To the Memory of Lars Hörmander (1931–2012)Moreover, Lars Gårding (1919–2014) and Åke Pleijel (1913–89) were appointed professors in mathematics in Lund during this period

To the Memory of LarsHörmander (1931–2012)Jan Boman and Ragnar Sigurdsson, Coordinating Editors

Lars Hörmander 1996.

The eminent mathematician LarsValter Hörmander passed away onNovember 25, 2012. He was theleading figure in the dramatic de-velopment of the theory of linearpartial differential equations duringthe second half of the twentiethcentury, and his L2 estimates for the∂ equation became a revolutionarytool in complex analysis of severalvariables. He was awarded the FieldsMedal in 1962, the Wolf Prize in1988, and the Leroy P. Steele Prizefor Mathematical Exposition in 2006.He published 121 research articlesand 9 books, which have had a

profound impact on generations of analysts.

Student in LundLars Hörmander was born on January 24, 1931, inMjällby in southern Sweden. His talents becameapparent very early. He skipped two years ofelementary school and moved to Lund in 1946to enter secondary school (gymnasium). He wasone of the selected students who were offeredthe chance to cover three years’ material in twoyears by staying only three hours per day in schooland devoting the rest of the day to individualstudies, a scheme he found ideal. His mathematicsteacher, Nils Erik Fremberg (1908–52), was alsodocent at Lund University and in charge of theundergraduate program there. When graduatingfrom gymnasium in the spring of 1948 Lars hadalready completed the first-semester universitycourses in mathematics.

Jan Boman is professor emeritus of mathematics, StockholmUniversity. His email address is [email protected].

Ragnar Sigurdsson is professor of mathematics, Universityof Iceland. His email address is [email protected].

Unless otherwise noted, all photographs are from Lars Hör-mander’s personal collection, courtesy of Sofia Broström(Hörmander).

DOI: http://dx.doi.org/10.1090/noti1274

Lars entered Lund University in the fall of 1948to study mathematics and physics. Marcel Riesz(1886–1969) became his mentor. Lars studied mostof the courses on his own, but followed the lectureson analysis given by Riesz during 1948–50. In theearly fall of 1949 he completed a bachelor’s degreeand a master’s degree in the spring of 1950,which could have allowed him to earn his livingas a secondary school teacher at the age of onlynineteen.

In October 1951 Lars completed the licentiatedegree with a thesis entitled “Applications ofHelly’s theorem to estimates of Tchebycheff type”,written under the supervision of Riesz. Shortlyafterwards Riesz retired from his professorshipand moved to the US to remain there during mostof the coming ten years. We have often been askedwho was the PhD thesis advisor of Lars Hörmander.The correct answer is no one and it needs a littlebit of explanation. As a young man Marcel Rieszworked mainly on complex and harmonic analysis,but later in life he became interested in partialdifferential equations and mathematical physics,so partial differential equations were certainlystudied in Lund in the early 1950s. Moreover, LarsGårding (1919–2014) and Åke Pleijel (1913–89)were appointed professors in mathematics in Lundduring this period. They had good internationalcontacts, and the young Lars concluded that partialdifferential equations would give him the bestopportunities in Lund.

Lars defended his doctorate thesis, “On thetheory of general partial differential operators,” onOctober 22, 1955, with Jacques-Louis Lions as op-ponent, and it was published in Acta Mathematicathe same year. His work was highly independent.Lars chose his problems himself and solved them.Gårding was often mentioned as the thesis advisorof Hörmander, but the fact is that he was no morethan a formal advisor, in the sense that he was animportant source of inspiration and that he servedas a chairman at Lars’s thesis defense. It is noexaggeration to say that the thesis opened a newera of the subject of partial differential equations.

890 Notices of the AMS Volume 62, Number 8

Some of the most important results were quiteoriginal and had not even been envisioned before.

Professor in Stockholm, Stanford, andPrincetonLars spent the year 1956 in the United States. InJanuary 1957 he became professor at StockholmUniversity. Since there was no activity in partialdifferential equations at Stockholm University atthis time, he had to start from the beginning andlecture on distribution theory, Fourier analysis, andfunctional analysis. In 1961–62 he lectured on thematerial that was to become his 1963 book, LinearPartial Differential Operators. He quickly gathereda large group of students around him. Among themwere Christer Kiselman and Vidar Thomée. One ofus (JB) had the privilege to be a member of thatgroup. Lars’s lectures were wonderful, and therewas excitement and enthusiasm around him. Hisvast knowledge, fast thinking, and overwhelmingcapacity for work inspired all of us, but alsosometimes scared his students. The clatteringfrom his typewriter, which was constantly heardthrough his door, is famous.

Lars spent the summer of 1960 in Stanford andthe following academic year at the Institute forAdvanced Study (IAS) in Princeton. The summer of1960 was both pleasant and productive for Lars,and both he and the members of the departmentin Stanford were interested in a continuation ona more permanent basis, but Lars was not readyto leave Sweden at that time. An arrangement wasmade so that he would be on leave from Stockholmfor April and May and combine the position inStockholm with a permanent professorship atStanford, where he would work for the springand summer quarters. In Stockholm 1962–63 hegave a lecture series on complex analysis which hedeveloped further in Stanford, 1964 and publishedas An Introduction to Complex Analysis in SeveralVariables in 1966. Revised editions were publishedin 1973 and 1990. It is remarkable that this bookhas kept its position as one of the main referenceson the subject for almost fifty years.

At Stanford in the summer of 1963 Lars receiveda letter which would change his life. RobertOppenheimer, the director of the IAS, made himan offer to become a professor and a permanentmember of the institute. It took Lars some timeto make up his mind. Attempts were made byLennart Carleson (b. 1928), Otto Frostman (1907–77), and Lars Gårding to arrange for a researchposition in Lund, but when this was declined bythe Swedish government, Lars decided to acceptthe offer. He spent the summer of 1964 at Stanfordand started his work at the institute in September.He soon found that ongoing conflicts had createda bad climate at the institute. He also felt strongpressure to produce a steady stream of high-quality research, and this he found paralyzing.

This appears especially paradoxical, since hisaccomplishments during his period in Princetonwere truly remarkable. Already in early 1966 hehad made up his mind to return to an ordinaryprofessorship in Sweden as soon as an opportunitycame up.

Back in Lund

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Student in Lundaround 1950.

Lars left Princeton in the spring of1968 and became a professor inLund, a position he held until hisretirement in 1996. He always keptgood contact with IAS and otheruniversities in the US. In 1977–78Lars was in charge of a special pro-gram on microlocal analysis at IAS.Among Lars’s prominent studentsin Lund were Johannes Sjöstrand,Anders Melin, Nils Dencker, andHans Lindblad.

During the years 1979–84 Larsworked on the four volumes ofThe Analysis of Partial DifferentialOperators published in 1983 and1985. This was the time of studyof the younger of us (RS) in Lund.Lars gave a wonderful lecture series on variousparts of the manuscript. The students corrected(rare) errors and in return got superb privatelessons on the parts they did not understand. Thefour volumes, written in the well-known compactHörmander style, contain enough material to filleight volumes rather than four. The amount ofwork needed to complete the project was clearlyformidable, and Lars later looked back at thisperiod as six years of slave labor.

In the fall of 1984 Lars succeeded Lennart Carle-son as director of the Institut Mittag-Leffler andbecame the managing editor for Acta Mathematica.He conducted a two-year program on nonlinearpartial differential equations. Back in Lund in thefall of 1986 Lars started a series of lectures onnonlinear problems. His notes from 1988 werewidely circulated and finally published in revisedform as the book Lectures on Nonlinear HyperbolicDifferential Equations in 1997. In 1991–92 Larsgave lectures which he later developed into thebook Notions of Convexity, published in 1994. Inthis book he shows again the depth and breadthof his knowledge of analysis.

Lars became emeritus on January 1, 1996. Fromthe beginning of the 1990s his research wasnot as focused on partial differential equationsas it had been before. He looked back on hiscareer and took up the study of various problemsthat he had dealt with and continued publishinginteresting papers. Lars was always interested inthe Nordic cooperation of mathematicians. Hepublished his first paper in the proceedings ofthe Scandinavian Congress of Mathematicians held

September 2015 Notices of the AMS 891

With John Forbes Nash, Jean Leray, and LarsGårding in Paris, 1958.

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With Bogdan Bojarski and Vidar Thomée inNorway, 1965.

in Lund in 1953, the second one in MathematicaScandinavica in 1954, and it is symbolic that hislast paper was published there in 2008. In 1997Mikael Passare (1959–2011) initiated the annualNordan conferences, which is a platform for Nordicresearchers in complex analysis and related topics.Lars participated the first few years. He gave hislast conference talk in Reykjavik in 2002. Thesubject was unusual for him: he talked about hisL2 method from a historical perspective. It waspublished in Journal of Geometric Analysis in 2003.

Lars Hörmander had a huge influence on ourdevelopment as mathematicians, first as a teacherand advisor and later as a colleague and friend.We always admired him for his great knowledge,his sharp mind, and his masterful way of commu-nicating mathematics in speech and writing. Untilhis death, he kept his great spirit and memory.His interests in mathematics, science, nature, andhistory were always the same. We kept regularcontact with him and it was always a pleasure totalk to him. We are very grateful for having hadthe opportunity to work with him.

In this series of memorial articles we have as-sembled nine contributions from Lars’s colleagues,friends, students, and his daughter, Sofia. NicolasLerner writes on Lars’s contributions to partial

differential equations, and Jean-Pierre Demailly onhis work in complex analysis.

Nicolas Lerner

On Lars Hörmander’s Work on PartialDifferential EquationsThe BeginningLars Hörmander wrote a PhD thesis under theguidance of L. Gårding, and the publication of thatthesis, “On the theory of general partial differentialoperators” [29], in Acta Mathematica in 1955 canbe considered as the starting point of a new era forpartial differential equations. Among other things,very general theorems of local existence were es-tablished without using an analyticity hypothesisof the coefficients. Hörmander’s arguments reliedon a priori inequalities combined with abstractfunctional analytic arguments. Let us cite L. Gård-ing in [26]: It was pointed out very emphaticallyby Hadamard that it is not natural to consider onlyanalytic solutions and source functions even for anoperator with analytic coefficients. This reduces theinterest of the Cauchy-Kowalevski theorem which…does not distinguish between classes of differen-tial operators which have, in fact, very differentproperties such as the Laplace operator and theWave operator.

L. Ehrenpreis in [23] and B. Malgrange in [58]had proven a general theorem on the existence of afundamental solution for any constant coefficientsPDE, and the work [30] by Hörmander providedanother proof along with some improvement onthe regularity properties, whereas [29] gave a char-acterization of hypoelliptic constant coefficientsPDE via properties of the algebraic variety

charP = ζ ∈ Cn, P(ζ) = 0.The operator P(D) is hypoelliptic if and onlyif

|ζ| → ∞ on charP =⇒ | Imζ| → ∞.

Here

hypoellipticity means

(1) singsuppu = singsuppPufor the C∞ singular support. The characterizationof hypoellipticity of the constant coefficient oper-ator P(D) by a simple algebraic property of thecharacteristic set is a tour de force, technicallyand conceptually: in the first place, nobody hadconjectured such a result or even remotely sug-gested a link between the two properties, and next,the proof provided by Hörmander relies on a verysubtle study of the characteristic set, requiring anextensive knowledge of real algebraic geometry.

In 1957, Hans Lewy made a stunning discovery[57]: the equation Lu = f with

(2) L = ∂∂x1

+ i ∂∂x2

+ i(x1 + ix2)∂∂x3

Nicolas Lerner is professor of mathematics at UniversitéParis VI, France. His email address is [email protected].


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With Lars Gårding and Tomas Claesson, 2008.

does not have local solutions for most right-handsides f . The surprise came in particular from thefact that the operator L is a nonvanishing vectorfield with a very simple expression and also, asthe Cauchy-Riemann operator on the boundaryof a pseudo-convex domain, it is not a cooked-up example. Hörmander started working on theLewy operator (2) with the goal to get a generalgeometric understanding of a class of operatorsdisplaying the same defect of local solvability. Thetwo papers [34], [33], published in 1960, achievedthat goal. Taking a complex-valued homogeneoussymbol p(x, ξ), the existence of a point (x, ξ) inthe cotangent bundle such that

(3) p(x, ξ) = 0, p, p (x, ξ) 6= 0

ruins local solvability at x (here ·, · stands forthe Poisson bracket). With this result, Hörmandernonetheless gave a generalization of the Lewyoperator, but above all provided a geometricexplanation for that nonsolvability phenomenon.

A.-P. Calderón’s 1958 paper [17] on the unique-ness in the Cauchy problem was somehow thestarting point for the renewal of singular integralsmethods in local analysis. Calderón proved in [17]that an operator with real principal symbol withsimple characteristics has the Cauchy uniquenessproperty; his method relied on a pseudodifferentialfactorization of the operator which can be handledthanks to the simple characteristic assumption.It appears somewhat paradoxical that Hörman-der, who later became one of the architects ofpseudodifferential analysis, found a generalizationof Calderón’s paper using only a local method,inventing a new notion to prove a Carleman es-timate. He introduced in [32], [31] the notion ofpseudoconvexity of a hypersurface with respectto an operator and was able to handle the caseof tangent characteristics of order two. A largearray of counterexamples, due to P. Cohen [18],A. Plis [73], and later to S. Alinhac [1] and S. Alinhacand M. S. Baouendi [2], showed the relevanceof the pseudoconvexity hypothesis for Cauchyuniqueness.

In 1962, at the age of thirty-one, Hörmanderwas awarded the Fields Medal. His impressivework on PDE, in particular his characterization of

Receiving the Fields Medal from King Gustav VIAdolf.

Opening ceremony of ICM in Stockholm, 1962.From left: Lars Gårding, Lars Hörmander, JohnMilnor, Hassler Whitney, Åke Pleijel, HaraldCramér, Otto Frostman. Standing at the rostrum:Rolf Nevanlinna. Gårding and Whitney presentedthe work of the Fields Medalists Hörmander andMilnor, resp. The organizing committeeconsisted of Frostman (chair), Cramér, Gårding,and Pleijel. Nevanlinna was president of IMU andchairman of the Fields Medal Committee.

hypoellipticity for constant coefficients and hisgeometrical explanation of the Lewy nonsolvabilityphenomenon, were certainly very strong argumentsfor awarding him the medal. Also his new point ofview on PDE, which combined functional analysiswith a priori inequalities, had led to very generalresults on large classes of equations which hadbeen out of reach in the early 1950s.

The Microlocal Revolution, Act I

Pseudodifferential Equations

The paper [17] of Calderón led to renewed interestin singular integrals, and the notion of pseudo-differential operators along with a symbolic cal-culus was introduced in the 1960s by severalauthors: J. J. Kohn and L. Nirenberg in [53], andA. Unterberger and J. Bokobza in [81]. Hörmanderwrote in 1965 a synthetic account of the nascentpseudodifferential methods with the article [36].A pseudodifferential operator A = a(x,D) with


symbol a is defined by the formula

(4) (Au)(x) =∫Rneix·ξa(x, ξ)u(ξ)dξ(2π)−n,

say for u ∈ C∞c (Rn). The symbol a is a smoothfunction on the phase space which should satisfysome estimates, e.g., ∃m,∀α,β,

supx,ξ(1+ |ξ|)−m+|β||(∂αx ∂

βξa)(x, ξ)| <∞.

This type of operator, initially used to constructparametrices of elliptic operators, soon became akey tool in the analysis of PDE.

Hypoellipticity

In 1934, A. Kolmogorov introduced the operatorin R3

t,x,v ,

(5) K = ∂t + v∂x − ∂2v ,

to provide a model for Brownian motion in onedimension. Hörmander took up the study of generaloperators

(6) H = X0 −∑

1≤j≤rX2j ,

where the (Xj)0≤j≤r are smooth real vector fieldswhose Lie algebra generates the tangent space ateach point: the rank of the Xj and their iteratedPoisson brackets is equal to the dimension of theambient space (forK, we have X0 = ∂t + v∂x, X1 =∂v , [X1, X0] = ∂x). These operators were proven tobe hypoelliptic in the 1967 article [37]: (1) holdswith P =H for theC∞ singular support. This paperwas the starting point of many studies, includingnumerous articles in probability theory, and theoperatorsH soon became known as Hörmander’ssum of squares. Their importance in probabilitycame from the fact that these operators appearedas a generalization of the heat equation where thediffusion term

∑1≤j≤r X2

j was no longer elliptic buthad instead some hypoelliptic behavior. ChapterXXII in Hörmander’s book [45] is concerned withhypoelliptic pseudodifferential operators: on theone hand, operators with a pseudodifferentialparametrix, such as the hypoelliptic constantcoefficient operators, and on the other handgeneralizations of the Kolmogorov operators (6).Results on lower bounds for pseudodifferentialoperators due to A. Melin [60] are a key toolin this analysis. Results of L. Boutet de Monvel[13], J. Sjöstrand [76], L. Boutet de Monvel, andA. Grigis and B. Helffer [14] are also given inthat chapter. Chapter XIX in [45] deals withelliptic operators on a manifold without boundaryand the index theorem. In the Notes of ChapterXVIII, Hörmander writes: It seems likely that itwas the solution by Atiyah and Singer [5] of theindex problem for elliptic operators which led tothe revitalization of the theory of singular integraloperators.

Spectral Asymptotics

The article [38], published in 1968, provides thebest possible estimates for the remainder term inthe asymptotic formula for the spectral functionof an arbitrary elliptic differential operator. Thisis achieved by means of a complete description ofthe singularities of the Fourier transform of thespectral function for low frequencies. The methodof proof for a positive elliptic operator P of ordermon a compact manifold is using the construction ofa parametrix for the hyperbolic equation i∂t+P1/m,which is formally exp itP1/m, an operator that canbe realized as a Fourier integral operator.

The Microlocal Revolution, Act II

Propagation of Singularities

The fact that singularities should be classifiedaccording to their spectrum was recognized first inthe early 1970s by three Japanese mathematicians:the Lecture Notes [75] by M. Sato, T. Kawai, andM. Kashiwara set the basis for the analysis in thephase space and microlocalization. The analyticwave-front set was defined in algebraic terms andelliptic regularity, and propagation theorems wereproven in the analytic category. The paper [15] byJ. Bros and D. Iagolnitzer gave a formulation ofthe analytic wave-front set that was more friendlyto analysts. The definition of the C∞ wave-frontset was given by Hörmander in [40]. For Ω anopen subset of Rn, u ∈ D′(Ω), (x0, ξ0) ∈ Ω× Sn−1

belongs to the complement of WFu means thatthere exists a neighborhood U ×V of (x0, ξ0) suchthat ∀χ ∈ C∞c (U),∀N ∈ N,

(7) supλ≥1,ξ∈V

|χu(λξ)|λN <∞.

The propagation of singularities theorem forreal principal-type operators (see [75] for theanalytic wave-front set and Hörmander’s [41] forthe C∞ wave-front set) represents certainly theapex of microlocal analysis. Since the seventeenthcentury, with the works of Huygens and Newton,the mathematical formulation for propagation oflinear waves lacked correct definitions. The wave-front set provided the ideal framework: for P a realprincipal-type operator with smooth coefficients(e.g., the wave equation) and u a function suchthat Pu ∈ C∞, WFu is invariant by the flow of theHamiltonian vector field of the principal symbolof P .

Fourier Integral Operators

The propagation results found new proofs viaHörmander’s articles on Fourier integral operators[39] and [20] (joint work with J. Duistermaat). It isinteresting to quote at this point the introductionof [39] (the reference numbers are those of ourreference list): The work of Egorov is actually anapplication of ideas from Maslov [59] who stated at


the International Congress in Nice that his book ac-tually contains the ideas attributed here to Egorov[22] and Arnold [4] as well as a more general andprecise operator calculus than ours. Since the bookis highly inaccessible and does not appear to bequite rigorous we can only pass this information onto the reader, adding a reference to the explana-tions of Maslov’s work given by Buslaev [16]. In thiscontext we should also mention that the “Maslovindex” which plays an essential role in Chapters IIIand IV was already considered quite explicitly byJ. Keller [51]. It expresses the classical observationin geometrical optics that a phase shift of π/2 takesplace at a caustic. The purpose of the present paperis not to extend the more or less formal methodsused in geometrical optics but to extract from thema precise operator theory which can be applied tothe theory of partial differential operators.

The simplest example of a Fourier integraloperator U is given by the formula

(8) (Uv)(x)=∫eiφ(x,η)c(x, η)v(η)dη(2π)−n,

where the real phaseφ is (positively) homogeneouswith degree 1 in η such that

det ∂2φ/∂x∂η 6= 0,and c is some amplitude behaving like a symbol.Some operators of this type were already intro-duced in 1957 in P. Lax’s paper [54] as parametricesof hyperbolic operators. A fundamental theoremdue to Y. V. Egorov [22] is that FIO are quantizingasymptotically canonical transformations in thesense that

(9) U∗a(x,D)U≡(a χ)(y,D) mod Pm−1,for any symbol a of order m, where χ is thecanonical transformation naturally attached to thephase φ and Pm−1 stands for pseudodifferentialoperators with order m− 1.

Local Solvability

After Lewy’s example (2) and Hörmander’s workon local solvability, L. Nirenberg and F. Trevesin 1970 [68], [69], [70], after a study of complexvector fields in [67] (see also the S. Mizohata paper[65]), introduced the so-called condition (Ψ) andprovided strong arguments suggesting that thisgeometric condition should be equivalent to localsolvability. The necessity of condition (Ψ) for localsolvability of principal-type pseudodifferentialequations was proved in two dimensions byR. Moyer in [66] and in general by Hörmander [44]in 1981. The sufficiency of condition (Ψ) for localsolvability of differential equations was proved byR. Beals and C. Fefferman [8] in 1973. They createda new type of pseudodifferential calculus, basedon a Calderón-Zygmund decomposition, and wereable to remove the analyticity assumption requiredby L. Nirenberg and F. Treves. The sufficiency ofthat geometric condition was proven in 1988 in

two dimensions by N. Lerner [55]. Later in 1994,Hörmander, in his survey article [47], went back tolocal solvability questions, giving a generalizationof Lerner’s article [56]. In 2006, N. Dencker [19]proved that condition (Ψ) implies local solvabilitywith a loss of two derivatives.

More on Pseudodifferential Calculus

A most striking fact in R. Beals and C. Fefferman’sproof was the essential use of a nonhomogeneouspseudodifferential calculus which allowed a finerlocalization than what could be given by conicmicrolocalization. The efficiency and refinementof the pseudodifferential machinery was suchthat the very structure of this tool attractedthe attention of several mathematicians, amongthem R. Beals and Fefferman [7], Beals [6], andA. Unterberger [80]. Hörmander’s 1979 paper [43],“The Weyl calculus of pseudodifferential operators,”represents an excellent synthesis of the mainrequirements for a pseudodifferential calculus tosatisfy; that article was used by many authorsin multiple circumstances, and the combinationof the symplectically invariant Weyl quantizationalong with the datum of a metric on the phasespace was proven to be a very efficient approach.

The thirty-page presentation of the Basic Cal-culus in Chapter XVIII of [45] is concerned withpseudodifferential calculus and is an excellentintroduction to the topic. R. Melrose’s totally char-acteristic calculus [62] and L. Boutet de Monvel’stransmission condition [12] are given a detailedtreatment in this chapter. The last sections aredevoted in part to results on new lower boundsby C. Fefferman and D. H. Phong [25]. Chapter XXin [45] is entitled “Boundary Problems for EllipticDifferential Operators.” It reproduces at the begin-ning elements of Chapter X in [35] and takes intoaccount the developments on the index problemfor elliptic boundary problems given by L. Boutet deMonvel [12], [11] and G. Grubb [27]. Chapter XXIVin [45] is devoted to the mixed Dirichlet-Cauchyproblem for second-order operators. Singularitiesof solutions of the Dirichlet problem arriving atthe boundary on a transversal bicharacteristic willleave again on the reflected bicharacteristic. Thestudy of tangential bicharacteristics required anew analysis and attracted the attention of manymathematicians. Among these works: the papersby R. Melrose [61], M. Taylor [78], G. Eskin [24],V. Ivriı [50], R. Melrose and J. Sjöstrand [63], [64],K. Andersson and R. Melrose [3], J. Ralston [74],and J. Sjöstrand [77].

Subelliptic Operators

A pseudodifferential operator of orderm is said tobe subelliptic with a loss of δ derivatives whenever

(10) Pu ∈ Hsloc =⇒ u ∈ Hs+m−δloc .


The elliptic case corresponds to δ = 0, whereasthe cases δ ∈ (0,1) are much more complicatedto handle. The first complete proof for operatorssatisfying condition (P) was given by F. Trevesin [79], using a coherent states method (see alsoSection 27.3 of Hörmander’s [46]). Although it isfar from an elementary proof, the simplificationsallowed by condition (P) permit a rather compactexposition. The last three sections of Chapter XXVIIin [46] are devoted to the much more involvedcase of subelliptic operators satisfying condition(Ψ), and one could say that the proof is extremelycomplicated. Let us cite Hörmander in [49]: Forthe scalar case, Egorov [21] found necessary andsufficient conditions for subellipticity with loss ofδ derivatives (δ ∈ [0,1)); the proof of sufficiencywas completed in [42]. The results prove that thebest δ is always of the form k/(k + 1) where kis a positive integer.…A slight modification of thepresentation of [42] is given in Chapter 27 of [46],but it is still very complicated technically. Anotherapproach which covers also systems operating onscalars has been given by Nourrigat [71], [72] (seealso the book [28] by Helffer and Nourrigat), but itis also far from simple so the study of subellipticoperators may not yet be in a final form.

Nonlinear Hyperbolic Equations

In 1996, Hörmander’s book appeared [48]. Thefirst subject which is treated is the problem oflong-time existence of small solutions for nonlinearwaves. Hörmander uses the original method ofS. Klainerman [52]. It relies on a weighted L∞Sobolev estimate for a smooth function in termsof L2 norms of ZIu, where ZI stands for an iterateof homogeneous vector fields tangent to the wavecone. The chapter closes with a proof of globalexistence in 3D when the nonlinearity satisfiesthe so-called “null condition,” i.e., a compatibilityrelation between the nonlinear terms and the waveoperator.

The last part of the book is concerned with theuse of microlocal analysis in the study of nonlinearequations. Chapter 9 is devoted to the study ofpseudodifferential operators lying in the “bad class”S0

1,1. The results of Chapter 9 are applied in Chapter10 to construct Bony’s paradifferential calculus[9], [10]. One associates to a symbol a(x, ξ), withlimited regularity in x, a paradifferential operatorand proves the basic theorems on symbolic calculus,as well as “Bony’s paraproduct formula.” Next,Bony’s paralinearization theorem is discussed: itasserts that if F is a smooth function and u belongsto Cρ(ρ > 0), F(u) may be written as Pu + Ru,where P is a paradifferential operator with symbolF ′(u) and R is a ρ-regularizing operator. This isused to prove microlocal elliptic regularity forsolutions to nonlinear differential equations. Thelast chapter is devoted to propagation of microlocalsingularities, where the author proves Bony’s

theorem on propagation of weak singularities forsolutions to nonlinear equations.

Final CommentsAfter this not-so-short review of Hörmander’sworks on PDE, we see in the first place that hewas instrumental in the mathematical setting ofFourier integral operators (achieved in part withJ. Duistermaat) and also in the elaboration of acomprehensive theory of pseudodifferential opera-tors. Fourier integral operators had a long heuristictradition, linked to quantum mechanics, but theirmathematical theory is indeed a major lastingcontribution of Lars Hörmander. He was also thefirst to study what’s now called Hörmander’s sumof squares of vector fields and their hypoelliptic-ity properties. These operators are important inprobability theory and geometry but also gained arenewed interest in the recent studies of regular-ization properties for Boltzmann’s equation andother nonlinear equations.

References[1] S. Alinhac, Nonunicité du problème de Cauchy, Ann.

of Math. (2), 117(1):77–108, 1983.[2] S. Alinhac and M. S. Baouendi, A nonuniqueness

result for operators of principal type, Math. Z.,220(4):561–568, 1995.

[3] K. G. Andersson and R. B. Melrose, The propaga-tion of singularities along gliding rays, Invent. Math.,41(3):197–232, 1977.

[4] V. I. Arnol′d, On a characteristic class enteringinto conditions of quantization, Funkcional. Anal. iPriložen., 1:1–14, 1967.

[5] M. F. Atiyah and I. M. Singer, The index of ellipticoperators on compact manifolds, Bull. Amer. Math.Soc., 69:422–433, 1963.

[6] R. Beals, A general calculus of pseudodifferentialoperators, Duke Math. J., 42:1–42, 1975.

[7] R. Beals and C. Fefferman, Classes of spatially inho-mogeneous pseudo-differential operators, Proc. Nat.Acad. Sci. USA, 70:1500–1501, 1973.

[8] , On local solvability of linear partial differentialequations, Ann. of Math. (2), 97:482–498, 1973.

[9] J.-M. Bony, Calcul symbolique et propagation dessingularités pour les équations aux dérivées par-tielles non linéaires, Ann. Sci. École Norm. Sup. (4),14(2):209–246, 1981.

[10] J.-M. Bony, Analyse microlocale des équations auxdérivées partielles nonlinéaires, in Microlocal Anal-ysis and Applications, Lecture Notes in Math., 1495,Springer, 1991, pp. 1–45.

[11] L. Boutet de Monvel, Comportement d’un opérateurpseudo-différentiel sur une variété à bord, C. R. Acad.Sci. Paris, 261:4587–4589, 1965.

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Jean-Pierre Demailly

Lars Hörmander and the Theory of L2L2L2

Estimates for the ∂∂∂ OperatorI met Lars Hörmander for the first time in the early1980s on the occasion of one of the “KomplexeAnalysis” conferences held in Oberwolfach underthe direction of Hans Grauert and Michael Schneider.My early mathematical education had already beengreatly influenced by Hörmander’s work on L2

estimates for the ∂-operator in several complexvariables. The most basic statement is that onecan solve an equation of the form ∂u = v for anygiven (n, q)-form v on a complex manifold X suchthat ∂v = 0, along with a fundamental L2 estimateof the form

∫X |u|2e−ϕdVω ≤

∫X γ−1

q |v|2e−ϕdVω.This holds true whenever ϕ is a plurisubharmonicfunction such that the right-hand side is finiteand X satisfies suitable convexity assumptions,e.g., when X possesses a weakly plurisubharmonicexhaustion function. Here dVω is the volume formof some Kähler metric ω on X, and γq(x), atany point x ∈ X, is the sum of the q smallesteigenvalues of i∂∂ϕ(x) with respect to ω(x). Thiswas in fact the main subject of a PhD coursedelivered by Henri Skoda in Paris during the year1976–77, and, to a great extent, the theory of L2

estimates was my entry point into complex analysisof several variables. At the same time, I followed agraduate course of Serge Alinhac on PDE theory,and Lars Hörmander appeared again as one of themain heroes. I was therefore extremely impressedto meet him in person a few years later—his tallstature and physical appearance did make for aneven stronger impression. I still remember thaton the occasion of the Wednesday afternoon walkin the Black Forest, Hörmander was in a groupof two or three that essentially left all the restbehind when hiking on the somewhat steep slopesleading to the Glaswaldsee, a dozen kilometersnorth of the Mathematisches ForschungsinstitutOberwolfach.

It seems that Lars Hörmander himself, at leastin the mid 1960s, did not consider his work on∂-estimates [13] to stand out in a particular wayamong his other achievements; after all, theseestimates appeared to him to be only a specialcase of Carleman’s technique, which also appliesto more general classes of differential operators.In his own words, Apart from the results involvingprecise bounds, this paper does not give any newexistence theorems for functions of several complexvariables. However, we believe that it is justifiedby the methods of proof. In spite of this rathermodest statement, the paper already permitted

Jean-Pierre Demailly is professor of mathematics at Uni-versité Grenoble Alpes, France. His email address [email protected].


one to bypass the difficult question of boundaryregularity involved in the Morrey-Kohn approach[22], [18], [19], [20]. For this, Hörmander observesthat the Friedrichs regularization lemma appliesto the particular situation he considers. Also, andperhaps more importantly, Hörmander’s techniquegives a new proof of the existence of solutions of∂-equations on pseudoconvex or q-convex domains,thus recovering the results of Andreotti-Grauert[1] by a more direct analytic approach. We shouldmention here that Andreotti-Vesentini [2] indepen-dently obtained similar results in the context ofcomplete Hermitian manifolds through a refine-ment of the Bochner-Kodaira technique [3], [16],[17]. One year after the publication of [13], LarsHörmander published his tremendously influentialbook An Introduction to Complex Analysis in Sev-eral Variables [14], which is now considered to beone of the foundational texts in complex analysisand geometry.

It took only a few years to realize that the veryprecise L2 estimates obtained by Hörmander forsolutions of ∂-equations had terrific applications toother domains of mathematics. Chapter VII of [14]already derives a deep existence theorem for solu-tions of PDE equations with constant coefficients.More surprisingly, there are also striking applica-tions in number theory. In 1970, Enrico Bombieriextended in this way an earlier result of Serge Langconcerning algebraic values of meromorphic mapsof finite order: if a system of such functions hastranscendence degree larger than the dimensionand satisfies algebraic differential equations, thenthe set of points where they simultaneously takevalues in an algebraic number field is containedin a certain algebraic hypersurface of boundeddegree. The proof combines use of Lelong’s theoryof positive currents with L2 estimates for arbitrarysingular plurisubharmonic weights; cf. [4]. It iscrucial here to allow ϕ to have poles, e.g., loga-rithmic poles of the form log

∑|gj |2 where the

gj ’s are holomorphic functions sharing commonzeroes (the degree bounds were later refined byHenri Skoda [30] and [21]). Using Hörmander’stechniques in a fundamental way, Henri Skoda gavefurther major applications to complex analysisand geometry. In [28] it is shown that analyticsubsets of Cn possessing a given growth of thearea at infinity can be defined by holomorphicfunctions with a precise control of the growth; suchquantitative versions of Cartan’s theorems A and Bhad also been anticipated in [14] (see e.g., section7.6, “Cohomology with bounds”). The article [29],on the other hand, gives an almost optimal L2

estimate for the solutions of Bézout equations∑gjhj = f in the ring of holomorphic functions (if

f and the gj ’s are given so that a certain quotient|f |2|g|−2(n+1+ε)e−ϕ is integrable, then h can befound such that |h|2|g|−2(n+ε)e−ϕ is integrable).This result, in its turn, yields profound results of

local algebra, e.g., in the form of a fine control ofthe integral closure of ideals in the ring of germs ofholomorphic functions [5]; see also [31] for morerelated geometric statements. It is remarkable thatalgebraic proofs of such results were found onlyafter the discovery of the analytic proof. Aboutthe same time, Yum-Tong Siu [26] gave a proof ofthe analyticity of sublevel sets of Lelong numbersof closed positive currents; Siu’s main argumentagain involves the Hörmander-Bombieri technique.

But the story does not stop there; Hörmander’sL2 estimates also have intimate connections withfundamental questions of algebraic geometry. Infact, Takeo Ohsawa and Kensho Takegoshi discov-ered in 1987 a very deep L2 extension theorem:every L2 holomorphic function defined on a subva-riety Y of a complex manifold X can be extendedas a holomorphic function F on X satisfying anL2 bound

∫X |F|2e−ϕdVX ≤ C

∫Y |f |2e−ϕdVY . This

holds true provided ϕ is plurisubharmonic andsuitable curvature assumptions are satisfied [24].The initial proof used a complicated, twistedBochner-Kodaira-Nakano formula, but it was re-cently discovered by Bo-Yong Chen [6], [7] that theMorrey-Kohn-Hörmander estimates were in factsufficient to prove it, while improving the estimatesalong the way. The Ohsawa-Takegoshi L2 extensiontheorem itself has quite remarkable consequencesin the theory of analytic singularities, for instancea basic regularization theorem for closed positivecurrents [9] or a proof of the semicontinuity ofcomplex singularity exponents [11]. Finally, [24]can be used to confirm the conjecture on the invari-ance of plurigenera in a deformation of projectivealgebraic varieties [27], [25]: this basic statement ofalgebraic geometry still has no algebraic proof asof this date! Another very strong link with algebraicgeometry occurs through the concept of multiplierideal sheaves: ifϕ is an arbitrary plurisubharmonicfunction, then the ideal sheaf of germs of holomor-phic functions such that |f |2e−ϕ is integrable is acoherent analytic sheaf; the proof is essentially astraightforward consequence of the Hörmander-Bombieri technique. In general, if (L, e−ϕ) is asingular hermitian line bundle on a compact Kählermanifold X, one has Hq(X,KX ⊗ L ⊗ I(ϕ)) = 0as soon as the curvature of ϕ is positive defi-nite (a generalization of the Kawamata-Viehwegvanishing theorem [15], [32]; cf. [8], [23], [10]).In case i∂∂ϕ is merely semipositive, one getsinstead a surjective Lefschetz morphism ωq ∧ • :H0(X,Ωn−qX ⊗ L ⊗ I(ϕ)) → Hq(X,KX ⊗ L ⊗ I(ϕ))[12]. All the arguments use Hörmander’s theory ofL2 estimates in one way or the other. Contrary tothe exceedingly modest words of Hörmander, theL2 existence theorem for solutions of ∂-equationsappears to be one of the most powerful theoremsof contemporary mathematics!


References[1] A. Andreotti and H. Grauert, Théorèmes

de finitude pour la cohomologie des espacescomplexes, Bull. Soc. Math. France 90 (1962),193–259.

[2] A. Andreotti and E. Vesentini, Carleman es-timates for the Laplace-Beltrami equation oncomplex manifolds, Publications Mathématiquesde l’I.H.É.S. 25 (1965), 81–130.

[3] S. Bochner, Curvature and Betti numbers (I) and(II), Ann. of Math. 49 (1948), 379–390; ibid. 50(1949), 77–93.

[4] E. Bombieri, Algebraic values of meromor-phic maps, Invent. Math. 10 (1970), 267–287;Addendum, Invent. Math. 11 (1970), 163–166.

[5] J. Briançon and H. Skoda, Sur la clôtureintégrale d’un idéal de germes de fonctions holo-morphes en un point de Cn, C. R. Acad. Sci. sérieA 278 (1974), 949–951.

[6] Bo-Yong Chen, A simple proof of the Ohsawa-Takegoshi extension theorem, arXiv:math.CV/1105.2430

[7] , The Morrey-Kohn-Hörmander formulaimplies the twisted Morrey-Kohn-Hörmanderformula, arXiv:math.CV/1310.3816.

[8] J.-P. Demailly, Transcendental proof of a general-ized Kawamata-Viehweg vanishing theorem, C. R.Acad. Sci. Paris Sér. I Math. 309 (1989), 123–126.

[9] , Regularization of closed positive currentsand intersection theory, J. Alg. Geom. 1 (1992),361–409.

[10] , A numerical criterion for very ample linebundles, J. Differential Geom. 37 (1993), 323–374.

[11] J.-P. Demailly and J. Kollár, Semicontinuityof complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Ec. Norm.Sup. 34 (2001), 525–556.

[12] J.-P. Demailly, Th. Peternell, and M. Schnei-der, Pseudo-effective line bundles on compactKähler manifolds, Internat. J. Math. 6 (2001),689–741.

[13] L. Hörmander, L2 estimates and existence theo-rems for the ∂ operator, Acta Math. 113 (1965),89–152.

[14] , An Introduction to Complex Analysis inSeveral Variables, 1st edition, Elsevier SciencePub., New York, 1966; 3rd revised edition, North-Holland Math. Library, vol. 7, Amsterdam, 1990.

[15] Y. Kawamata, A generalization of Kodaira-Ramanujan’s vanishing theorem, Math. Ann. 261(1982), 43–46.

[16] K. Kodaira, On a differential geometric methodin the theory of analytic stacks, Proc. Nat. Acad.Sci. USA 39 (1953), 1268–1273.

[17] , On Kähler varieties of restricted type,Ann. of Math. 60 (1954), 28–48.

[18] J. J. Kohn, Regularity at the boundary of the ∂-Neumann problem, Proc. Nat. Acad. Sci. USA 49(1963), 206–213.

[19] , Harmonic integrals on strongly pseudo-convex manifolds I, Ann. of Math. 78 (1963), 206–213.

[20] , Harmonic integrals on strongly pseudo-convex manifolds II, Ann. of Math. 79 (1964), 450–472.

[21] P. Lelong, Sur les cycles holomorphes à coef-ficients positifs dans Cn et un complément authéorème de E. Bombieri, C. R. Math. Rep. Acad.Sci. Canada 1 (4) (1978/79), 211–213.

[22] C. B. Morrey, The analytic embedding of abstractreal analytic manifolds, Ann. of Math. 68 (1958),159–201.

[23] A. M. Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Proc.Nat. Acad. Sci. USA 86 (1989), 7299–7300; andAnn. of Math. 132 (1990), 549–596.

[24] T. Ohsawa and K. Takegoshi, On the extensionof L2 holomorphic functions, Math. Z. 195 (1987),197–204.

[25] M. Paun, Siu’s invariance of plurigenera: a one-tower proof, J. Differential Geom. 76 (2007), 485–493.

[26] Y. T. Siu, Analyticity of sets associated to Lelongnumbers and the extension of closed positivecurrents, Invent. Math. 27 (1974), 53–156.

[27] , Extension of twisted pluricanonicalsections with plurisubharmonic weight and in-variance of semipositively twisted plurigenera formanifolds not necessarily of general type, Com-plex Geometry (Göttingen, 2000), Springer, Berlin,2002, pp. 223–277.

[28] H. Skoda, Sous-ensembles analytiques d’ordrefini ou infini dans Cn, Bull. Soc. Math. France 100(1972), 353–408.

[29] , Application des techniques L2 à l’étudedes idéaux d’une algèbre de fonctions holomor-phes avec poids, Ann. Sci. École Norm. Sup. 5(1972), 545–579.

[30] , Estimations L2 pour l’opérateur ∂ et ap-plications arithmétiques, Séminaire P. Lelong(Analyse), année 1975/76, Lecture Notes in Math.,538, Springer-Verlag, Berlin, 1977, pp. 314–323.

[31] , Morphismes surjectifs de fibrés vecto-riels semi-positifs, Ann. Sci. École Norm. Sup. 11(1978), 577–611.

[32] E. Viehweg, Vanishing theorems, J. Reine Angew.Math. 335 (1982), 1–8.

Michael Atiyah

I must have met Lars Hörmander in the early 1960s,probably on one of many trips to Lund, where Ialso collaborated with the other Lars (Gårding). Atvarious stages Hörmander and I discussed ellipticdifferential equations and index theory. From thesediscussions and from his book, with its excellenttreatment of distributions, I got my education inmodern analysis.

Later we overlapped for a year at the Institute forAdvanced Study in Princeton. By the time I arrivedhe had already decided to return permanently toLund. But before he left he instructed me in theart of log-splitting with wedges, something whichevery Swedish boy picks up and which I found

Michael Atiyah is Honorary Professor at the University ofEdinburgh and Fellow of Trinity College, Cambridge, UK.His email address is [email protected].


useful later when I acquired a Scandinavian logcabin in the Scottish Highlands.

I think the transition from Princeton to Lund wasnot easy for Lars. He was at home in the Swedishcountryside and culture, but mathematically hewas somewhat isolated, despite his fame and hisinternational connections.

Lars had one of the most penetrating intellectsI have ever come across. I remember when Iwas struggling with the linear algebra of Novikovadditivity for the signature of manifolds, Lars justdisposed of the problem in minutes.

Lars and I met at numerous internationalconferences. but I remember particularly a jointtrip to Japan, where the tall blond Scandinaviantowered over all the Japanese (and me), attractinguniversal attention, particularly from parties ofJapanese schoolgirls.

François TrevesThis is much too short a space, for me, in whichto condense a lifetime of complicated interactionwith Lars Hörmander. We met for the first time inFebruary 1958. I had finished the writing of my thèsede Doctorat and convinced a (nonmathematician)friend to drive us from Paris to Stockholm, where Iwas eager to meet Hörmander for a week of vacationin the northern snows. We arrived Saturday nightand went directly to Drottninggatan (Queen Street,not pedestrian at the time), where the mathematicsdepartment was then located. I wanted to have afeel for the place, but what we saw was a not verywide, brightly lit street in downtown Stockholmwith a surprising number of very drunk menricocheting along the sidewalks, from the tightlyparked cars to the walls. On Sunday morningHörmander picked me up in his Volvo and droveus to the Mittag-Leffler Institute in Djursholm,where he was living. Unbeknownst to me I was thebearer of bad news from Paris: Lojasiewicz had justproved the divisibility of distributions by analyticfunctions (and much more), while Hörmander hadjust finished writing his paper on the division ofdistributions by polynomials (making use of theSeidenberg principle and implying the existenceof a tempered fundamental solution for any PDEwith constant coefficients).

At the time Hörmander was twenty-six, one yearyounger than I and already famous. Outside thecircle of the participants in the Laurent Schwartzseminar at the Institut Henri Poincaré (a small circleindeed: Lions, Malgrange, Malliavin, Martineau, anda few others), nobody knew me—nobody, that is,except Hörmander. We had exchanged letters since1956, and Schwartz had asked me to give a coupleof lectures at the quarterly Bourbaki seminar on

François Treves is professor emeritus of mathematicsat Rutgers University. His email address is [email protected].

With Atiyah and three Japanese ladies in Japan,1969.

Hörmander’s PhD thesis, recently published inActa Mathematica. Those lectures (my first) hadbeen a revelation to me, mainly about the standingof analysis in the mathematical firmament: theyhad been preceded by lectures in algebra andtopology. With my arrival at the podium there wasa mass exodus from the large Hermite auditorium.A few people stayed; I remember Claude Chevalleyand Henri Cartan among my sparse public—out ofpity, I guess, for the novice at the blackboard.

In 1959 Hörmander visited Berkeley (whereI had my first job, an assistant professor posi-tion), bringing his very novel explanation of HansLewy’s counterexample to the local solvabilityof a smooth, nowhere null, complex vector fieldL—in modern language, the nonvanishing of the

Levi form 12i

[L, L

]at a characteristic point. This

breakthrough was destined to have a determinativeimpact on my future research, more than I couldhave imagined at the time. More importantly, fromabout that date on Hörmander’s work was to havean overarching influence on the development ofthe modern theories of several complex variablesand of linear PDE (with ramifications in probabilitytheory). Prior sections of this article by otherauthors describe the scope of Hörmander’s oeuvre,so I will limit myself here to a couple of verypersonal recollections.

My relations with Hörmander were alwayscourteous and often cordial, even warm, I wouldsay, in the last decades. There was one exception: ata conference in Nice in 1972 on the topic of Fourierintegral operators and symplectic geometry, hisattitude towards me seemed uncharacteristicallycold, and I asked friends what could be thecause. Hörmander had recently submitted a paperto the Annals in which he proved a necessaryand sufficient condition for a PDE with constantcoefficients to have an analytic solution for everyanalytic right-hand side. That the heat equation in3D did not have this property had been discoveredearlier by E. De Giorgi, who, incidentally, openly


refused to seek a necessary and sufficient condition,so as not to discourage other mathematicians fromgetting interested in the problem. Hörmander’spaper had been returned to him for revisionon the basis of some—rather mild I was told—objections by the referee. Apparently, Lars hadconvinced himself that I was the referee, which Iwas not, and he was showing to common friendsa list of twenty points indicative of my personalstyle. I was never shown the list (nor the refereereport!), but someone told me what point #1 was:that my correspondence was always typewritten,which was true. (I am, to this day, curious about thenineteen other “qualities” I shared with the culprit.)Nirenberg told Hörmander that I had too muchrespect for him to be the referee (a well-intentionedexaggeration, perhaps), while B. Malgrange put tome a sweet request: May I tell Lars that the refereewas not born in Europe?. Bernard, I, and the othersknew this to be true.

My last stay in Lund was part of a series of visitsby mathematicians to celebrate the final settling,by N. Dencker, of the circle of solvability problemsstarted fifty years earlier with the Lewy example(to which Nirenberg and I had made contributions,as well as R. Beals and Ch. Fefferman, R. Moyer,and, later, N. Lerner). During that visit, Lars tookmy wife and me on a day tour by car of Skåne: firstto Ystad, which I had wished to visit because ofWallander’s travails and which turned out to be anold, lovely small town, nothing like what I expectedfrom reading Mankel’s novels, and afterwards toTrelleborg, where my wife had worked in her earlytwenties. She left us to revisit old places. As inevery one of my not-so-rare stays in Lund, theweather was splendid. We sat under the trees, twoold professors indulging in a bit of nostalgia. Thatis when I learned that Hörmander had writtena number of historical essays on the twentieth-century mathematical developments in which hehad participated; they were in the safekeeping ofLund University for publication after his death.One of them, devoted to the ∂ Neumann problem,had already circulated. Having read it, I was prettyconfident that the unpublished ones would be asseverely impartial and as valuable to interestedhistorians and mathematicians.

Lars insisted on taking us to the train toCopenhagen the next morning. We both knew thatthis was the last time, for, though he appeared tobe well, it was the tenth year of his illness and hehad decided to terminate all medication.

During lunch in Ystad I had mentioned that inthe diet to which celiac disease confined me, whatI missed most was beer. A few weeks after ourreturn home I received an email from his daughterwith a list of gluten-free beers; I have been enjoyingone of the brands ever since, for which I am verygrateful to both him and his daughter. A minor,

but still pleasurable, addition to my mathematicaldebt to Lars Hörmander.

Sigurdur Helgason

My contact with Hörmander started with my useof his 1963 book, which I used several timesin a course on distributions. Already his workwas having major impact on the large field ofpartial differential equations. At that time I wasinvolved in the study of differential equationson a homogeneous space G/K invariant underthe natural action of G. Constant coefficientdifferential operators are the first natural example.While invariant operators in the above sense areplentiful in nature, one’s optimism is quicklydampened by Levy’s example,

δx + iδy − 2i(x+ iy)δz ,of an operator which is not locally solvable yetclosely related to an operator which is left invarianton the Heisenberg group.

At the Institute for Advanced Study in 1964 I hadthe opportunity of discussing these matters withHörmander. The same afternoon he came up withthe result that if G is a Lie group for which everyfirst-order invariant operator is locally solvable,then G is either abelian or has a normal abeliansubgroup of codimension one. This he derivedquickly from his necessary condition of localsolvability, specialized to first-order operators.The result was also proved by Cerezo and Rouvièrewith further explicit details.

This was typical of my experience with Hörman-der. When asking him a mathematical question, onegot back much more than just the answer. Anotherexample of this is my questioning him about theproperty of a constant coefficient differential oper-ator being a homeomorphism of the space of testfunctions onto its image. Ehrenpreis’s treatment ofthis question did not convince me. Since I neededthis result in my research, I consulted Hörmander.He was optimistic, but the full proof did notemerge until the second volume of his magnificentfour-volume opus The Analysis of Linear PartialDifferential Operators. Theorem 15.4.2 there givesan explicit intrinsic description of the topology ofthe Fourier transform space D. Since Schwartz’stopology of D is rather complicated, the topologyof D is fairly complicated too. However, the resultis powerful and the proof shows the touch of amaster, also in the analysis of several complexvariables. Since the proof can be reduced so as toneed Cauchy’s theorem in just one variable, I havefound it ideally suited to a course on distributiontheory. For example, it leads very quickly to the

Sigurdur Helgason is professor emeritus of mathematics atthe Massachusetts Institute of Technology. His email addressis [email protected].


existence theorem LD′ = D′ for any constantcoefficient differential operator L.

Gerd GrubbIt was during my PhD studies at Stanford University(completed in 1966) when I first met Lars Hörman-der. I did not follow his lectures then, but did solater during summer visits to Stanford in 1967 and1971. The first time I asked him a question wason the Calderón projector put forward by RobertSeeley in [6], where he kindly explained to me theessentially equivalent construction in his Annalspaper [3].

Back in Denmark around 1970 I organizedan interuniversity study group with the aim ofunderstanding the new results on hypoellipticity; itculminated with Lars’s participation in a workshopwe organized in Århus in spring 1972 (where someFrench colleagues also came: A. and J. Unterberger,C. Zuily, M. Derridj). That spring I was also occupiedwith an honorable task Lars had given me: to bethe faculty opponent on the thesis [7] of JohannesSjöstrand in Lund. (Here he kidded me with thefact that the part I found most strange was theone he had insisted on.)

Besides producing wonderful mathematics him-self, Lars had a great influence on the work of allwith whom he had contact. He inspired everyonearound him to sharpen their efforts and formula-tions in research questions in analysis, to makesimplifications by use of functional analysis suchthat one would reach the really hard questionsthat demanded original tricks or new theories. Butbeing in his vicinity did not increase one’s feelingof self-importance.

The Danish “hypoelliptic study group” wasrevived later when we were invited to read prelimi-nary versions of his four-volume treatise [4]. I wasallowed a small influence on the chapter on ellipticboundary problems. A particular effort was madeby Niels J. Kokholm (who also received specialthanks) at the same time as he was working ona thesis under Lars’s guidance in Lund. Kokholmand I later carried a large project through [2], buthe quit mathematics for more concrete IT jobsafter this.

In 1985 we started the Danish-Swedish AnalysisSeminar, where Lars and I, together with AndersMelin, organized one-day meetings, alternatingbetween Lund and Copenhagen, once or twiceper semester. This made it possible to have asufficiently large audience for a lot of interestingguests, and it served the whole PDE community inDenmark.

In the 1980s it became possible to writemanuscripts in TEX; Richard Melrose brought

Gerd Grubb is professor emerita of mathematics at theUniversity of Copenhagen, Denmark. Her email address [email protected].

macros from MIT, and Lars passed them on tome, introducing me to this clever way to mastermathematical formulation.

I have consulted Lars on many details in myongoing research and refereeing jobs and havelearned enormously from him; I have also givencomments on some of his writings. We madeone joint publication [1], on pseudodifferentialoperators satisfying the transmission condition(preserving smoothness up to a boundary) withsymbols inSm%,δ-spaces, including results on Poissonoperators of this type. The work was quite difficult;whenever I wrote something, a formulation ordeduction by Lars would usually win over it, butafter a lot of work back and forth it ended as anice informative piece, I think.

A bit uncommonly, Lars detested being thecenter of a celebration. Therefore he often fledfrom his home in Lund when a special birthdaywas approaching. On the other hand, to organize ameeting gives one the chance to invite people inreturn for their hospitalities. In 1995, when Lars’sretirement was approaching, he and I, jointly withAnders Melin and Johannes Sjöstrand, used theframework of the Danish-Swedish analysis seminarto arrange two 3-day meetings [5], where we fit inas many of his good international colleagues aspossible, having feasts in Copenhagen and Lundeach time. In 2006 (a birthday year) again manyvisitors came informally to Lund.

On the private side, we shared an interest in thewild flowers of Skåne and Blekinge, in particularrare Nordic orchids. This is a very sad occasion tothink back on those years.

References[1] G. Grubb and L. Hörmander, The transmission

property, Math. Scand. 67 (1990), 273–289.[2] G. Grubb and N. J. Kokholm, A global calculus

of parameter-dependent pseudodifferential boundaryproblems in Lp Sobolev spaces, Acta Math. 171 (1993),165–229.

[3] L. Hörmander, Pseudo-differential operators andnonelliptic boundary problems, Ann. of Math. 83 (1966),129–209.

[4] , The Analysis of Linear Partial DifferentialOperators I–IV, Springer Verlag, Berlin, 1983–85.

[5] L. Hörmander and A. Melin (eds.), Partial differen-tial equations and mathematical physics, Papers fromthe Danish-Swedish Analysis Seminar held at the Uni-versities of Copenhagen and Lund 1995, Progress inNonlinear Differential Equations and Their Applications,21, Birkhäuser, Boston, 1996.

[6] R. T. Seeley, Singular integrals and boundary valueproblems, Amer. J. Math. 88 (1966), 781–809.

[7] J. Sjöstrand, Operators of principal type with interiorboundary conditions, Acta Math. 130 (1973), 1–51.


Jean-Michel BonyI first met Lars Hörmander at the InternationalCongress of Mathematicians (Nice, 1970), but Iwas already quite familiar with his work. One ofmy first papers, devoted to sums of squares ofvector fields, relied on his result of hypoellipticity.Above all, it is in his book Linear Partial DifferentialOperators that I learned the theory of PDE when Iwas a student in École Normale Supérieure. I couldappreciate then the precision and the concision ofhis style: a complete exposition in eighteen pagesof the theory of distributions, which was at thattime the subject of a one-semester course in Paris.

Even though the words were not spoken, one cansay that microlocal analysis was born at this ICM,with the lectures of Mikio Sato and Youri Egorov,and the plenary lecture of Lars Hörmander. For thefirst time, Hörmander defined the wave-front set,stated and proved his fundamental theorem on thepropagation of singularities, and announced hismonumental work (partly with J. J. Duistermaat)on Fourier integral operators.

A few months later I had the surprise of receivinga preprint of L. Hörmander, giving another proof,and an improvement, of my own contribution tothe congress (an extension of Holmgren’s theorem).I should not have been surprised. If L. Hörmanderhad been for forty years the foremost contributor tothe theory of linear PDE, I think that it is due to threereasons: many outstanding theorems, of course,but also the fact that he gave us fundamentaltools such as Fourier integral operators or hissuccessive extensions of the pseudodifferentialcalculus. The third reason is that quite frequentlyhe rewrote the results of other mathematicians,trying to extend their generality and above all togive proofs which link them to a small numberof fundamental concepts and results. All this wasconverging towards a new treatise on the subject,which he had in mind as early as in the beginningof the 1970s.

I met L. Hörmander rather frequently afterthat, particularly during two long stays at theMittag-Leffler Institute, where he organized twoone-year thematic programs on PDE. During thefirst one, in 1974, he looked quite interested inhyperfunction theory, and I can find traces of ourtalks on this topic in Chapter 9 of his treatise. Themanuscript was already quite advanced, carefullystored in three binders (the last volume was dividedlater), and I remember the clatter of his typewriterbetween the seminars. Ten years later the secondprogram was mainly devoted to nonlinear PDE, thetreatise was completed, and the typewriter wasreplaced by a noiseless computer.

Jean-Michel Bony is professor emeritus of mathematics at

l’École polytechnique, Palaiseau, France. His email addressis [email protected].

My talks with Lars Hörmander were not limitedto mathematics. He was fond of French literature,reading in French, for instance, novels by Stendhaland more recently A la recherche du temps perdu.He did not like to speak French, probably becausehis French, though quite good, was not perfect.However, he made an exception for his answerwhen he was made Doctor Honoris Causa of theUniversity of Paris-Sud at Orsay.

The development of the theory of PDE has beenand still is a wonderful journey. Without LarsHörmander, this development would have beencertainly much slower, and it could have produceda messy and tangled heap of results and specificmethods. We are all indebted to him and willremain so for a long time.

Christer O. KiselmanLars Hörmander was appointed professor at Stock-holm University College effective January 1, 1957,not yet twenty-six years old. I started my studiesthere in the fall of 1957 and soon became aware ofhis existence. The student newspaper Gaudeamuspublished a report from the installation ceremonywith the heading “Twenty-six-year-old mathematicsmachine solemnly installed.”

During my first years I had no contact withHörmander, but I had several teachers who werestudents of his: Benny Brodda, Vidar Thomée,Göran Björck, Stephan Schwarz, and Lars Nystedt.In October 1960 I talked with Olof Hanner abouta possible continuation of my studies, and Imentioned that I would like to have Lars Hörmanderas my advisor. Olof was not surprised; he justremarked that the reputation of his young colleaguehad spread efficiently.

Lars was always available for consultations. Oneknocked at his door—he could answer any questionimmediately. I never experienced any difficulty intalking mathematics with him. Most often he wastyping articles or chapters of his book, the onewhich was to appear in 1963. The clattering wasintense. He also typed lecture notes, which weremimeographed using the technology of the time.When I left his office the clattering resumed afterzero seconds.

Lars lectured on partial differential equationsduring the academic year 1961–62. These lecturesforeshadowed the Springer book that came out in1963. We, his students, read the manuscript andcommented on it. I wrote a licentiate thesis on aproblem proposed by him concerning approxima-tion of solutions to partial differential equationswith constant coefficients.

During the academic year 1962–63, Lars gave aseries of lectures on analysis in several complex

Christer O. Kiselman is guest professor in the Departmentof Information Technology at Uppsala University, Sweden.His email address is [email protected].


variables. In this way he started the creation of hisbook that came out with Van Nostrand in 1966 andwhich is now one of the most quoted in the area. Theorganization of the lectures was exactly the one wenow see in the book. Holomorphic functions wereconsidered as solutions to differential equations,which was then a new approach for me and whichgave powerful construction methods and madegeneralizations possible. It was a great experienceto witness the birth of this book.

During the fall semester of 1963, Lars gave aseries of seminars on convex and subharmonicfunctions. More than thirty years later, in 1994,he published his book Notions of Convexity, whichtakes up these topics with a unified treatmentof convex, subharmonic, and plurisubharmonicfunctions.

Most important were three seminars on L2-methods for the ∂ operator, which he gave in thefall of 1963. The results then appeared in ActaMathematica in 1965 in a groundbreaking article.

In 1964 Lars left Stockholm for Stanford andPrinceton. He visited Stockholm in May 1965 andgave four lectures on pseudodifferential operators,an area that was then rather new.

Robert Oppenheimer offered me a membershipin the Institute for Advanced Study in Princeton,NJ, for the academic year 1965–66. Lars was therethen, and of course it was he who had arrangedeverything for me.

It became a most valuable year in every respect,both mathematically and culturally. I was therewith my wife Astrid and our son Dan, who turnedtwo during the fall. We arrived early on one of thefirst days in July while Lars and Viveka were inSweden. They let us stay in their house during thesummer. As a small service in return, we took careof their dog Shilly-Shally.

Lars gave a series of lectures at the institutewith the title “Pseudo-differential operators andboundary problems.” This was an elaboration andextension of the lectures he had given at StockholmUniversity in May 1965. Furthermore, Lars gavea seminar within the framework of the CurrentLiterature Seminar on “The Lefschetz fixed pointformula for elliptic complexes.” I was in constantcontact with Lars during that year and wrote apaper on the growth of entire functions and onanalytic functionals. Another valuable contact wasMiguel Herrera (1938–84), with whom I studiedresidue theory.

After that year I had many contacts with Larsconcerning complex convexity and fundamentalsolutions as well as many other topics, especiallyafter his return to Lund in 1968. I was the facultyopponent when his students Arne Enqvist andRagnar Sigurdsson presented their PhD theses.On the last day of 2010, he sent me a new,strong theorem on the regularity of fundamental

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With wife, Viveka, and Airedale terrier Shilly attheir house in Princeton, 1965.

solutions, an excellent addition to the results inhis four-volume book.

Nobody has been so important for my intellec-tual and scientific development as Lars.

Sofia BroströmMy father was born on January 24, 1931, in asmall fishing village in southern Sweden as theyoungest of five children. His father was a schoolteacher—originally trained as a painter, he hadpaid his own way to a teaching degree at twenty-one—who had married a bright farmer’s daughterwho recalled her few years in school as a highlightof her life. Together they created an atmospherecharacterized by a strong thirst for knowledge andeducation, and all five children graduated quicklyfrom secondary school and then continued withhigher education.

My father went through school even quicker thanthe others: he skipped two years of elementaryschool, with the result that he graduated withthe usual interval of two years between thesiblings, although in reality he was four yearsyounger than his youngest sister. Thanks to anenthusiastic teacher in secondary school, Nils-ErikFremberg, he had also finished the first semesterof university mathematics when graduating atseventeen. Despite previous plans to become anengineer like his older brother, this made himdecide to continue with more mathematics andphysics, receiving his master’s degree at nineteen

Sofia Broström is an interaction designer at Sony Mobile,and Lars Hörmander’s daughter. She lives in Lund, Sweden.Her email address is [email protected].


With daughter Gisela in Princeton, 1960.

and his licentiate degree (PhD) a year and a halflater in October 1951 at age twenty.

At the department Christmas party of that yearhe met my mother, Viveka. They married whilemy father was in military service and had theirfirst daughter, Gisela, in October 1954. One yearlater my father received his doctorate throughthe dissertation that made him famous. Since mymother had only been able to stay at home to takecare of Gisela for the first few months, much of thedissertation was actually written with my sister inhis (pulled-out) desk drawer (which was just theright size for the carriage insert). He apparentlydidn’t find this distracting, which was typical ofmy father—the only effect an interruption seemedto have on his work was the few minutes lost tothe actual distraction, nothing more.

It is not for me, however, to give a full biographyof my father’s life, so let me skip to my own

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With daughter Sofia, 1971.

memories of him. I was born in 1965, almost elevenyears after my sister, during the Princeton years. Iadored my father as a child, and we spent countlesshours together building with Legos, doing jigsawpuzzles, playing backgammon and Scrabble (forthe latter my mother would also join), or doingcarpentry. My father loved to work with his hands,especially with wood, having contemplated a futureas a carpenter as a child.

Among other things he built a Barbie dollhousefor me with four rooms, each fitted with electricityand completely furnished—my sister also helpingwith some of the smaller details, like textilesor earthenware. And I loved the weeks beforeChristmas when he would spend long hours inthe basement crafting presents for me—it wasfrustrating that I couldn’t join, but at the sametime exciting. One year I was deeply intrigued byscreeching sounds coming from the basement. Itturned out that he had built the kitchen for thedoll house, complete with refrigerator, stove, sink,cupboards, drawers, even a broom closet, and thatthe strange sounds came from sanding down fourcoins to make them into stove plates.

I was very fortunate that my father always had alot of spare time, so much so that my mother usedto complain that she couldn’t keep up with him,and this was even more so in the summers whenhe would take long periods away from work. Onthe island of Askerön off the west coast of Swedenmy parents had bought a summer house by the sea,and there we spent much time outdoors together.Either in the forest—a highlight was an ospreynest that we would visit every day for years—oron the sea—my father loved sailing. When I wasvery young we had a SeaCat, which, to my greatchagrin, was later traded in for a much smallerboat for day trips only. We also fished for plaice,with nets, together with a friend and his children.

When not actively engaged with the family orthe house, my father would spend his spare timereading. He always read a lot, fiction as well as fact,mostly history or science. He had a very inquisitivemind and loved to learn new things up until thelast days of his life. In all this he was much aidedby his excellent memory; he seemed unable toforget anything he had learned and, unfortunately,unable to realize that other people did not havequite his powers of memory…he could be quiteimpatient with me. And I had to be careful whenasking him a question: there was always the riskof a long and enthusiastic lecture, containing alot more information than I wanted. For betteror for worse, this seems to have been a heritabletrait—my son complains about the same behaviorin me.

My father set exceptionally high standards notonly for himself but for everyone around him,including of course his children. Needless to say,this was difficult for us, especially until I was


With grandson Sander, 2003.

old enough to realize that he himself was theprimary victim of his drive for perfection, notanyone else. He never felt that anything he did wasquite good enough and was devastated if he madea mistake. Despite his demands for excellence,he did not, however, have career ambitions forus, just as he lacked such ambitions himself. Hisfirst goal when studying mathematics had been tobecome a teacher in secondary school (somethinghe famously said to my mother on the evening theymet), and the promotions and prizes he receivedin his career were never what he strived for—onthe contrary, they weighed him down. He trulywanted us to do whatever we fancied, as long aswe did what we did well and were able to supportourselves. He had chosen mathematics solely outof love for the subject and wanted us to do thesame in our lives.

He was also a very loyal father. Many timesI chose to do things he advised me not to do,but once my mind was made up he was alwayscompletely loyal with my decision. For example,my former husband and I bought a rundown housewith a large garden despite his warnings. But oncehe realized that we were really going ahead, hequickly arranged for a generous loan and fromday one was a faithful caretaker of the house andgarden, effectively sheltering me from realizingwhat a crazy decision I had made. When he was nolonger able to help, the house of course becameimpossible to keep and has now been sold.

In 1996 my son, Sander, was born, and Larsbecame a grandfather, something which delightedhim no end. I think even more so since my sisterhad decided to end her life in 1978 at the ageof twenty-three. When my son was born he couldagain see a continuation, a path into the future,and I think that this at least partially restored hispeace of mind. And he was overjoyed that he wasable to follow Sander all the way to sixteen, evento the publication of his first book (Sander is aphoto artist). Before his death my father often saidto me that he was content with his life, that it hadbeen a good life. I do believe he died a happy man.

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To the Memory of Lars Hörmander (1931–2012)Moreover, Lars Gårding (1919–2014) and Åke Pleijel (1913–89) were appointed professors in mathematics in Lund during this period