##
**The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis.
2nd ed.**
*(English)*
Zbl 0712.35001

Grundlehren der Mathematischen Wissenschaften, 256. Berlin etc.: Springer-Verlag. xi, 440 p. DM 69.00/pbk; DM 128.00/hbk (1990).

Editorial remark: The following review, written in 2013, treats all four volumes.

When I was asked to write about Lars Hörmander’s treatise “Analysis of Linear Partial Differential Operators” I accepted with some hesitation only after clarifying for myself what is not asked for: a history of partial differential equations since 1983/85 on a few pages. For such a study time is not yet ready and a few pages will not suffice. Moreover there is no need to praise once more these volumes and their impact, and in particular the outstanding achievements and the scholarship of Lars Hörmander. This has been done several times and by more competent authors. What I try to do is to add a few, hopefully acceptable lines of interpretation of the impact that Hörmander’s work, which he cumulated in these volumes, has had. Some emphasis will be on developments after the publication of these volumes.

The starting point must be the publication of Hörmander’s thesis in 1955 [“On the theory of general partial differential operators”, Acta Math. 94, 161–248 (1955; Zbl 0067.32201)] which was published in the same volume of Acta Mathematica as J. L. Lions’s thesis [“Problèmes aux limites en théorie des distributions”, Acta Math. 94, 13–153 (1955; Zbl 0068.30902)]. Both theses put into their centre of investigation the use of functional analysis, for example the one or the other form of (L. Schwartz’s) distributions or generalized functions, in the study of partial differential equations. The extent to which this was done was absolutely new and transformed the field. While Lions dealt with (elliptic) boundary value problems, Hörmander set out a new programme “The Theory of General Partial Differential Operators” (influenced by his studies of I. G. Petrowsky’s work and arguably initiated by L. Gårding). The key features of Hörmander’s thesis are, besides the consequent use of more (abstract) functional analysis, that general questions are posed, for example when is a solution of a given partial differential equation smooth (arbitrarily often differentiable) given locally smooth data, and that the Fourier transform as a tool is systematically applied to transform a problem related to partial differential equations into a problem for the corresponding symbol (in the language we use now). Clearly there are predecessors to this idea, for example L. Gårding’s work on hyperbolic equations with constant coefficients, and not least I. G. Petrowsky’s work. But in Hörmander’s thesis we see a clear pattern: from the differential equation to the differential operator, from the differential operator to the symbol, use properties of the symbol to investigate the operator (mainly by deriving appropriate estimates), and then return to the equation. The characterization of hypoelliptic operators with constant coefficients can be seen as a key example. In the thesis some new tools were introduced such as norms and function spaces adapted to the operator, better its symbol, or comparisons of symbols entailing the notion of operators of equal strength and comparisons (in terms of estimates) of such operators.

Much of this programme and many more results proven along these lines were presented in Hörmander’s book “Linear Partial Differential Operators” published by Springer in 1963 [Die Grundlehren der mathematischen Wissenschaften 116 (1963; Zbl 0108.09301)], now 50 years ago and this justifies in addition to stay with that book for some time. Compared with other texts on partial differential equations of that time the book was unusual, but immediately recognised as a ground breaking contribution. The trinity of ellipticity, hyperbolicity, and parabolicity is not anymore the classification scheme, but questions are posed such as: Which operators have locally smooth solutions given locally smooth data, when is a solution always an analytic function, for which operators is the non-characteristic Cauchy problem well-posed, etc. Moreover classical tools like special solutions, series expansions and separation of variables, or ad hoc constructions of Green’s functions, were not treated. But we see for constant coefficient operators topics such as fundamental solutions (Malgrange-Ehrenpreis theorem) and their properties, interior regularity results, the non-characteristic and the characteristic Cauchy problem for general operators, global uniqueness and the Holmgren theorem, etc., treated. Interesting for later purpose is to note here that besides the notions of elliptic, hypoelliptic and hyperbolic operators the 1963 text still contains the notion of parabolic operators (much in the sense of I. G. Petrowsky). The four-volume treatise will not see anymore the notion of parabolic operators in the index. The way of treating operators with variable coefficients was quite surprising and new at the time. The first chapter is devoted to operators with no solutions, of course much influenced by the example of H. Lewy. It is followed by a discussion of operators with constant strength and local existence problems depending on the smallness of the domain or the coefficients. Of interest here is how the local solution operator is, via the Schwartz kernel theorem, related to Green’s functions and in some sense approximate fundamental solutions. Interesting is also how, in the case of smooth coefficients, bounds for the solution operator are derived. Special attention is paid to ellipticity. Already in the case of constant coefficient operators ellipticity was linked to analyticity, now this topic is taken up in the case of variable coefficient operators – again an answer to a general question posed to establish a theory of general differential operators. It follows a treatment of the Cauchy problem for strictly hyperbolic operators with smooth coefficients based on \(L^2\)-techniques. Finally elliptic boundary value problems along the lines proposed by Ya. B. Lopatinski and Z. Ya. Shapiro are discussed.

Spending time on the book published 50 years ago is important in order to understand how much the landscape changed in the 1960s and 1970s leading to the theory discussed in the treatise “The Analysis of Linear Partial Differential Operators, I–IV”. Volumes I and II were published jointly in 1983 followed by Volumes III and IV in 1985. At first sight Volumes I and II look like an extension and update of the 1963 book. But this is correct only to some extent. The emergence of micro-local analysis changed the field, already when dealing with distribution theory, and some deeper results for variable coefficient operators using micro-local analysis are easier understood when first discussed for constant coefficient operators. Let us be more systematical. We can look at Volume I, which has the subtitle “Distribution Theory and Fourier Analysis”, as a rather advanced textbook for a core area of modern analysis. Hörmander clearly had the intention to write such a book, and by supplying problems and solutions in the second edition of this volume this is made even clearer. Calculus in higher dimensions is presented in such a way that later on we can immediately turn to manifolds, but more importantly the key objects needed are carefully introduced: test functions, cut-off functions and partitions of unity, as well as convolutions of functions. The distributions are introduced, rather down-to-earth, no abstract theory of topological vector spaces is needed, and we can find essentially all about distributions that is needed later on (or even beyond) and often illustrated by carefully chosen examples. This is followed by a chapter on basic operations for distributions and again many, many illuminating examples are provided. The emphasis is on what is needed in concrete studies (for example homogeneous distributions) and already here we encounter many concrete fundamental solutions for differential operators. The chapter on convolution is not just devoted to extend a certain operation from functions to distributions. It has as a clear aim the support theorem in full generality in mind and most of all the relation to fundamental solutions for operators with constant coefficients. Tensor products and the kernel theorem follow with a rather “concrete” proof. Micro-local analysis is in some sense analysis related to objects defined on the co-tangent bundle. Chapter VI introduces distributions on manifolds and then discusses the tangent and the co-tangent bundle, again having in mind later needs when dealing with differential operators.

The approximately 90 pages of Chapter VII on the Fourier transform constitute a masterpiece in itself. We are introduced to basic results on the Schwartz space and on the space of tempered distributions, a lot of examples are discussed, but also topics such as the Malgrange Preparation Theorem, the method of stationary phase, or oscillatory integrals are all dealt with in detail. We will need this later on. In particular a paragraph completely devoted to the Fourier transform of Gaussians is provided and further the use of the Fourier transform to obtain fundamental solutions is investigated. Here we also find a discussion of the fundamental kernel of the Kolmogorov operator, an operator with variable coefficients, preparing ground for the famous “sum of squares” result which we will meet in Volume III. Hörmander is usually working in spaces of Schwartz distributions or in an \(L^2\)-context. The \(L^p\)-theory of the Fourier transform and its applications is touched upon only briefly. However for this we have the E. M. Stein school and Stein’s monumental monographs. What was covered in the 1963 book on 33 pages is now covered on 250 pages. Meanwhile (since 1983) new books on distribution theory and the Fourier transform have been published, and some are really good, for example [J.-M. Bony, Cours d’analyse. Théorie des distributions et analyse de Fourier. Les Éditions de l’École Polytechnique (2001; Zbl 1229.46002)], or [J. J. Duistermaat and J. A. C. Kolk, Distributions. Theory and applications. Birkhäuser, Basel (2010; Zbl 1213.46001)], or [G. Grubb, Distributions and operators. Graduate Texts in Mathematics 252. Springer (2009; Zbl 1171.47001)]. These are great books adding one or the other new aspect, allowing some new insights, or giving some original presentations. However Hörmander’s treatment of the topic stays unsurpassed.

We are not yet at the end of Volume I. Chapter VIII introduces us to what caused really the change in the analysis of partial differential operators in the 1970s: micro-local analysis as a tool for the study of singularities of solutions of partial differential equations. The wave front set of a distribution is introduced (not using characteristic sets of differential operators as originally, but using the Fourier analytic approach, i.e., looking at directions of non-decay of the Fourier transform of the localized distribution), the behaviour of wave front sets under operations of distributions is discussed, and then we see the modern machinery working: a first discussion of wave front sets of solutions of partial differential equations with constant coefficients. We learn already how the characteristic set of a differential operator enters into the picture, we get a micro-local understanding of ellipticity, we see interesting examples, we start to understand the rule of the principal part of a differential operator, we have a first glance at the propagation of singularities, and micro-hyperbolicity is introduced. Here we start to realize how central it is to turn from the operator to its symbol. The symbol as an object defined on the co-tangent bundle “interacts” with wave front sets and characteristic sets both naturally considered as subsets of the co-tangent bundle. All this is first treated in the category of smooth functions, but it is then also treated in the context of \(C^L\)-functions, i.e., functions where the derivatives have a certain decay (an extension of Gevrey classes), and the Holmgren uniqueness theorem is discussed. While the emergence of micro-local analysis was a highlight of analysis in the 1970s, this chapter is a further highlight in Volume I. On a technical level which should be accessible to graduate students, key ideas and results are presented, but much more is achieved: we are prepared to study and understand the deep results in the later volumes.

The final chapter in Volume I treats hyperfunctions. The theory is developed as much as it is needed to understand (later on) analytic regularity.

I deliberately spent much time on Volume I. Nowadays distribution theory, Fourier analysis (of distributions), wave front sets and micro-local analysis are standard tools. These tools are used in many areas, not least in mathematical physics. Needless to say that also the study of (certain) non-linear problems requires a mastery in using these tools. Hörmander’s Volume I make these tools accessible to many mathematicians. If 30 years after the publication of this treatise someone wants to start to learn micro-local analysis, this is still the best source. The 1983 research monograph is an advanced textbook in 2013, a witness how the field has developed, but also a witness of the lasting impact of the author.

Volume II is quite adequately described in its preface: “This volume is an expanded version of Chapter III, IV, V and VII of my 1963 book. In addition there is an entire new chapter on convolution equations, one on scattering theory, and one on methods from analytic functions of several variables.” And further, “Although the field is no longer very active – perhaps because of its advanced state of development – and although it is possible to pass directly from Volume I to Volume III, the material presented here should not be neglected by the serious student.”

First a remark on the three new chapters. The inclusion of a chapter on scattering theory (Chapter XIV) looks at first glance surprising. Obviously an important topic, but not really a part of the “theory of general partial differential operators”. There are however good reasons to include such a chapter. We see how one of the major tools, the Fourier (Laplace) transform, works also for these types of problems, and moreover Chapter XXX, in some sense also Chapter XXIX are prepared. We are reminded that in the mid 1970s and early 1980s scattering theory was a hot topic and micro-local analysis as a tool in spectral theory was much used. Further Enss’s papers (the “Enss method”) were seen to be promising to lead to serious progress in the problem of asymptotic completeness. Chapter XV on analytic function theory is an addition, in Hörmander’s own words “limited in scope” due to the existence of more specialized monographs such as those by L. Ehrenpreis [Fourier analysis in several complex variables. Pure and Applied Mathematics 17. Wiley-Interscience Publishers, New York (1970; Zbl 0195.10401)] or V. P. Palamodov [Linear differential operators with constant coefficients. Die Grundlehren der mathematischen Wissenschaften 168. Springer (1970; Zbl 0191.43401)]. The aim of this chapter is to develop some analytical techniques and to supply some simplified proofs. Again, the Fourier-Laplace transform and \(L^2\)-estimates are the basic tools. From these three new chapters, Chapter XVI on convolution equations is the most interesting one. It is a comprehensive presentation in the spirit of the entire treatise of results known on this topic until the early 1980s. These equations are so naturally linked to partial differential operators that they have a natural place in this volume. Of course, it is the Fourier transform which makes these links more apparent. In addition, having in mind Hörmander’s monograph “Notions of Convexity” [Birkhäuser, Basel (1994; Zbl 0835.32001)], this chapter prepares more. The complex of geometry (especially convexity), analyticity and solving certain classes of equations is in some sense scattered over the whole treatise.

Back to the revised chapters of the 1963 book. Chapter X (Existence and Approximation of Solutions of Differential Equations) and Chapter XI (Interior Regularity of Solutions of Differential Equations) are updates of the corresponding chapters. Chapter XII (The Cauchy and Mixed Problem, Constant Coefficients) is more. We first have a very detailed discussion of the wave equation including the oscillatory Cauchy problem and these two paragraphs dealing with this topic skilfully prepare core topics in micro-local analysis. The “classical results” are rearranged, the singularities of fundamental solutions are investigated in detail. The investigations of the characteristic Cauchy problems are considerably extended. It is interesting to note that the notion of “parabolic operator” has disappeared. In fact all four volumes have no place for dissipative operators generating one-parameter semi-groups. Having in mind the importance of heat kernels in spectral theory or in index theory this is a bit surprising. The mixed problem studied for constant coefficient operators considers Cauchy data on one hypersurface and boundary data (Lopatinski-Shapiro type) on a further hypersurface. We see the interplay of hyperbolic theory (Cauchy data) and elliptic theory (boundary data). Again a systematic use of the Fourier transform “reduces” much to the study of the corresponding symbols. These “updates” of the constant coefficient operator case is a wonderful presentation of a now “classical” subject, very valuable for serious students, and much helpful when moving to variable coefficient operators.

Having in mind that Volumes III and IV are devoted to variable coefficient operators, it is not surprising that Part III of the 1963 book “Differential Operators with Variable Coefficients” features in Volume II of the 1983/85 treatise only with a chapter on operators of constant strength. The basic definitions and basic local existence results are discussed, but for regularity questions the micro-local aspect is added, i.e., the behaviour of wave front sets. Global existence results are included as is a discussion of non-uniqueness of the Cauchy problem. Some interesting examples are provided.

The impact of Volume II is difficult to judge. The topic itself is central and absolutely important for a general theory. The presentation is full of insights, it is difficult to imagine that it will be surpassed in the next decades. It presents most of all a lot of ideas explored in the coming volumes by looking at simple examples, and it familiarizes us with certain techniques, however this volume could not have stimulated much new research.

When now turning to Volume III (with the subtitle “Pseudo-differential Operators”) and IV (with the subtitle “Fourier Integral Operators”) we have to change our point of view and our expectations, as Hörmander changed and had to change his intentions. These two volumes try to summarize and partly to clarify in a comprehensive manner the developments of the previous 20 years, say starting with the Kohn-Nirenberg paper [J. J. Kohn and L. Nirenberg, “An algebra of pseudo-differential operators”, Commun. Pure Appl. Math. 18, 269–305 (1965; Zbl 0171.35101)] and his own paper [L. Hörmander, “Pseudo-differential operators”, Commun. Pure Appl. Math. 18, 501–517 (1965; Zbl 0125.33401)]. How close these volumes are at cutting edge research of that time is made clear in an indirect way by Hörmander himself in the preface when pointing out that while he often had to rewrite (rearrange, reformulate) the content of his papers for the monograph, from four lengthy, very influential papers he could keep larger passages unchanged. These papers are “The Weyl calculus of pseudo-differential operators” [Commun. Pure Appl. Math. 32, 359–443 (1979; Zbl 0388.47032)], “Subelliptic operators” [Seminar on singularities of solutions of linear partial differential equations, Inst. Adv. Study, Princeton 1977–78, Ann. Math. Stud. 91, 127–208 (1979; Zbl 0446.35086)], “Pseudo-differential operators of principal type” [Singularities in boundary value problems, Proc. NATO Adv. Study Inst., Maratea/Italy 1980, 69–96 (1981; Zbl 0459.35096)] and “Uniqueness theorem for second order elliptic equations” [Commun. Partial Differ. Equations 8, 21–64 (1983; Zbl 0546.35023)]. Thus we are not dealing anymore with a text for advanced graduates (in the mid 1980s), we are dealing with a first rank research monograph at its time written for specialists with a solid background in ongoing research. Often results are polished, new insights added and ideas or the connections were made clearer than they were in the research papers. For the active researcher these volumes became immediately invaluable, newcomers were given some guidance.

Although Hörmander is still developing a theory of general partial differential operators, he starts Volume III with a chapter on second order elliptic equations, partly to set the scene how to switch from constant to variable coefficients, partly to introduce some techniques and ideas, however and most of all to appreciate that sometimes, especially in the context of second order operators, more specialized methods are appropriate and successful. Nonetheless, the subtitles outline the topics which will recurrently show up in the general theory: local existence, inner regularity, uniqueness problems, boundary value problems (especially the Dirichlet problem), parametrix construction, spectral problems.

The next chapter, Chapter XVIII “Pseudo-differential Operators” is with approximately 120 pages the longest in the whole treatise, followed by Chapter XXVI “Pseudo-differential Operators of Principal Type” in Volume IV with approximately 110 pages. Sometimes such a “quantitative analysis” is helpful to reveal more. The “revolution” in the analysis of linear partial differential operators was the emergence of pseudo-differential operators. Singular integrals and singular integral operators were studied since the 1940s in relation to partial differential equations. S. G. Michlin, (who coined the name “symbol”), M. S. Agranovich and I. N. Vekua and others in Russia, and A.P. Calderón and A. Zygmund in the U.S.A. did pioneering work. M. F. Atiyah and I. M. Singer while working on the index theorem now bearing their names, and stimulated partly by a paper of I. M. Gel’fand, made in their analytical considerations more use of these approaches and this triggered the early work on pseudo-differential operators in the mid 1960s.

From the first symbolic calculi which allowed to construct a parametrix for certain elliptic operators it took some hard, very concentrated efforts in a short period of time to clarify central issues and to find the core of the underlying techniques. In Chapter XVIII Hörmander has chosen an approach combining the historical developments with later insights. The basic calculus (Section 18.1) gives us the machinery for dealing already with some concrete problems; note that the micro-local aspect (action on wave front sets) is already included. Then the micro-local aspect is even more emphasized by introducing the co-normal distribution (recall that micro-local analysis is analysis on the co-tangent bundle), and in this context also the transmission condition (introduced by L. Boutet de Monvel) is discussed. Thus tools for handling boundary value problems are introduced. With these tools at hand in 18.3 a detailed analysis of totally characteristic equations is given (relying much on R. Melrose’s work). It turned out quite soon that these first calculi were not adequate for dealing with certain questions arising for some classes of degenerate elliptic or hypoelliptic operators. In his 1967 paper “Pseudo-differential operators and hypoelliptic equations” [Proc. Sympos. Pure Math. 10, 138–183 (1967; Zbl 0167.09603)] Hörmander made an attempt to overcome some of these problems, the Beals-Fefferman calculus was a step further. The unifying approach was however Hörmander’s Weyl calculus and Sections 18.4–18.6 are devoted to this following closely the original paper from 1979. The Weyl calculus gave a further push to the theory, on the one hand it was unifying and covered a lot of results, on the other hand it provided the machinery for new investigations in topics not approachable so far. Knowledge of (at least major parts) of Chapter XVIII is meanwhile expected as standard by every researcher in partial differential equations.

The next two chapters are treating topics related to the index theorem: “Elliptic Operators on Compact Manifolds without Boundary” and “Boundary Problems for Elliptic Differential Operators”. These are key topics and Hörmander provides essentially the analytic part of the theory. There have been a lot of developments since 1985, for example the case of non-smooth manifolds, and we refer to the many accounts written on this central part of 20th century mathematics.

Elliptic differential operators have no (nontrivial) characteristics. This makes their analysis in some sense special and, understood properly, simple. While the first success of pseudo-differential analysis was in the treatment of elliptic operators, introducing the micro-local point of view in the study of non-elliptic operators changed the field completely; a proper understanding of the meaning and impact of characteristics, characteristic manifolds, etc., was achieved. Key to this is the understanding of analysis on the co-tangent bundle, and for this symplectic geometry is a basic tool. Historically, in the context of partial differential equations, symplectic geometry is not as new as it looks at first glance. Hamilton-Jacobi theory (either viewed as part of mechanics, or as part of the theory of differential equations), and, in our context more importantly, geometric optics (especially when discussing the relation of geometric to wave optics) already used notions from symplectic geometry. But it took a long time for clarifying the situation. Sections 21.1–21.3 form a beautiful, almost self-contained introductory course in symplectic geometry, while Sections 21.4–21.6 must be seen more as providing preparatory material needed later on when dealing with Fourier integral operators. The value of this chapter is that it makes important (geometric) background material accessible to a more general audience, in 1985 as well as today.

Clearly, once Hörmander’s book was published, the mathematical community started to quote his book and less often his papers. Thus seeing his 1967 paper “Hypoelliptic second order differential equations” [Acta Math. 119, 147–171 (1967; Zbl 0156.10701)] as his most cited paper might be a distortion with respect to real impact. However this paper had indeed an outstanding influence not only in analysis. The observation that the Kolmogorov equation is hypoelliptic, but not of constant strength and degenerate on a hypersurface, and the hypoellipticity of further second order operators showing up in the theory of several complex variables (of course we have to mention the work of J. J. Kohn and E. M. Stein and their students) led Hörmander to a thorough investigation cumulating in the famous bracket condition for the hypoellipticity of “sums of squares of vector fields”. Stein observed that non-isotropic metrics are key for the understanding of certain problems in several complex variables; in particular his papers [E. M. Stein and L. P. Rothschild, “Hypoelliptic differential operators and nilpotent groups”, Acta Math. 137 (1976), 247–320 (1977; Zbl 0346.35030)] and [A. Nagel, E. M. Stein and S. Wainger, “Balls and metrics defined by vector fields. I: Basic properties”, Acta Math. 155, 103–147 (1985; Zbl 0578.32044)] led to a deeper understanding of Hörmander’s result. These ideas were extended by C. L. Fefferman and D. H. Phong and led R. Strichartz to develop sub-Riemannian geometry. In parallel, P. Malliavin introduced in 1976 his “Stochastic Calculus of Variations” to derive the smoothness of the transition function of diffusions generated by Hörmander-type operators. This calculus, baptized by D. W. Stroock the Malliavin calculus, caused a fundamental change in probability theory, transforming parts of it to stochastic analysis. Yes, this 1967 paper of Hörmander is a seminal contribution with far-reaching consequences and influence. The surprise is that in Chapter XXII “Some Classes of (Micro-)Hypoelliptic Operators” we learn almost nothing about these developments. In Section 22.1 operators with (micro-) pseudo-differential parametrix are discussed, and Section 22.2 gives (an extension of) the famous result under the title “Generalized Kolmogorov Equations”. The final two Sections 22.3, 22.4 deal with more special problems related to hypoellipticity with variable coefficients (A. Melin’s inequality, hypoellipticity with loss of one derivative).

Chapter XXIII now turns to the strictly hyperbolic Cauchy problem for operators with variable coefficients. In Chapter XII this problem was investigated for constant coefficient operators. The first section deals with first order operators which are hyperbolic with respect to a hyperplane. The basic techniques are energy integral estimates. Furthermore the wave front sets of solutions are discussed. In 23.2 higher order operators are treated and the guiding idea is that of factorization. A proper notion of strict hyperbolicity is introduced, corresponding symbols are analysed and existence and regularity results for the (non-characteristic) Cauchy problem are proved. In addition a description of the wave front set of solutions is provided. Section 23.3 gives a necessary condition (due to P. Lax and S. Mizohata) for the correctness of the Cauchy problem. Section 23.4 deals with hyperbolic operators of principal type, proving existence and regularity results (wave front sets, propagation of singularities). The next chapter continues the discussion of the Cauchy problem, more precisely we are now dealing with the mixed Dirichlet-Cauchy problem for second order operators, one of the central topics of classical mathematical physics. This chapter introduces also further techniques and notions needed for a more detailed discussion of the propagation of singularities of solutions. First energy estimates are derived leading to existence and regularity results. It follows a discussion of the singularities (wave front sets) in the elliptic and the hyperbolic regions and the role of bicharacteristics emerges. The bicharacteristic flow is covered in 24.3 followed by a discussion of the diffractive case. Eventually in 24.5 the main results on the propagation of singularities are proved. The chapter closes with a detailed look at the Tricomi equation and a section on parameter dependent operators.

Volume III closes with two appendices, one on some spaces of distributions, the other one covers material from differential geometry.

To some extent the results in Chapter XXIII and XXIV are the most beautiful achievements of micro-local analysis to which many authors have contributed. They give formulations and proofs to very central and classical problems in the theory of partial differential equations. Micro-local analysis, i.e., working on the co-tangent bundle (where symbols are defined) to study existence and to analyse wave front sets and the propagation of singularities, turns out to be the appropriate frame. It is however noteworthy that certain underlying ideas are classical. They are rooted, for example, in the complementary relations of wave and geometric optics. These two chapters provide an example of how Hörmander deviates from the historical development when at the time of writing the book a much better understanding and hence a more insightful approach was available for treating certain problems. Wave front sets and the propagation of singularities were a topic which emerged and was developed with the introduction of Fourier integral operators in the early 1970s. Meanwhile, however, these results are completely understood while working just with pseudo-differential operators.

Volume IV starts with introducing Fourier integral operators. Historically one must have in mind P. Lax’s paper [“Asymptotic solutions of oscillatory initial value problems”, Duke Math. J. 24, 627–646 (1957; Zbl 0083.31801)] where a systematic attempt was made to solve initial value problems by integrals (or integral operators) allowing a phase function different from that used for the Fourier transform. Similar ideas were also behind V. P. Maslov’s work in the 1960s. Hörmander’s paper [“Fourier integral operators I”, Acta Math. 127, 79–183 (1971; Zbl 0212.46601)] and his joint paper with J. J. Duistermaat [“Fourier integral operators II”, Acta Math. 128, 183–269 (1972; Zbl 0232.47055)] developed a complete, self-contained global theory. Chapter XXV introduces first Lagrangian distributions and then the calculus of Fourier integral operators and it gives some applications, for example a further proof of the propagation of singularity result. Fourier integral operators with complex phase functions are also treated.

We have already mentioned that Chapter XXVI “Pseudo-differential Operators of Principal Type” is the second longest in the whole treatise. The basic problem which led to this rather involved part of the theory is H. Lewy’s example of a partial differential equation with arbitrarily often differentiable (complex-valued) coefficients which has no solution. Lewy’s example is related to complex analysis (of several variables) and has a certain geometric background. Pseudo-differential operators of principal type are operators which allow a homogenous principal symbol satisfying some additional geometric conditions (geometry here refers to characteristics, or better bicharacteristics). The case of a real principal symbol is treated in 26.1 and compared with the complex-valued case its treatment is quite straightforward. The central problem for the complex-valued case is that of local solvability. Besides Hörmander’s contribution we must mention the seminal work of F. Treves and L. Nirenberg (condition (\(\Psi\))), Yu. Egorov (conjugating with Fourier integral operators) and R. Beals and C. L. Fefferman (localization). The results in the area form a further success story of micro-local analysis. At the time of writing the books condition (\(\Psi\)) was conjectured to be necessary for solvability, Section 26.4 gives a proof of this conjecture using ideas of R.D.Moyer. Whether condition (\(\Psi\)) is sufficient was an open question in 1985. Hörmander is using a different condition, condition (P) introduced earlier by F. Treves, and shows that (P) implies local solvability; see 26.11 for the result, 26.5–26.10 provide the proof. Only in 2006 could N. Dencker prove that (\(\Psi\)) implies local solvability [“The resolution of the Nirenberg-Treves conjecture”, Ann. Math. (2) 163, No. 2, 405–444 (2006; Zbl 1104.35080)].

Hörmander’s thesis gave a characterization of hypoelliptic operators with constant coefficients. Hypoellipticity can be defined as preserving the singular support: \(\mathrm{sing\,supp}\, u = \mathrm{sing\,supp}\, Pu\). Ellipticity can be defined for an operator of order \(m\) as follows: For every \(s\), the solution to \(Pu = f\) is locally in the Sobolev space of order \(m+s\) provided \(f\) is locally in the space of order \(s\). Sub-ellipticity of loss of derivatives of order \(\delta\) (\(0<\delta<1\)) is the condition that \(f\) locally in the Sobolev space of order \(s\) implies that \(u\) is locally in the Sobolev space of order \(m+s-\delta\). This notion was introduced by Hörmander, note that \(\delta < 1\) implies that the condition depends on the principal symbol only. Chapter XXVII is devoted to a complete study of subelliptic operators. Here Hörmander follows essentially and very closely his paper “Subelliptic Operators” from 1979 cited above [Zbl 0446.35086]. The work on the main result, Theorem 27.1.11, starts with F. Treves, owes much to Yu. Egorov, and was completed by Hörmander. The entire chapter is essentially devoted to the proof of Theorem 27.1.11.

In Chapter XXVIII the uniqueness for the Cauchy problem is discussed, the starting point is A.P. Calderón’s uniqueness result, Section 28.1. Basic tools are Carleman estimates which are further investigated (for operators with real principal symbol) in Section 28.2 and this leads to further uniqueness results for which convexity or pseudo-convexity conditions are needed.

Partial differential equations are important tools in mathematical physics. Often their spectral theory is of more interest than explicit solutions, or sophisticated regularity results, etc. The machinery of micro-local analysis, in particular Fourier integral operators, provided further tools and led to new deep insights in this area. At the end of his treatise Hörmander picks up two topics related to spectral theory: in Chapter XXIX spectral asymptotics, more precisely asymptotic properties of eigenvalues, and in Chapter XXX long range scattering, where a central question is that of asymptotic completeness. Spectral analysis has developed dramatically since the 1970s, and many authors have made significant contributions. Hörmander’s presentation was at its time a helpful synthesis for the two topics being treated, i.e., asymptotic properties of eigenvalues and long range scattering.

I doubt that even such a long report can do full justice to Hörmander’s achievement and the impact of these volumes. We may refer further to the citation when he was awarded the Steele Prize in 2006 for mathematical exposition. In the last 30 years the analysis of partial differential operators has developed enormously. Clearly, nonlinear problems are now in the centre of research, Hörmander himself made important contributions to the study of nonlinear hyperbolic equations, J.-M. Bony’s work on micro-local analysis (propagation of singularities) must be mentioned. In mathematical physics, for example due to the work of B. Helffer and J. Sjöstrand, much progress has been made by using methods highlighted in Hörmander’s treatise. In the theory of elliptic operators non-smooth problems were treated, for example symbolic calculi for boundary value problems on manifolds with corners or edges (R. Melrose, B.-W. Schulze). This list can easily be extended. We have to leave it to scholars in the history of mathematics to come up with a more careful and better justified appraisal of Hörmander’s treatise in 50 or 100 years. But I am convinced that these volumes will withstand the test of time as will Hörmander’s contribution to mathematics.

When I was asked to write about Lars Hörmander’s treatise “Analysis of Linear Partial Differential Operators” I accepted with some hesitation only after clarifying for myself what is not asked for: a history of partial differential equations since 1983/85 on a few pages. For such a study time is not yet ready and a few pages will not suffice. Moreover there is no need to praise once more these volumes and their impact, and in particular the outstanding achievements and the scholarship of Lars Hörmander. This has been done several times and by more competent authors. What I try to do is to add a few, hopefully acceptable lines of interpretation of the impact that Hörmander’s work, which he cumulated in these volumes, has had. Some emphasis will be on developments after the publication of these volumes.

The starting point must be the publication of Hörmander’s thesis in 1955 [“On the theory of general partial differential operators”, Acta Math. 94, 161–248 (1955; Zbl 0067.32201)] which was published in the same volume of Acta Mathematica as J. L. Lions’s thesis [“Problèmes aux limites en théorie des distributions”, Acta Math. 94, 13–153 (1955; Zbl 0068.30902)]. Both theses put into their centre of investigation the use of functional analysis, for example the one or the other form of (L. Schwartz’s) distributions or generalized functions, in the study of partial differential equations. The extent to which this was done was absolutely new and transformed the field. While Lions dealt with (elliptic) boundary value problems, Hörmander set out a new programme “The Theory of General Partial Differential Operators” (influenced by his studies of I. G. Petrowsky’s work and arguably initiated by L. Gårding). The key features of Hörmander’s thesis are, besides the consequent use of more (abstract) functional analysis, that general questions are posed, for example when is a solution of a given partial differential equation smooth (arbitrarily often differentiable) given locally smooth data, and that the Fourier transform as a tool is systematically applied to transform a problem related to partial differential equations into a problem for the corresponding symbol (in the language we use now). Clearly there are predecessors to this idea, for example L. Gårding’s work on hyperbolic equations with constant coefficients, and not least I. G. Petrowsky’s work. But in Hörmander’s thesis we see a clear pattern: from the differential equation to the differential operator, from the differential operator to the symbol, use properties of the symbol to investigate the operator (mainly by deriving appropriate estimates), and then return to the equation. The characterization of hypoelliptic operators with constant coefficients can be seen as a key example. In the thesis some new tools were introduced such as norms and function spaces adapted to the operator, better its symbol, or comparisons of symbols entailing the notion of operators of equal strength and comparisons (in terms of estimates) of such operators.

Much of this programme and many more results proven along these lines were presented in Hörmander’s book “Linear Partial Differential Operators” published by Springer in 1963 [Die Grundlehren der mathematischen Wissenschaften 116 (1963; Zbl 0108.09301)], now 50 years ago and this justifies in addition to stay with that book for some time. Compared with other texts on partial differential equations of that time the book was unusual, but immediately recognised as a ground breaking contribution. The trinity of ellipticity, hyperbolicity, and parabolicity is not anymore the classification scheme, but questions are posed such as: Which operators have locally smooth solutions given locally smooth data, when is a solution always an analytic function, for which operators is the non-characteristic Cauchy problem well-posed, etc. Moreover classical tools like special solutions, series expansions and separation of variables, or ad hoc constructions of Green’s functions, were not treated. But we see for constant coefficient operators topics such as fundamental solutions (Malgrange-Ehrenpreis theorem) and their properties, interior regularity results, the non-characteristic and the characteristic Cauchy problem for general operators, global uniqueness and the Holmgren theorem, etc., treated. Interesting for later purpose is to note here that besides the notions of elliptic, hypoelliptic and hyperbolic operators the 1963 text still contains the notion of parabolic operators (much in the sense of I. G. Petrowsky). The four-volume treatise will not see anymore the notion of parabolic operators in the index. The way of treating operators with variable coefficients was quite surprising and new at the time. The first chapter is devoted to operators with no solutions, of course much influenced by the example of H. Lewy. It is followed by a discussion of operators with constant strength and local existence problems depending on the smallness of the domain or the coefficients. Of interest here is how the local solution operator is, via the Schwartz kernel theorem, related to Green’s functions and in some sense approximate fundamental solutions. Interesting is also how, in the case of smooth coefficients, bounds for the solution operator are derived. Special attention is paid to ellipticity. Already in the case of constant coefficient operators ellipticity was linked to analyticity, now this topic is taken up in the case of variable coefficient operators – again an answer to a general question posed to establish a theory of general differential operators. It follows a treatment of the Cauchy problem for strictly hyperbolic operators with smooth coefficients based on \(L^2\)-techniques. Finally elliptic boundary value problems along the lines proposed by Ya. B. Lopatinski and Z. Ya. Shapiro are discussed.

Spending time on the book published 50 years ago is important in order to understand how much the landscape changed in the 1960s and 1970s leading to the theory discussed in the treatise “The Analysis of Linear Partial Differential Operators, I–IV”. Volumes I and II were published jointly in 1983 followed by Volumes III and IV in 1985. At first sight Volumes I and II look like an extension and update of the 1963 book. But this is correct only to some extent. The emergence of micro-local analysis changed the field, already when dealing with distribution theory, and some deeper results for variable coefficient operators using micro-local analysis are easier understood when first discussed for constant coefficient operators. Let us be more systematical. We can look at Volume I, which has the subtitle “Distribution Theory and Fourier Analysis”, as a rather advanced textbook for a core area of modern analysis. Hörmander clearly had the intention to write such a book, and by supplying problems and solutions in the second edition of this volume this is made even clearer. Calculus in higher dimensions is presented in such a way that later on we can immediately turn to manifolds, but more importantly the key objects needed are carefully introduced: test functions, cut-off functions and partitions of unity, as well as convolutions of functions. The distributions are introduced, rather down-to-earth, no abstract theory of topological vector spaces is needed, and we can find essentially all about distributions that is needed later on (or even beyond) and often illustrated by carefully chosen examples. This is followed by a chapter on basic operations for distributions and again many, many illuminating examples are provided. The emphasis is on what is needed in concrete studies (for example homogeneous distributions) and already here we encounter many concrete fundamental solutions for differential operators. The chapter on convolution is not just devoted to extend a certain operation from functions to distributions. It has as a clear aim the support theorem in full generality in mind and most of all the relation to fundamental solutions for operators with constant coefficients. Tensor products and the kernel theorem follow with a rather “concrete” proof. Micro-local analysis is in some sense analysis related to objects defined on the co-tangent bundle. Chapter VI introduces distributions on manifolds and then discusses the tangent and the co-tangent bundle, again having in mind later needs when dealing with differential operators.

The approximately 90 pages of Chapter VII on the Fourier transform constitute a masterpiece in itself. We are introduced to basic results on the Schwartz space and on the space of tempered distributions, a lot of examples are discussed, but also topics such as the Malgrange Preparation Theorem, the method of stationary phase, or oscillatory integrals are all dealt with in detail. We will need this later on. In particular a paragraph completely devoted to the Fourier transform of Gaussians is provided and further the use of the Fourier transform to obtain fundamental solutions is investigated. Here we also find a discussion of the fundamental kernel of the Kolmogorov operator, an operator with variable coefficients, preparing ground for the famous “sum of squares” result which we will meet in Volume III. Hörmander is usually working in spaces of Schwartz distributions or in an \(L^2\)-context. The \(L^p\)-theory of the Fourier transform and its applications is touched upon only briefly. However for this we have the E. M. Stein school and Stein’s monumental monographs. What was covered in the 1963 book on 33 pages is now covered on 250 pages. Meanwhile (since 1983) new books on distribution theory and the Fourier transform have been published, and some are really good, for example [J.-M. Bony, Cours d’analyse. Théorie des distributions et analyse de Fourier. Les Éditions de l’École Polytechnique (2001; Zbl 1229.46002)], or [J. J. Duistermaat and J. A. C. Kolk, Distributions. Theory and applications. Birkhäuser, Basel (2010; Zbl 1213.46001)], or [G. Grubb, Distributions and operators. Graduate Texts in Mathematics 252. Springer (2009; Zbl 1171.47001)]. These are great books adding one or the other new aspect, allowing some new insights, or giving some original presentations. However Hörmander’s treatment of the topic stays unsurpassed.

We are not yet at the end of Volume I. Chapter VIII introduces us to what caused really the change in the analysis of partial differential operators in the 1970s: micro-local analysis as a tool for the study of singularities of solutions of partial differential equations. The wave front set of a distribution is introduced (not using characteristic sets of differential operators as originally, but using the Fourier analytic approach, i.e., looking at directions of non-decay of the Fourier transform of the localized distribution), the behaviour of wave front sets under operations of distributions is discussed, and then we see the modern machinery working: a first discussion of wave front sets of solutions of partial differential equations with constant coefficients. We learn already how the characteristic set of a differential operator enters into the picture, we get a micro-local understanding of ellipticity, we see interesting examples, we start to understand the rule of the principal part of a differential operator, we have a first glance at the propagation of singularities, and micro-hyperbolicity is introduced. Here we start to realize how central it is to turn from the operator to its symbol. The symbol as an object defined on the co-tangent bundle “interacts” with wave front sets and characteristic sets both naturally considered as subsets of the co-tangent bundle. All this is first treated in the category of smooth functions, but it is then also treated in the context of \(C^L\)-functions, i.e., functions where the derivatives have a certain decay (an extension of Gevrey classes), and the Holmgren uniqueness theorem is discussed. While the emergence of micro-local analysis was a highlight of analysis in the 1970s, this chapter is a further highlight in Volume I. On a technical level which should be accessible to graduate students, key ideas and results are presented, but much more is achieved: we are prepared to study and understand the deep results in the later volumes.

The final chapter in Volume I treats hyperfunctions. The theory is developed as much as it is needed to understand (later on) analytic regularity.

I deliberately spent much time on Volume I. Nowadays distribution theory, Fourier analysis (of distributions), wave front sets and micro-local analysis are standard tools. These tools are used in many areas, not least in mathematical physics. Needless to say that also the study of (certain) non-linear problems requires a mastery in using these tools. Hörmander’s Volume I make these tools accessible to many mathematicians. If 30 years after the publication of this treatise someone wants to start to learn micro-local analysis, this is still the best source. The 1983 research monograph is an advanced textbook in 2013, a witness how the field has developed, but also a witness of the lasting impact of the author.

Volume II is quite adequately described in its preface: “This volume is an expanded version of Chapter III, IV, V and VII of my 1963 book. In addition there is an entire new chapter on convolution equations, one on scattering theory, and one on methods from analytic functions of several variables.” And further, “Although the field is no longer very active – perhaps because of its advanced state of development – and although it is possible to pass directly from Volume I to Volume III, the material presented here should not be neglected by the serious student.”

First a remark on the three new chapters. The inclusion of a chapter on scattering theory (Chapter XIV) looks at first glance surprising. Obviously an important topic, but not really a part of the “theory of general partial differential operators”. There are however good reasons to include such a chapter. We see how one of the major tools, the Fourier (Laplace) transform, works also for these types of problems, and moreover Chapter XXX, in some sense also Chapter XXIX are prepared. We are reminded that in the mid 1970s and early 1980s scattering theory was a hot topic and micro-local analysis as a tool in spectral theory was much used. Further Enss’s papers (the “Enss method”) were seen to be promising to lead to serious progress in the problem of asymptotic completeness. Chapter XV on analytic function theory is an addition, in Hörmander’s own words “limited in scope” due to the existence of more specialized monographs such as those by L. Ehrenpreis [Fourier analysis in several complex variables. Pure and Applied Mathematics 17. Wiley-Interscience Publishers, New York (1970; Zbl 0195.10401)] or V. P. Palamodov [Linear differential operators with constant coefficients. Die Grundlehren der mathematischen Wissenschaften 168. Springer (1970; Zbl 0191.43401)]. The aim of this chapter is to develop some analytical techniques and to supply some simplified proofs. Again, the Fourier-Laplace transform and \(L^2\)-estimates are the basic tools. From these three new chapters, Chapter XVI on convolution equations is the most interesting one. It is a comprehensive presentation in the spirit of the entire treatise of results known on this topic until the early 1980s. These equations are so naturally linked to partial differential operators that they have a natural place in this volume. Of course, it is the Fourier transform which makes these links more apparent. In addition, having in mind Hörmander’s monograph “Notions of Convexity” [Birkhäuser, Basel (1994; Zbl 0835.32001)], this chapter prepares more. The complex of geometry (especially convexity), analyticity and solving certain classes of equations is in some sense scattered over the whole treatise.

Back to the revised chapters of the 1963 book. Chapter X (Existence and Approximation of Solutions of Differential Equations) and Chapter XI (Interior Regularity of Solutions of Differential Equations) are updates of the corresponding chapters. Chapter XII (The Cauchy and Mixed Problem, Constant Coefficients) is more. We first have a very detailed discussion of the wave equation including the oscillatory Cauchy problem and these two paragraphs dealing with this topic skilfully prepare core topics in micro-local analysis. The “classical results” are rearranged, the singularities of fundamental solutions are investigated in detail. The investigations of the characteristic Cauchy problems are considerably extended. It is interesting to note that the notion of “parabolic operator” has disappeared. In fact all four volumes have no place for dissipative operators generating one-parameter semi-groups. Having in mind the importance of heat kernels in spectral theory or in index theory this is a bit surprising. The mixed problem studied for constant coefficient operators considers Cauchy data on one hypersurface and boundary data (Lopatinski-Shapiro type) on a further hypersurface. We see the interplay of hyperbolic theory (Cauchy data) and elliptic theory (boundary data). Again a systematic use of the Fourier transform “reduces” much to the study of the corresponding symbols. These “updates” of the constant coefficient operator case is a wonderful presentation of a now “classical” subject, very valuable for serious students, and much helpful when moving to variable coefficient operators.

Having in mind that Volumes III and IV are devoted to variable coefficient operators, it is not surprising that Part III of the 1963 book “Differential Operators with Variable Coefficients” features in Volume II of the 1983/85 treatise only with a chapter on operators of constant strength. The basic definitions and basic local existence results are discussed, but for regularity questions the micro-local aspect is added, i.e., the behaviour of wave front sets. Global existence results are included as is a discussion of non-uniqueness of the Cauchy problem. Some interesting examples are provided.

The impact of Volume II is difficult to judge. The topic itself is central and absolutely important for a general theory. The presentation is full of insights, it is difficult to imagine that it will be surpassed in the next decades. It presents most of all a lot of ideas explored in the coming volumes by looking at simple examples, and it familiarizes us with certain techniques, however this volume could not have stimulated much new research.

When now turning to Volume III (with the subtitle “Pseudo-differential Operators”) and IV (with the subtitle “Fourier Integral Operators”) we have to change our point of view and our expectations, as Hörmander changed and had to change his intentions. These two volumes try to summarize and partly to clarify in a comprehensive manner the developments of the previous 20 years, say starting with the Kohn-Nirenberg paper [J. J. Kohn and L. Nirenberg, “An algebra of pseudo-differential operators”, Commun. Pure Appl. Math. 18, 269–305 (1965; Zbl 0171.35101)] and his own paper [L. Hörmander, “Pseudo-differential operators”, Commun. Pure Appl. Math. 18, 501–517 (1965; Zbl 0125.33401)]. How close these volumes are at cutting edge research of that time is made clear in an indirect way by Hörmander himself in the preface when pointing out that while he often had to rewrite (rearrange, reformulate) the content of his papers for the monograph, from four lengthy, very influential papers he could keep larger passages unchanged. These papers are “The Weyl calculus of pseudo-differential operators” [Commun. Pure Appl. Math. 32, 359–443 (1979; Zbl 0388.47032)], “Subelliptic operators” [Seminar on singularities of solutions of linear partial differential equations, Inst. Adv. Study, Princeton 1977–78, Ann. Math. Stud. 91, 127–208 (1979; Zbl 0446.35086)], “Pseudo-differential operators of principal type” [Singularities in boundary value problems, Proc. NATO Adv. Study Inst., Maratea/Italy 1980, 69–96 (1981; Zbl 0459.35096)] and “Uniqueness theorem for second order elliptic equations” [Commun. Partial Differ. Equations 8, 21–64 (1983; Zbl 0546.35023)]. Thus we are not dealing anymore with a text for advanced graduates (in the mid 1980s), we are dealing with a first rank research monograph at its time written for specialists with a solid background in ongoing research. Often results are polished, new insights added and ideas or the connections were made clearer than they were in the research papers. For the active researcher these volumes became immediately invaluable, newcomers were given some guidance.

Although Hörmander is still developing a theory of general partial differential operators, he starts Volume III with a chapter on second order elliptic equations, partly to set the scene how to switch from constant to variable coefficients, partly to introduce some techniques and ideas, however and most of all to appreciate that sometimes, especially in the context of second order operators, more specialized methods are appropriate and successful. Nonetheless, the subtitles outline the topics which will recurrently show up in the general theory: local existence, inner regularity, uniqueness problems, boundary value problems (especially the Dirichlet problem), parametrix construction, spectral problems.

The next chapter, Chapter XVIII “Pseudo-differential Operators” is with approximately 120 pages the longest in the whole treatise, followed by Chapter XXVI “Pseudo-differential Operators of Principal Type” in Volume IV with approximately 110 pages. Sometimes such a “quantitative analysis” is helpful to reveal more. The “revolution” in the analysis of linear partial differential operators was the emergence of pseudo-differential operators. Singular integrals and singular integral operators were studied since the 1940s in relation to partial differential equations. S. G. Michlin, (who coined the name “symbol”), M. S. Agranovich and I. N. Vekua and others in Russia, and A.P. Calderón and A. Zygmund in the U.S.A. did pioneering work. M. F. Atiyah and I. M. Singer while working on the index theorem now bearing their names, and stimulated partly by a paper of I. M. Gel’fand, made in their analytical considerations more use of these approaches and this triggered the early work on pseudo-differential operators in the mid 1960s.

From the first symbolic calculi which allowed to construct a parametrix for certain elliptic operators it took some hard, very concentrated efforts in a short period of time to clarify central issues and to find the core of the underlying techniques. In Chapter XVIII Hörmander has chosen an approach combining the historical developments with later insights. The basic calculus (Section 18.1) gives us the machinery for dealing already with some concrete problems; note that the micro-local aspect (action on wave front sets) is already included. Then the micro-local aspect is even more emphasized by introducing the co-normal distribution (recall that micro-local analysis is analysis on the co-tangent bundle), and in this context also the transmission condition (introduced by L. Boutet de Monvel) is discussed. Thus tools for handling boundary value problems are introduced. With these tools at hand in 18.3 a detailed analysis of totally characteristic equations is given (relying much on R. Melrose’s work). It turned out quite soon that these first calculi were not adequate for dealing with certain questions arising for some classes of degenerate elliptic or hypoelliptic operators. In his 1967 paper “Pseudo-differential operators and hypoelliptic equations” [Proc. Sympos. Pure Math. 10, 138–183 (1967; Zbl 0167.09603)] Hörmander made an attempt to overcome some of these problems, the Beals-Fefferman calculus was a step further. The unifying approach was however Hörmander’s Weyl calculus and Sections 18.4–18.6 are devoted to this following closely the original paper from 1979. The Weyl calculus gave a further push to the theory, on the one hand it was unifying and covered a lot of results, on the other hand it provided the machinery for new investigations in topics not approachable so far. Knowledge of (at least major parts) of Chapter XVIII is meanwhile expected as standard by every researcher in partial differential equations.

The next two chapters are treating topics related to the index theorem: “Elliptic Operators on Compact Manifolds without Boundary” and “Boundary Problems for Elliptic Differential Operators”. These are key topics and Hörmander provides essentially the analytic part of the theory. There have been a lot of developments since 1985, for example the case of non-smooth manifolds, and we refer to the many accounts written on this central part of 20th century mathematics.

Elliptic differential operators have no (nontrivial) characteristics. This makes their analysis in some sense special and, understood properly, simple. While the first success of pseudo-differential analysis was in the treatment of elliptic operators, introducing the micro-local point of view in the study of non-elliptic operators changed the field completely; a proper understanding of the meaning and impact of characteristics, characteristic manifolds, etc., was achieved. Key to this is the understanding of analysis on the co-tangent bundle, and for this symplectic geometry is a basic tool. Historically, in the context of partial differential equations, symplectic geometry is not as new as it looks at first glance. Hamilton-Jacobi theory (either viewed as part of mechanics, or as part of the theory of differential equations), and, in our context more importantly, geometric optics (especially when discussing the relation of geometric to wave optics) already used notions from symplectic geometry. But it took a long time for clarifying the situation. Sections 21.1–21.3 form a beautiful, almost self-contained introductory course in symplectic geometry, while Sections 21.4–21.6 must be seen more as providing preparatory material needed later on when dealing with Fourier integral operators. The value of this chapter is that it makes important (geometric) background material accessible to a more general audience, in 1985 as well as today.

Clearly, once Hörmander’s book was published, the mathematical community started to quote his book and less often his papers. Thus seeing his 1967 paper “Hypoelliptic second order differential equations” [Acta Math. 119, 147–171 (1967; Zbl 0156.10701)] as his most cited paper might be a distortion with respect to real impact. However this paper had indeed an outstanding influence not only in analysis. The observation that the Kolmogorov equation is hypoelliptic, but not of constant strength and degenerate on a hypersurface, and the hypoellipticity of further second order operators showing up in the theory of several complex variables (of course we have to mention the work of J. J. Kohn and E. M. Stein and their students) led Hörmander to a thorough investigation cumulating in the famous bracket condition for the hypoellipticity of “sums of squares of vector fields”. Stein observed that non-isotropic metrics are key for the understanding of certain problems in several complex variables; in particular his papers [E. M. Stein and L. P. Rothschild, “Hypoelliptic differential operators and nilpotent groups”, Acta Math. 137 (1976), 247–320 (1977; Zbl 0346.35030)] and [A. Nagel, E. M. Stein and S. Wainger, “Balls and metrics defined by vector fields. I: Basic properties”, Acta Math. 155, 103–147 (1985; Zbl 0578.32044)] led to a deeper understanding of Hörmander’s result. These ideas were extended by C. L. Fefferman and D. H. Phong and led R. Strichartz to develop sub-Riemannian geometry. In parallel, P. Malliavin introduced in 1976 his “Stochastic Calculus of Variations” to derive the smoothness of the transition function of diffusions generated by Hörmander-type operators. This calculus, baptized by D. W. Stroock the Malliavin calculus, caused a fundamental change in probability theory, transforming parts of it to stochastic analysis. Yes, this 1967 paper of Hörmander is a seminal contribution with far-reaching consequences and influence. The surprise is that in Chapter XXII “Some Classes of (Micro-)Hypoelliptic Operators” we learn almost nothing about these developments. In Section 22.1 operators with (micro-) pseudo-differential parametrix are discussed, and Section 22.2 gives (an extension of) the famous result under the title “Generalized Kolmogorov Equations”. The final two Sections 22.3, 22.4 deal with more special problems related to hypoellipticity with variable coefficients (A. Melin’s inequality, hypoellipticity with loss of one derivative).

Chapter XXIII now turns to the strictly hyperbolic Cauchy problem for operators with variable coefficients. In Chapter XII this problem was investigated for constant coefficient operators. The first section deals with first order operators which are hyperbolic with respect to a hyperplane. The basic techniques are energy integral estimates. Furthermore the wave front sets of solutions are discussed. In 23.2 higher order operators are treated and the guiding idea is that of factorization. A proper notion of strict hyperbolicity is introduced, corresponding symbols are analysed and existence and regularity results for the (non-characteristic) Cauchy problem are proved. In addition a description of the wave front set of solutions is provided. Section 23.3 gives a necessary condition (due to P. Lax and S. Mizohata) for the correctness of the Cauchy problem. Section 23.4 deals with hyperbolic operators of principal type, proving existence and regularity results (wave front sets, propagation of singularities). The next chapter continues the discussion of the Cauchy problem, more precisely we are now dealing with the mixed Dirichlet-Cauchy problem for second order operators, one of the central topics of classical mathematical physics. This chapter introduces also further techniques and notions needed for a more detailed discussion of the propagation of singularities of solutions. First energy estimates are derived leading to existence and regularity results. It follows a discussion of the singularities (wave front sets) in the elliptic and the hyperbolic regions and the role of bicharacteristics emerges. The bicharacteristic flow is covered in 24.3 followed by a discussion of the diffractive case. Eventually in 24.5 the main results on the propagation of singularities are proved. The chapter closes with a detailed look at the Tricomi equation and a section on parameter dependent operators.

Volume III closes with two appendices, one on some spaces of distributions, the other one covers material from differential geometry.

To some extent the results in Chapter XXIII and XXIV are the most beautiful achievements of micro-local analysis to which many authors have contributed. They give formulations and proofs to very central and classical problems in the theory of partial differential equations. Micro-local analysis, i.e., working on the co-tangent bundle (where symbols are defined) to study existence and to analyse wave front sets and the propagation of singularities, turns out to be the appropriate frame. It is however noteworthy that certain underlying ideas are classical. They are rooted, for example, in the complementary relations of wave and geometric optics. These two chapters provide an example of how Hörmander deviates from the historical development when at the time of writing the book a much better understanding and hence a more insightful approach was available for treating certain problems. Wave front sets and the propagation of singularities were a topic which emerged and was developed with the introduction of Fourier integral operators in the early 1970s. Meanwhile, however, these results are completely understood while working just with pseudo-differential operators.

Volume IV starts with introducing Fourier integral operators. Historically one must have in mind P. Lax’s paper [“Asymptotic solutions of oscillatory initial value problems”, Duke Math. J. 24, 627–646 (1957; Zbl 0083.31801)] where a systematic attempt was made to solve initial value problems by integrals (or integral operators) allowing a phase function different from that used for the Fourier transform. Similar ideas were also behind V. P. Maslov’s work in the 1960s. Hörmander’s paper [“Fourier integral operators I”, Acta Math. 127, 79–183 (1971; Zbl 0212.46601)] and his joint paper with J. J. Duistermaat [“Fourier integral operators II”, Acta Math. 128, 183–269 (1972; Zbl 0232.47055)] developed a complete, self-contained global theory. Chapter XXV introduces first Lagrangian distributions and then the calculus of Fourier integral operators and it gives some applications, for example a further proof of the propagation of singularity result. Fourier integral operators with complex phase functions are also treated.

We have already mentioned that Chapter XXVI “Pseudo-differential Operators of Principal Type” is the second longest in the whole treatise. The basic problem which led to this rather involved part of the theory is H. Lewy’s example of a partial differential equation with arbitrarily often differentiable (complex-valued) coefficients which has no solution. Lewy’s example is related to complex analysis (of several variables) and has a certain geometric background. Pseudo-differential operators of principal type are operators which allow a homogenous principal symbol satisfying some additional geometric conditions (geometry here refers to characteristics, or better bicharacteristics). The case of a real principal symbol is treated in 26.1 and compared with the complex-valued case its treatment is quite straightforward. The central problem for the complex-valued case is that of local solvability. Besides Hörmander’s contribution we must mention the seminal work of F. Treves and L. Nirenberg (condition (\(\Psi\))), Yu. Egorov (conjugating with Fourier integral operators) and R. Beals and C. L. Fefferman (localization). The results in the area form a further success story of micro-local analysis. At the time of writing the books condition (\(\Psi\)) was conjectured to be necessary for solvability, Section 26.4 gives a proof of this conjecture using ideas of R.D.Moyer. Whether condition (\(\Psi\)) is sufficient was an open question in 1985. Hörmander is using a different condition, condition (P) introduced earlier by F. Treves, and shows that (P) implies local solvability; see 26.11 for the result, 26.5–26.10 provide the proof. Only in 2006 could N. Dencker prove that (\(\Psi\)) implies local solvability [“The resolution of the Nirenberg-Treves conjecture”, Ann. Math. (2) 163, No. 2, 405–444 (2006; Zbl 1104.35080)].

Hörmander’s thesis gave a characterization of hypoelliptic operators with constant coefficients. Hypoellipticity can be defined as preserving the singular support: \(\mathrm{sing\,supp}\, u = \mathrm{sing\,supp}\, Pu\). Ellipticity can be defined for an operator of order \(m\) as follows: For every \(s\), the solution to \(Pu = f\) is locally in the Sobolev space of order \(m+s\) provided \(f\) is locally in the space of order \(s\). Sub-ellipticity of loss of derivatives of order \(\delta\) (\(0<\delta<1\)) is the condition that \(f\) locally in the Sobolev space of order \(s\) implies that \(u\) is locally in the Sobolev space of order \(m+s-\delta\). This notion was introduced by Hörmander, note that \(\delta < 1\) implies that the condition depends on the principal symbol only. Chapter XXVII is devoted to a complete study of subelliptic operators. Here Hörmander follows essentially and very closely his paper “Subelliptic Operators” from 1979 cited above [Zbl 0446.35086]. The work on the main result, Theorem 27.1.11, starts with F. Treves, owes much to Yu. Egorov, and was completed by Hörmander. The entire chapter is essentially devoted to the proof of Theorem 27.1.11.

In Chapter XXVIII the uniqueness for the Cauchy problem is discussed, the starting point is A.P. Calderón’s uniqueness result, Section 28.1. Basic tools are Carleman estimates which are further investigated (for operators with real principal symbol) in Section 28.2 and this leads to further uniqueness results for which convexity or pseudo-convexity conditions are needed.

Partial differential equations are important tools in mathematical physics. Often their spectral theory is of more interest than explicit solutions, or sophisticated regularity results, etc. The machinery of micro-local analysis, in particular Fourier integral operators, provided further tools and led to new deep insights in this area. At the end of his treatise Hörmander picks up two topics related to spectral theory: in Chapter XXIX spectral asymptotics, more precisely asymptotic properties of eigenvalues, and in Chapter XXX long range scattering, where a central question is that of asymptotic completeness. Spectral analysis has developed dramatically since the 1970s, and many authors have made significant contributions. Hörmander’s presentation was at its time a helpful synthesis for the two topics being treated, i.e., asymptotic properties of eigenvalues and long range scattering.

I doubt that even such a long report can do full justice to Hörmander’s achievement and the impact of these volumes. We may refer further to the citation when he was awarded the Steele Prize in 2006 for mathematical exposition. In the last 30 years the analysis of partial differential operators has developed enormously. Clearly, nonlinear problems are now in the centre of research, Hörmander himself made important contributions to the study of nonlinear hyperbolic equations, J.-M. Bony’s work on micro-local analysis (propagation of singularities) must be mentioned. In mathematical physics, for example due to the work of B. Helffer and J. Sjöstrand, much progress has been made by using methods highlighted in Hörmander’s treatise. In the theory of elliptic operators non-smooth problems were treated, for example symbolic calculi for boundary value problems on manifolds with corners or edges (R. Melrose, B.-W. Schulze). This list can easily be extended. We have to leave it to scholars in the history of mathematics to come up with a more careful and better justified appraisal of Hörmander’s treatise in 50 or 100 years. But I am convinced that these volumes will withstand the test of time as will Hörmander’s contribution to mathematics.

Reviewer: Niels Jacob (Swansea) (2013)

### MathOverflow Questions:

Existence of solutions to first-order PDE involving convolutionProve that a given distribution is tempered

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35L99 | Hyperbolic equations and hyperbolic systems |

42Bxx | Harmonic analysis in several variables |

46Fxx | Distributions, generalized functions, distribution spaces |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |