Jozef Marcinkiewicz Zhuyu Ye

Figure 1: Jozef Marcinkiewicz Jozef Marcinkiewicz’s life

According to the limited document, Jozef Marcinkiewicz was born on March 30th, 1910, in Cimoszka, a small village near Poland. Marcinkiewicz’s parents were Klemens and Aleksandra. In Cimoszka, the house which Marcinkiewicz family lived and the surrounding buildings have been well preserved. Klemens Marcinkiewicz bought the house mainly from the money he had earned in America based on the published Jozef’s biography. Jozef Marcinkiewicz had a sister Stanilawa, the eldest of the children, two older brothers, Mieczyslaw and Edward, and a younger brother Kazimierz.

Similar to lots of famous mathamaticians, Marcinkiewicz grew up with some health problems, in par- ticular he had lung trouble. However, surprisedly, this did not stop him taking an active part in sports. Swimming and skiing were two sports at which he became particularly proficient. Because of his poor health, Marcinkiewicz first took private lessons at home and then he finished elementary school in Janw.

Later on, he entered the fourth grade of the King Sigismund Augustus Gymnasium in Bialystok. In Bi- alystok ,Marcinkiewicz lived in lodgings, not far from his school-friend Zygmunt Krassowski. Zygmunt played an important impact on Jozef’s life and also people are able to know more about Marcinkiewicz because of his friend as well. The father of the latter, Zenon Krassowski, was a mathematics teacher at the Gymnasium but Marcinkiewicz was taught by another teacher, Konstanty Kosinski. However, Zenon Krassowski had great influence on the development of Marcinkiewicz’s mathematical interests. As a pupil, Marcinkiewicz did not care very much for school mathematics; in his School Leaving Ex- amination (Matura) he got a C+. He was awarded his Matura certificate on June 22nd, 1930.

Figure 2: The first page of gradebook

” In 1930 ,Marcinkiewicz entered the Division of Mathematical and Natural Sciences of the University of Stefan Batory in Wilno. The town was then in Poland but it had been known by its Russian name of Vilnius when it was the capital of Lithuania. The university there had been named after Stefan Batory who was king of Poland from 1575 to 1586. The university had three professors of mathematics, the most famous of them being Antoni Zygmund who was appointed in the year that Marcinkiewicz began his undergraduate course. Zygmund was undertaking research on trigonometric series and in 1931- 32 he gave a course on this topic at Wilno for the first time. It was, based on his own description, an ambitious course, with an course on Lebesgue integration preceding it. Zygmund writes that the course was : ” ... too difficult for the average student of the second year ...” ( JZEF MARCINKIEWICZ (1910?1940) ? ON THE CENTENARY OF HIS BIRTH by LECH MALIGRANDA)

Marcinkiewicz graduated in 1933, after only three years of study and on 20 June 1933 he obtained a Master of Science degree (in mathematics) at USB. The title of his master thesis was Convergence of the Fourier.

2 Jozef Marcinkiewicz’s mathematical works

Even tough Jozef Marcinkiewicz only spanned on math research for six years, he made a huge contri- bution on his field. Math’s result proved by Marcinkiewicz are in the following areas of mathematics: - Functional Analysis (interpolation of operators, Marcinkiewicz spaces and vector- valued inequalities) - Probability Theory (independent random variables, Khintchine type inequalities, characteristic func- tions, Brownian motion) -Theory of Real Functions -Trigonometric Series, Power Series, Orthogonal and Fourier Series -Approximation Theory -Theory of Functions of Complex Variables. In the period of six years ,Jzef Marcinkiewicz wrote 55 papers (while spending one year in the army). 19 were published with co-authors (14 with A. Zygmund, two with S. Bergman and one with S. Kaczmarz, R. Salem, B. Jessen12 and A. Zygmund). Despite the brevity of his period of mathematical activity, it has nonetheless left a define mark on mathematics. Marcinkiewicz’s papers besides the original and important results, contain a lot of ideas. They are still used today and continue to inspire mathematicians. Meanwhile, lots of math theories are related to his name. Here is a piece of letter Antoni wrote to his pupil Jozef Marcinkiewicz:

”I was one of his professors at the University in Wilno; I introduced him to mathematical research and interested him in problems with which I was then concerned. Later on we collaborated and wrote several joint papers; but his scientific development was so rapid and the originality of his ideas so great that in certain parts of my own field of work I may only consider myself as his pupil. ”

Marcinkiewicz was regarded as one of the most outstanding younger generation in Polish mathematics and together with Stanisaw Zaremba, Leon Lichtenstein, Juliusz Schauder and Antoni Zygmund, the most prominent in classical analysis. Zygmund considered him his best pupil, although he had many students in Poland and the USA. it is also important for us to realise that the reason Marcinkiewicz could become famous partly because of Zygmund.

A description of Marcinkiewicz’s achievements was written in Polish by Antoni Zyg- mund in 1960 and

3 then translated into English and published in the Collected Papers . Achievements of Marcinkiewicz in analysis were described in Japanese by Satoru Igari . Moreover, that Philip Holgate delivered on 25 February 1989 a lecture on Independent functions: probability and analysis in Poland between the wars, which was published in 1997 , and in the third part of this work some important achievements of Marcinkiewicz and Zygmund were discussed.

Figure 3: Jzef Marcinkiewicz and Antoni Zygmund

Collaboration with other scholars

First, I want to mention about the background story or can be called the start of collaboration either.In his second year of studies he was in school, in the academic year 1931/32, at Zygmund’s course on orthogonal series preceded by an introduction to the theory of Lebesgue integration. This course was too difficult for the average second year student and Marcinkiewicz asked Zygmund for permission to take this course. That was the beginning of their fruitful mathematical collaboration. While in Lww Marcinkiewicz also collaborated with Stefan Kaczmarz (1895-1939) and Wladyslaw Orlicz (1903-1990) he became interested in problems of general orthogonal systems and wrote a series of papers on this subject. He published joint paper with Kaczmarz on multipliers of Fourier series and was working in Lww on general orthogonal series. On 11 October 1938 Marcinkiewicz presented a talk in Pozna. The development of the probability theory for the last 25 years. This lecture was probably connected with his application for a professor position in Pozna. After this visit he went to Paris, where he stayed six months (October 1938 March 1939). In this period Marcinkiewicz collaborated with Stefan Bergman and Raphal Salem.6 With

4 Bergman he wrote two joint papers in the theory of complex functions of two variables and with Salem one paper on Riemann sums. Morever, before his murder happened, the Soviets discovered how brilliant their captive was. They offered him some form of collaboration. Marcinkiewicz politely asked in a letter for his mathematical books and a copy of his PhD certificate to be sent to him at the camp. It is supposed that, in the end, Marcinkiewicz declined the Soviet offer.

Historical events that marked Marcinkiewicz’s life.

In the periods September 1933 -August 1934 and September 1934 or August 1935 he was assistant to the Zygmund chair of mathematics at USB. In the interim he did one year military service in 5th Infantry Regiment of Legions in Wilno. He finished the military course with an excellent score. On 17 September 1934 he was transferred to the reserves. Marcinkiewicz took his soldiering duties seriously, but not without a sense of humour as far as the disadvantages of military service were concerned. He received the following evaluation:

Figure 4:

During the years 1934-1938 Marcinkiewicz was taken on six-weeks military exercises (25 June-16 September 1934, 12 August 21 September 1935, 1 July 10 August 1936, 1938 before travel to France). This was related to World War II. The spent the academic year 1935/1936 at the Jan Kazimierz Uni- versity in Lww. This was a one year Fellowship from the Fund for National Culture and the assistant position at the chair of Stefan Banach in the period 1 December 1935-31 August 1936 with 12 hours of teaching weekly . Marcinkiewicz visited the Scottish Caf. He solved problems 83 of Auerbach, 106 of Banach and 131 of Zygmund from the Scottish Book. Moreover, he posed his own problem number . World War II did have a huge damage on most Polish family included Marcinkiwwicz’s. According to MALIGRANDA, Marcinkiewicz’s parents Klemens and Aleksandra were transported in June 1941 to Uzbekistan by the NKVD and six months later they died of hunger in Bukhara on 24 December 1941. Jzef was executed in Kharkov in Spring 1940. Edward, who was later transported to Siberia, joined the Polish army of General Anders and took part in the battle of Monte Cassino (Italy). He then lived in Argentina, Italy and Switzerland. The youngest brother, Kazimierz, one of the defenders of Lww,

5 returned to his family?s house. In 1946 , he was killed by security officers in Janw. Mieczyslaw was forced by communist authorities to sell the farm and move to a different place.

Figure 5:

Significant historical events around the world during Marcinkiewicz’s life

The most significant historical even during Jozef’s life is the World War II. Besides that was the main reason why he joined the army. Marcinkiewicz was not only a great mathematician, but he was also a magnificent Polish patriot returning to Poland. He could have stayed in England or go to United States, but in his opinion, this meant desertion. Instead, he chose to fight and thus became a martyr. The invasion of Poland, also known as the Battle of Bode or Debord, is the starting point of the World War II Europe Theater, also known as the ”Blitzkrieg” in the history of world war. Poland called the ”Battle of 1939” or ”Battle of September 1939”, while Germany called the ”Battle of Poland”, combat code-named ”white program.” The Battle of Poland was the invasion of Poland by Germany, Slovakia and the Soviet Army in September 1939, which was generally regarded as the beginning of the Second World War. A week after the signing of the German-Soviet Non-Aggression Pact, Germany launched an attack on September 1, 1939, while the Soviet Union invaded Poland on 17 September, and on 6 October the two countries occupied the territory of Poland , The Polish campaign ended. Then it raised the question that why Germany was captured in only 28 days. And here are causes I personally think: 1. German made up reasons to start the war.A radio station on the German-Polish border was broad- casting anti-German remarks in Polish, and the Germans launched the war and blamed that Poland would invade. 2. Suitable combat time.The war occurred around three in the morning. The Polish

6 army was asleep. There is no way the army was able to fight back.3. Advanced combat methods.The Germans were inspired by de Gaulle’s book ”The Future of the Army,” developing armored units, developing blitzkriegs in tanks and planes, destroying Polish heavy armor on the ground, and then making it difficult for Poland to counter German armored forces. (I am not a military fan so that I am not really familiar with other details about the war) 4. British and French appeasement policyPoland has to other countries for help, but led by the British and French European giants ignored.5. Russia was really sneaky about the war strategy.The German invasion, Stalin to protect Polish territory of the Soviet Union on the grounds of ethnic minorities invaded Poland, Poland is difficult to resist the invasion of two powerful countries. There are ”Soviet non-aggression pacts”, Sood divided Poland after the exchange of fire, but in the Katyn forest massacre on the Polish people. World War II, the German army only spent four weeks to occupy Poland.. Most of the description is that Poland was so weak that it couldnt even fight back. The Polish Air Force plane was too late to take off on the Germans were all destroyed. Polish cavalry recklessly cut German tanks with swords. However, is this the historical truth? Around the Second World War there are many ”myths”, one of the most famous is the September 1939 Germany launched the ”Blitzkrieg” (Blitzkrieg). Through the implementation of the ”revolutionary” theory of war, Germany occupy Poland in four weeks. Due to old equipment, Poland air force was attacked by Germany even before flying the plane. The Polish cavalry recklessly to the German large-scale mechanized forces launched an impact in order not to feel that embarrassed. The Polish policy reflects that the country is geographically and geographically precarious. Poland has no danger on the eastern and western borders (Poland is on a large plain extending from northern Ger- many to the Russian grassland), and the two neighbors of the East and the West are historical enemies, States posed a clear and real threat to Poland’s national integrity. Unfortunately, the illusion prevailed in the European political propositions, and Poland’s foreign policy was doomed to never escape this illusion. Moreover, in the event of war, the Soviet army must pass through Polish territory in order to provide assistance to Western allies. Poland will oppose the Soviet Union to do so, they believe that the Soviet army once the occupation of Polish territory can not be evacuated. The idea of Poland is correct, as evidenced by the events of 1945 (the demarcation of the Polish border by the Soviet Union, the United States, and the United Kingdom after the end of World War II, and the partial transfer of Polish territory to the Soviet Union). Poland’s lack of modern equipment, it is beyond doubt. Poland has done a number of remedial efforts

7 to make up for deficiencies, but it is too late. In addition, the Polish army in the mechanization of almost no achievements, most of their artillery traction with horses. Polish armor is very limited. Although there are four cavalry divisions in Poland have mechanized on the agenda, but by the outbreak of war, Poland is only a tank brigade available. Despite the large num- ber of light tanks and the performance of the 37mm ”Bofors” (Bofors) anti-tank gun, but the limited financial resources, the Polish Army did not take any further modernization measures. In 1939, Poland can send the battlefield of nearly 2.5 million people, including the reserve personnel and national army volunteers, but such a large army organization form is no different from 1914. Poland’s Plan Z was based on the assurances of Western allies, especially France, that Poland would be assisted before it was defeated. The Polish border is almost defenseless, so this fragile plan is becoming more unsure. Poland is located in the plains, to facilitate the attack on the tank forces, the German occupation of Slovakia, you can attack from three directions to Poland, the Polish border defense situ- ation deteriorating. Although some rivers throughout Poland can provide some natural barriers to the Germans, relying on these rivers for defense means abandoning some important economic assets. The lack of strategic planning reflects the failure of the Polish Army to gain experience from the evolution of modern warfare, especially in the use of armored forces in rapid offensive maneuvers.

Figure 6:

Significant mathematical progress during the Marcinkiewicz’s lifetime

Based on provided information on Jozef Marcinkiewicz, in the period of six years (1933?1939) Jzef

8 Marcinkiewicz ”wrote 55 papers (while spending one year in the army). 19 were published with co- authors (14 with A. Zygmund, two with S. Bergman and one with S. Kaczmarz, R. Salem, B. Jessen12 and A. Zygmund). Despite the brevity of his period of mathematical activity, it has nonetheless left a define mark on mathematics.” Functional Analysis can be regarded as the most famous math contribution Jozef’ made. 1.Marcinkiewicz interpolation theorem: (a) The Hardy operator H is defined by 1??x f(t)dt, x?I, where I = (0,a), 0 ¡ a ? ?. The operator H is not bounded from L1(I) to L1(I) (for Hf(x)= x example, f (x) = 1 ? ? L1(0,1), but Hf (x) = ?1/(xlnx) on (0,1/2) so that ?0 0 xln2x (0,1/2) 0 Hf0 ?/ L1(0, 1)), but it is bounded from L1(I) to weak-L1(I) and is bounded from L?(I) to L?(I). ?(b) The maximal operator M is defined by 1?? Mf(x) = sup —I— —f(t)—dt, I = (a,b) ? [0,1].

2. Approximation Theory Based on my understanding, The approximation theory is mainly about finding the best approximation with a simpler function, and the resulting errors can be quantified. The ”best” and ”simpler” practical meanings mentioned above will vary with the application . The goal of the approximation theory is to approximate the actual function as close as possible to the accuracy of the computer floating point operations. Generally, the polynomials of higher order are used, and / or the interval of the polynomial approximation function is reduced. The reduction interval may be achieved using a number of different coefficients and gains for the function to be approximated. Now the mathematical library will be divided into a number of intervals between the interval, each interval with a number of polynomials is not high.

Theory.jpg

Figure 7:

Connections between history and the development of mathematics

Based on Zygmund, Jozef’s first mathematical paper appeared in 1933; the last one he sent for pub-

9 lication in the Summer of 1939. This short period of mathematical activity left, however, a definite imprint on Mathematics, and but for his premature death he would probably have been one of the most outstanding contemporary mathematicians. Considering what he did during his short life and what he might have done in normal circumstances one may view his early death as a great blow to Polish Mathematics, and probably its heaviest individual loss during the second world war.

Remarks

Marcinkiewicz was probably the first who used the word ?interpolation of operators” . Riesz and Thorin spoke on ” convexity theorems” . Marcinkiewicz’s proof is based on an idea of decomposition of a function which generates later on the concept of the K-functional playing a central role in modern interpolation theory.

References

1. http://www.math.uic.edu/\someone/notes.pdf

2. Bak, Newman: Complex Analysis. Springer 1989

3. http://www-history.mcs.st-and.ac.uk/Biographies/Marcinkiewicz.html

4.http://www.math.technion.ac.il/hat/fpapers/Dabrhens.pdf

5.J. Marcinkiewicz, Collected Papers, A. Zygmund (ed.), PWN, Warszawa, 1964.

6.http://www.norbertwiener.umd.edu/seminars/abstracts11-12/abstract06.pdf

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