ p0 p a mapping of L to L ; indeed, if |T f |p,∞ 6 Ck f kp, then well. The latter operator acting from L to L is bounded; it is natural to introduce the norm (or quasi-norm) of T as here p0 ∈ [2,∞), and so p = p0/(p0 − 1) ∈ (1,2]. the infimum over all possible values of C. Now we are in a (2) In studies of conjugate Fourier series, the following position to formulate. singular integral operator (the periodic Hilbert transform) Theorem 2. Let p , q , p , q ∈ [1,∞] satisfy the inequal- 1 Z t 0 0 1 1 H( f )(s) = lim f (s −t)cot dt ities p0 6 q0, p1 6 q1 and q0 , q1, and let p, q ∈ [1,∞] be 2π ε→0 ε6|t|6π 2 such that p 6 q and the equalities plays an important role. Indeed, by linearity it is sufficient 1 1 − θ 1 1 1 − θ 1 to define H on a basis in L2(−π,π), and the relations = + and = + p p p q q q 0 1 0 1 H(cosnt) = sinns for n > 0, H(sinnt) = −cosns for n > 1 hold for some θ ∈ (0,1). If T is a linear operator, which show that it expresses passing from a trigonometric series p q ,∞ maps L 0 into L 0 and its norm is N0 and simultaneously to its conjugate. Moreover, these formulae show that H is p q ,∞ p q 2 T : L 1 → L 1 has N1 as its norm, then T maps L into L bounded on L (−π,π) and its norm is equal to one. and its norm N satisfies the estimate In the mid-1920s, Marcel Riesz obtained his celebrated 1−θ θ result about this operator; first, he announced it in a brief N 6 CN N (1) 0 1 note in the Comptes rendus de l’Academie´ des Sciences with C depending on p0, q0, p1, q1 and θ. Paris, and three years later published his rather long proof The convexity inequality (1) is a characteristic feature that H is bounded on Lp(−π,π) for p ∈ (1,∞), i.e. for of the interpolation theory. The general form of this the- every finite p > 1 there exists Ap > 0 such that orem (it is valid for quasi-additive operators, whose special kH( f )k 6 A k f k for all f ∈ L (−π,π). (2) case are sub-additive ones described prior to Theorem 1) is p p p p proved in [23], ch. XII, sect. 4. In particular, it is shown that However, (2) does not hold for p = 1 and ∞; see [23], vol. I, one can take ch. VII, sect. 2, for the corresponding examples and a proof of this inequality. q q 1/q p(1−θ)/p0 pθ/p1 C = 2 + 0 1 ; There are several different proofs of this theorem; the 1/p |q − q0| |q − q1| p original proof of M. Riesz was reproduced in the first edition of the Zygmund’s monograph [23] which appeared in see [23], vol. II, p. 114, formula (4.18), where, unfortu- 1935. In the second edition published in 1959, this proof nately, the notation differs from that adopted here. Special was replaced by that of Calderon´ obtained in 1950. Let us cases of Theorem 2 and diagrams illustrating them can be outline another proof based on the Marcinkiewicz interpo- found in [9], pp. 155–156. It should be emphasised that lation theorem analogous to Theorem 1, but involving Lp- p q the restriction 6 is essential; indeed, as early as 1964, spaces on (−π,π) instead of the spaces on R. R. A. Hunt [6] constructed an example demonstrating that First we notice that it is sufficient to prove (2) only for Theorem 2 is not true without it; for a description of this p ∈ (1,2]. Indeed, assuming that this is established, then example see, e.g., [1], pp. 16–17. for f ∈ Lp and g ∈ Lp0 we have It was Marcinkiewicz himself who proposed an exten- Z π sion of his interpolation theorem to other function spaces. [H( f )(s)]g(−s)ds 6 Apk f kp kgkp0 . Namely, the so-called diagonal case (when p = q and −π 0 0 0 p = q ) of his theorem is formulated for Orlicz spaces in by the Holder¨ inequality (as above p = p/(p − 1), and so 1 1 0 [12]. References to papers containing further results on in- p > 2 when p 6 2). Since Z π Z π terpolation in these and other spaces (e.g., Lorentz and Mϕ ) [H( f )(s)]g(−s)ds = f (−s)[H(g)(s)]ds, can be found in [1], pp. 128–129, and [9], pp. 163–166. −π −π −1 Applications of the interpolation theorems. (1) In his the inequality kH(g)kp0 6 Ap kgkp0 is a consequence of the monograph [23], Zygmund gave a detailed study of the assertion converse to the Holder¨ inequality.

4 2 It was mentioned above that H is bounded in L . Hence, operator is in terms of the family {Aθ,p} and another family in order to apply Theorem 1 for p ∈ (1,2], it is sufficient of spaces {Bθ,p} constructed by using some Banach spaces to show that this operator is of weak type (1,1), and this B0 and B1 in the same way as A0 and A1. is an essential part of Calderon’s´ proof; see [23], vol. I, Let T : A0 + A1 → B0 + B1 be a linear operator such ch. IV, sect. 3. Moreover, an improvement of the latter proof that its norm as the operator mapping A0 (A1) to B0 (B1) allowed S. K. Pichorides [16] to obtain the least value of is equal to M0 (M1), then the operator T : Aθ,p → Bθ,p is the constant A in (2). It occurs that A = tanπ/(2p) and 1−θ θ p p also bounded and its norm is less than or equal to M0 M1 . cotπ/(2p) is this value for p ∈ (1,2] and p > 2, respectively. Along with the method based on the K-functional, there is There are many other applications of interpolation the- an equivalent method (also developed by Peetre) involving orems in analysis; see, e.g., [1], ch. 1, [23], ch. XII, and the so-called J-functional. Further details concerning this references cited in these books. approach to interpolation theory can be found in [1], chap- Further development of interpolation theorems. Re- ters 3 and 4. sults constituting the interpolation space theory were ob- The Marcinkiewicz function tained in the early 1960’s and are classical now. This theory was created in the works of Nachman Aronszajn, Alberto In the Annales de la Societ´ e´ Polonaise de Mathematique´ , Calderon,´ Mischa Cotlar, Emilio Gagliardo, Selim Grig- volume 17 (1938), Marcinkiewicz published two short pa- orievich Krein, Jacques-Louis Lions and Jaak Peetre to list pers. Two remarkable integral operators were considered in a few. We leave aside several versions of complex interpola- the first of these notes (see [10] and [13], pp. 444–451); they tion spaces developed from the Riesz–Thorin theorem (see, and their numerous generalisations became indispensable e.g., [1], ch. 4), and concentrate on ‘espaces de moyennes’ tools in analysis. One of these operators is always called introduced by Lions and Peetre in their celebrated article the ‘Marcinkiewicz integral’; see [23], ch. IV, sect. 2, for [8]. These ‘real method interpolation spaces’ usually de- its definition and properties. In particular, it is used for in- noted (A0,A1)θ,p are often considered as ‘descendants’ of vestigation of the structure of a measurable set near ‘almost the Marcinkiewicz interpolation theorem. arbitrary’ point; see [18], sections 2.3 and 2.4, whereas fur- Prior to describing these spaces, it is worth mention- ther references to papers describing some its generalisations ing another germ of interpolation theory originating from can be found in the monographs [18] and [23]. The second Lwow.´ The problem 87 in the Scottish Book [14] posed by operator is usually referred to as the ‘Marcinkiewicz func- Banach demonstrates his interest in nonlinear interpolation. tion’ (see, e.g., [9], pp. 192–194), but it also appears as the Presumably, it was formulated during Marcinkiewicz’s stay ‘Marcinkiewicz integral’. Presumably, the mess with names in Lwow;´ indeed, he solved problems 83 and 106 in [14], began as early as 1944, when Zygmund published the exten- which were posed before and after, respectively, the Ba- sive article [22], section 2 of which was entitled “On an in- nach’s problem on interpolation. A positive solution of the tegral of Marcinkiewicz”. In fact, this 14-pages long section latter problem (due to L. Maligranda) is presented in [14], is devoted to a detailed study of the Marcinkiewicz function pp. 163–170. µ, whose properties were just outlined by Marcinkiewicz himself in [10]. It is not clear whether Zygmund had al- Let us turn to defining the family of spaces {(A0,A1)θ,p} involved in the real interpolation method; here θ ∈ (0,1) ready received information about Marcinkiewicz’s death, when he decided to present in detail the results from [10] and p ∈ [1,∞]. In what follows, we write Aθ,p instead (the discovery of mass graves in the Katyn´ forest was an- of (A0,A1)θ,p for the sake of brevity. Let A0 and A1 be two Banach spaces, both continuously embedded in some nounced by the Nazi government in April 1943). (larger) Hausdorff topological vector space, then for a pair Zygmund begins his presentation with a definition of the Littlewood–Paley function g( ; f ), which is a nonlinear op- (θ, p) the space Aθ,p with p < ∞ consists of all a ∈ A0 +A1 θ for which the following norm erator applied to an integrable, 2π-periodic f . The purpose of introducing g( ; f ) was to provide a characterisation of Z ∞ p 1/p θ h −θ i dt p kakθ,p = t K(t,a) the L -norm k f kp in terms of the Poisson integral of f . 0 t After describing some properties of g(θ), Zygmund notes. is finite. Here K(t,a) is defined on A0 +A1 for t ∈ (0,∞) by It is natural to look for functions analogous to inf {ka0kA0 +tka1kA1 : a0 ∈ A0, a1 ∈ A1 and a0 +a1 = a}. a0,a1 g(θ) but defined without entering the interior of This K-functional was introduced by Peetre. If p = ∞, then the unit circle. −θ the expression supt>0{t K(t,a)} gives the norm kakθ,∞ when finite. After a reference to [10], Zygmund continues: Every A is an intermediate space with respect to the θ,p Marcinkiewicz had the right idea of introducing pair (A ,A ), i.e., 0 1 the function A0 ∩ A1 ⊂ Aθ,p ⊂ A0 + A1. µ(θ) = µ(θ; f ) Moreover, if A0 ⊂ A1, then nZ π [F(θ +t) + F(θ −t) − 2F(θ)]2 o1/2 A0 ⊂ Aθ ,p ⊂ Aθ ,p ⊂ A1 = 3 dt 0 0 1 1 0 t provided either θ0 > θ1 or θ0 = θ1 and p0 6 p1. For any nZ π hF(θ +t) + F(θ −t) − 2F(θ)i2 o1/2 p, it is convenient to put A = A and A = A . Now = t 2 dt 0,p 0 1,p 1 0 t we are in a position to explain what the interpolation of an where F(θ) is the integral of f , 5 Z θ F(θ) = C + f (u)du. to higher dimensions. Indeed, this can be written as 0 Z ∞ F(x +t) + F(x −t) − 2F(x) 2 dt More generally, he considers the functions 0 t Z π r /r n |F(θ +t) + F(θ −t) − 2F(θ)| o1 which resembles the expression for µ(τ; f ), and so Stein, µr(θ) = r+1 dt 0 t in his own words, was Z π r 1/r n r−1 F(θ +t) + F(θ −t) − 2F(θ) o = t dt , guided by the techniques used by A. P. Calderon´ t2 0 and A. Zygmund [2] in their study of the n- dimensional generalizations of the Hilbert trans- so that µ2(θ) = µ(θ). He proves the following facts which are clearly analogues of the corre- form; connected with this are some earlier ideas sponding properties of g(θ). of Marcinkiewicz. These facts are the estimates The definition of singular integral given in [2], to which Stein refers, involves a function Ω(x) defined for x ∈ Rn kµ k 6 A k f k and k f k 6 A kµ k q q q q p p p p and assumed: (i) to be homogeneous of degree zero, i.e. valid for q > 2 and 1 < p 6 2, respectively, where f has the to depend only on x0 = x/|x|; (ii) to satisfy the Holder¨ zero mean value in the second inequality, and the assertion: condition with exponent α ∈ (0,1]; (iii) to have the zero For every p ∈ (1,2] there exists a continuous, 2π-periodic mean value over the unit sphere in Rn. Then function f such that µp(θ; f ) = ∞ for almost every θ. Z Ω(y0) Furthermore, Marcinkiewicz conjectured that for p > 1 S( f )(x) = lim n f (x − y)dy ε→0 |y|> |y| the inequalities ε exists almost everywhere provided f ∈ Lp(Rn), p ∈ [1,∞). Apk f kp 6 kµkp 6 Bpk f kp (3) Furthermore, this singular integral operator is bounded in p n hold, where again f must have the zero mean value in the L (R ) for p > 1, i.e. the inequality kS( f )kp 6 Apk f kp second inequality. Moreover, he foresaw that it would not holds with Ap independent of f . be easy to prove these inequalities; indeed, the proof given Moreover, in the section dealing with background facts, by Zygmund in his article [22] is more than 11 pages long. Stein notes that µ is a nonlinear operator and writes (see The first step towards generalisation of the Marcinkie- [17], p. 433): wicz function was made by Daniel Waterman; his paper An “interpolation” theorem of Marcinkiewicz is [21] was published seven (!) years after presentation of the very useful in this connection. work to the AMS. However, its abstract appeared in the Proceedings of the International Congress of Mathema- In quoting the result of Marcinkiewicz, [. . . ] we ticians held in 1954 in Amsterdam. Waterman considered shall not aim at generality. For the sake of sim- the following µ-function plicity we shall limit ourselves to the special case that is needed. nZ ∞ [F(τ +t) + F(τ −t) − 2F(τ)]2 o1/2 µ(τ; f ) = 3 dt , 0 t After that the required form of the interpolation theorem (see Theorem 1 above) is formulated and used later in the where τ ∈ (−∞,∞) and F is a primitive of f ∈ Lp(−∞,∞), paper, thus adding one of the first items in the now long list p > 1. His proof of inequalities (3) for µ(τ; f ) heavily relies of its applications. Since the term interpolation was novel, on the M. Riesz theorem about conjugate functions on R1 quotation marks are used by Stein in the quoted piece; in- (see [21], p. 130, for the formulation), and its proof involves deed, Zygmund’s proof of the Marcinkiewicz theorem had the Marcinkiewicz interpolation theorem described above. appeared in 1956, just two years earlier than Stein’s article. Another consequence of inequalities (3) for µ(τ; f ) is a His generalization of the Marcinkiewicz function µ(τ; f ) characterization of the Sobolev space W 1,p(R), p ∈ (1,∞). Stein begins with the case when f ∈ Lp(Rn), p ∈ [1,2]. Re- Indeed, putting alising the analogy described above, he puts nZ ∞ [ f (τ +t) + f (τ −t) − 2 f (τ)]2 o1/2 0 M(τ; f ) = 3 dt Z Ω(y ) 0 t n Ft (x) = n−1 f (x − y)dy, x ∈ R , (4) |y|6t |y| for f ∈ W 1,p(R), we have that M(τ; f ) = µ(τ; f 0). Then (3) can be written as where Ω satisfies conditions (i)–(iii), and notes that if n = 1 A k f 0k kM(·; f )k B k f 0k , and Ω(y) = signy, then p p 6 p 6 p p Z x which implies the following assertion. Let p ∈ (1,∞), then Ft (x) = F(x+t)+F(x−t)−2F(x) with F(x) = f (s)ds. 0 f ∈ W 1,p(R) if and only if f ∈ Lp(R) and M(·; f ) ∈ Lp(R). Stein extended these results to higher dimensions in Therefore, it is natural to define the n-dimensional Marcin- the late 1950s and early 1960s (it is worth mentioning kiewicz function as follows: Z ∞ 2 / that µ is referred to as the Marcinkiewicz integral in his n [Ft (x)] o1 2 µ(x; f ) = 3 dt . (5) paper [17]). For this purpose he applied the real-variable 0 t technique used in the generalisation of the Hilbert transform His investigation of properties of this function Stein be- Z ∞ f (x +t) − f (x −t) gins by proving that kµ(·; f )k2 6 Ak f k2, where A is inde- P.V. dt , pendent of f , and his proof involving Plancherel’s theorem 0 t

6 is not elementary at all. Even less elementary is his proof closed in L1(0,1). In each of four theorems which differ by that µ(·; f ) is of weak type (1,1). Then the Marcinkiewicz the ranges of p and q involved, certain conditions are im- ∞ interpolation theorem (see Theorem 1 above) implies that posed on {mn}n=1 and these conditions are necessary and p n kµ(·; f )kp 6 Ak f kp for p ∈ (1,2] provided f ∈ L (R ). For sufficient for the sequence to define a multiplier operator all p ∈ (1,∞) this inequality is proved in [17] with assump- T : Lp → Lq. tions (i)–(iii) changed to the following ones: Ω(x0) is ab- After returning to Wilno, Marcinkiewicz kept on his stud- solutely integrable on the unit sphere and is odd there, i.e. ies of multipliers initiated in Lwow,´ and in May 1938, he Ω(−x0) = −Ω(x0). A few years later, A. Benedek, A. P. Cal- submitted (again to the Studia Mathematica) the seminal deron´ and R. Panzone demonstrated that for a C1-function paper [11], in which the main results are presented in a cu- Ω condition (iii) implies the last inequality for all p ∈ (1,∞). rious way. Namely, Theorems 1 and 2, concerning multi- In another note, Stein obtained the following generalisa- pliers of Fourier series and double Fourier series, are for- tion of the one-dimensional result. Let p ∈ (2n/(n + 2),∞) mulated in the reverse order. Presumably, the reason for and n > 2, then f belongs to the Sobolev space W 1,p(Rn) if this is the importance of multiple Fourier series for applica- and only if f ∈ Lp(Rn) and tions and generalisations. Let us formulate Theorem 1 in a Z 2 1/2 n [ f (· + y) + f (· − y) − 2 f (·)] o p n slightly updated form. n+ dy ∈ L (R ). p Rn |y| 2 Let f ∈ L (0,2π), p ∈ (1,∞), be a real-valued function For n > 2 this does not cover p ∈ (1,2n/(n + 2)], and so is and let its Fourier series be weaker than the assertion formulated above for n = 1. ∞ a0/2 + ∑ An(x), where An(x) = an cosnx + bn sinnx. In the survey article [9], pp. 193–194, one finds a list of n=1 papers concerning the Marcinkiewicz function. In particu- ∞ If a bounded sequence {λn} ⊂ R is such that lar, further properties of µ were considered by A. Torchin- n=1 2k+1 sky and S. Wang [19] in 1990, whereas T. Walsh [20] pro- |λn − λn+1| 6 M f or all k = 0,1,2,..., (6) posed a modification of the definition (4), (5) in 1972. ∑ n=2k Multipliers of Fourier series and integrals where M is a constant independent of k, then the mapping f 7→ ∞ λ A is a bounded operator in Lp(0,2π). During his stay in Lwow,´ Marcinkiewicz collaborated with ∑n=1 n n Stefan Kaczmarz and Juliusz Schauder,∗ due to whom his It is well-known that for p = 2 this theorem is true with interest in multipliers of orthogonal series had arisen. condition (6) omitted, but this is not mentioned in [11]. The ∞ Studies in this area of analysis were initiated by Hugo assumptions that f is real-valued and {λn}n=1 ⊂ R were Steinhaus in the 1920s; in its general form, the problem of not stated in [11] explicitly, but used in the proof. This multipliers is as follows. Let B1 be a Banach space with a was noted by Solomon Grigorievich Mikhlin [15], who ∞ extended this theorem to complex-valued multipliers and Schauder basis {gn}n=1, the (linear) operator T is called ∞ functions; also, he used the exponential from of the Fourier multiplier when there is a sequence {mn}n=1 of scalars of this space and T acts as follows: expansion: ∞ ∞ ∞ f (x) = cn expinx. B 3 f = cngn → T f ∼ mncngn . ∑ 1 ∑ ∑ n=−∞ n=1 n=1 Here ∼ means that the second sum assigned as T f can be- The trigonometric form was used by Marcinkiewicz for double Fourier series as well, and his sufficient conditions long to the same space B1 or be an element of another Ba- on bounded real multipliers {λmn} look rather awkward. nach space B2; this depends on properties of the sequence. Multipliers of Fourier series are of paramount interest and Now, the restrictions on {λmn} ⊂ C are usually expressed this was the topic of the remarkable paper [11] published by in a rather condensed form by using the so-called dyadic Marcinkiewicz in 1939. intervals; see, e.g., [18], sect. 5.1. Applying these condi- Not long before Marcinkiewicz’s visit to Lwow´ started, tions to multipliers acting on the expansion ∞ Kaczmarz investigated some properties of multipliers in the c expi{mx + ny} p ∑ mn function spaces (mainly L (0,1) and C[0,1]) under rather m,n=−∞ general assumptions about the system {g }∞ . Further re- n n=1 of f ∈ Lp((0,2π)2), p ∈ (1,∞), one obtains an updated for- sults about multiplier operators were obtained in the joint mulation of the multiplier theorem; see, e.g., [9], p. 201. paper [7] of Kaczmarz and Marcinkiewicz; it was submit- A simple corollary derived by Marcinkiewicz from this ted to the Studia Mathematica in June 1937, i.e., their col- theorem is as follows (see [11], p. 86). The fractions laboration lasted for another year after Marcinkiewicz left 2 2 Lwow.´ This paper has the same title as that of Kaczmarz m n |mn| 2 2 , 2 2 , 2 2 (7) and concerns the case when Lp(0,1) with p , ∞ is mapped m + n m + n m + n to Lq(0,1), q ∈ [1,∞]; it occurs that the case q = ∞ is provide examples of multipliers in Lp for double Fourier the simplest one. In this paper, it is assumed that every series. The reason to include these examples was to an- ∞ function gn is bounded, whereas the sequence {gn}n=1 is swer a question posed by Schauder and this is specially mentioned in a footnote. Moreover, after remarking that ∗ Both perished in World War II. Being in the reserve, Kaczmarz was his Theorem 2 admits an extension to multiple Fourier se- drafted and killed during the first week of war; the circumstances of his death are unclear. Schauder was on hiding in occupied Lwow´ and the ries, Marcinkiewicz added a straightforward generalisation Gestapo killed him in 1943 while he was trying to escape arrest. of formulae (7) to higher dimensions again referring to

7 Schauder’s question. This is an evidence that the question 2 ∂ u ξ j ξ j was an important stimulus for Marcinkiewicz in his work. F (ξ) = 1 2 F(∆u)(ξ), 1 6 j , j 6 n, ∂x ∂x |ξ|2 1 2 A natural way to generalise Marcinkiewicz’s theorems is j1 j2 to consider multipliers of Fourier integrals. Study of these 2 and the fact that the function ξ j1 ξ j2 /|ξ| is homogeneous operators was initiated by Mikhlin in 1956; see note [15] of degree zero. in which the first result of that kind was announced. Sev- Acknowledgements. The author thanks Irina Egorova for eral years later, Mikhlin’s theorem was improved by Lars the photo of Marcinkiewicz’s plaque and Alex Eremen- Hormander¨ [5], and since than it is widely used for various ko, whose comments helped to improve the original manu- purposes. To formulate this theorem we need the n-dimen- script. sional Fourier transform Z References F( f )(ξ) = (2π)−n/2 f (x) exp{−iξ · x}dx, ξ ∈ Rn, Rn [1]J.B ERGH,J.LOFSTR¨ OM¨ , Interpolation Spaces. An Intro- duction, Springer-Verlag, Berlin et al., 1976. defined for f ∈ L2(Rn) ∩ Lp(Rn), p ∈ (1,∞). It is clear [2] A.P.CALDERON´ ,A.ZYGMUND, On the existence of cer- Rn that any bounded measurable function Λ on defines the tain singular integrals, Acta Math. 88 (1952), 85–139. mapping [3]R.D UDA, Pearls from a Lost City: The Lvov School of Math- −1 n TΛ( f )(x) = F [Λ(ξ)F( f )(ξ)](x), x ∈ R , ematics, AMS, Providence RI, 2014. 2 n p n [4]L.G RAFAKOS,L.SLAVÍKOVA´ , The Marcinkiewicz multi- such that TΛ( f ) ∈ L (R ). If TΛ( f ) is also in L (R ) and plier theorem revisited, Arch. Math. 112 (2019), 191–203. TΛ is a bounded operator, i.e., [5]L.H ORMANDER¨ , Estimates for translation invariant opera- p n kTΛ( f )kp 6 Bp,nk f kp for all f ∈ L (R ) (8) tors in Lp spaces, Acta Math. 104 (1960), 93–140. with Bp independent of f , then Λ is called a multiplier [6]R.A.H UNT, An extension of the Marcinkiewicz interpola- for Lp. tion theorem to Lorentz spaces, Bull. Amer. Math. Soc. 70 (1964), 803–807. The description of all multipliers for L2 is known as well as for L1 and L∞ (it is the same for these two spaces); see [7]S.K ACZMARZ,J.MARCINKIEWICZ, Sur les multiplica- [18], pp. 94–95. However, the question about characterisa- teurs des series´ orthogonales, Studia Math., 7 (1938), 73–81. Also [13], 389–396. tion of the whole class of multipliers for other values of p is far from its final solution. The following assertion gives [8]J.-L.L IONS,J.PEETRE, Sur une classe d’espaces d’inter- ´ widely used sufficient conditions. polation Publications Math. IHES 19 (1964), 5–68. [9]L.M ALIGRANDA,Jozef´ Marcinkiewicz (1910–1940) – on Theorem (Mikhlin, Hormander).¨ Let Λ be a function of the Marcinkiewicz Centenary Volume. k n the centenary of his birth, C -class in the complement of the origin of R ; here k is the Banach Center Publ. 95 (2011), 133–234. least integer greater than n/2. If there exists B > 0 such that [10]J.M ARCINKIEWICZ, Sur quelques integrales´ du type de ∂ `Λ(ξ) Dini, Ann. Soc. Polon. Math. 17 (1938), 42–50. Also [13], |ξ|` B, 1 j < j < ··· < j n, 6 6 1 2 ` 6 444–451. ∂ξ j1 ∂ξ j2 ...∂ξ j` [11]J.M ARCINKIEWICZ, Sur les multiplicateurs des series´ de for all ξ ∈ Rn, ` = 0,...,k and all possible `-tuples, then Fourier, Studia Math. 8, no. 2 (1939), 78–91. Also [13], 501– inequality (8) holds, i.e., is a multiplier for Lp. Λ 512.

In various versions of this theorem, different assump- [12]J.M ARCINKIEWICZ, Sur linterpolation d’operations,´ C. R. tions are imposed on the differentiability of Λ. In partic- Acad. Sci. Paris 208 (1939), 1272–1273. Also [13], 539– ular, Hormander¨ [5], pp. 120–121, replaced the pointwise 540. inequality for weighted derivatives of Λ by a weaker one [13]J.M ARCINKIEWICZ, Collected Papers, PWN, Warsaw, involving certain integrals (see also [18], p. 96). Recently, 1964. Loukas Grafakos and Lenka Slav´ıkova´ [4] obtained new [14]R.D.M AULDIN (ed.), The Scottish Book. Mathematics from sufﬁcient conditions for Λ in the multiplier theorem, thus the Scottish Cafe´, Birkhauser,¨ Boston, 1981. improving Hormander’s¨ result. Their conditions are opti- mal in a certain sense explicitly described in [4]. [15]S.G.M IKHLIN, On the multipliers of Fourier integrals, Do- klady Akad. Nauk SSSR (N.S.) 109 (1956), 701–703 (in Rus- Corollary. Every function, which is smooth everywhere ex- sian). cept at the origin and is homogeneous of degree zero, is a [16]S.K.P ICHORIDES, On the best values of the constants in p Fourier multiplier for L . the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165–179. Its immediate consequence is the Schauder estimate 2 [17]E.M.S TEIN, On the functions of Littlewood–Paley, Lusin, ∂ u 6 Cp,nk∆ukp , 1 6 j1, j2 6 n, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430– ∂x j1 ∂x j2 p 466. valid for u belonging to the Schwartz space of rapidly de- [18]E.M.S TEIN, Singular Integrals and Differentiability Prop- caying inﬁnitely differentiable functions. For this purpose erties of Functions, Princeton Univ. Press, Princeton NJ, one has to use the equality 1970.

8 [19]A.T ORCHINSKY,S.L.WANG, A note on the Marcinkie- wicz integral, Colloq. Math. 60/61 (1990), 235–243. [20] T. WALSH, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203–217. [21]D.W ATERMAN, On an integral of Marcinkiewicz, Trans. Amer. Math. Soc. 91 (1959), 129–138. [22]A.Z YGMUND, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170–204. [23]A.Z YGMUND, Trigonometric Series, I, II, Cambridge Univ. Press, Cambridge, 1959. [24]A.Z YGMUND,Jozef´ Marcinkiewicz, in: [13], pp. 1–33. −−−−−−−−−−−−−−−−−−−−−−−− N. Kuznetsov is a Principal Research Scientist in the Labo- ratory for Mathematical Modelling of Wave Phenomena at the Institute for Problems in Mechanical Engineering, Rus- sian Academy of Sciences, St. Petersburg. This laboratory was founded by him in 1997, and he headed it until 2016. e-mail: [email protected]