##
**The analysis of linear partial differential operators. II: Differential operators with constant coefficients.**
*(English)*
Zbl 0521.35002

Grundlehren der Mathematischen Wissenschaften, 257. Berlin Heidelberg-New York - Tokyo: Springer-Verlag. VIII, 391 p. DM 124.00; $ 49.60 (1983).

This second volume of the author’s treatise is an expanded version of Chapters III, IV, V and VII of his classic “Linear partial differential operators” (1963; Zbl 0108.09301) but, as it contains 3 for other entirely new chapters (one on scattering theory, another on convolution equations and one on methods from the theory of analytic functions of several complex variables), it is certainly better to review it on his own merits, without reference to the earlier book, even if sometimes comparison will be made with its predecessor. The volume is mainly devoted to the study of linear differential operators with constant coefficients, and the main tool is Fourier-Laplace transformation. Particular cases of the topics treated here have been examined in the first volume and one can see the extreme care of the author to give motivations, and also how the treatise has been thought as a whole like if we may speak so – a symphony.

Chapter X (the first of this volume), “Existence and approximation of solutions of differential equations”, introduces the spaces \(B_{p,k}\), where \(k\) is a “temperate weight function”, i.e., a positive function defined on \(\mathbb{R}^n\) such that \(k(\xi+\eta)\leq(1+C|\xi|)^Nk(\eta)\), \(\forall\xi\), \(\eta\in\mathbb{R}^n\) and studies their properties (without references to restrictions to subspaces of \(\mathbb{R}^n)\); the next section starts with an existence theorem (th. 10.8.1) of fundamental solutions belonging to \(B^{\text{loc}}_{\omega,\widetilde p}\) for any not zero differential operator with constant coefficient \(P(D)\). Such a fundamental solution is called regular. The new feature is the study of singularities of the fundamental solution \(E(P)\) constructed precedently. For this the author considers the sets \[ \Lambda(P)= \{\eta\in\mathbb{R}^n,P(\xi+i\eta)= P (\xi)\},\quad \Lambda'(P)=\{x\in\mathbb{R}^n,\langle x,\eta\rangle=0,\eta\in \Lambda(P)\}, \]

\[ L(P)=\left\{\text{set of limits of }\xi-\frac{P(\xi+ \eta)} {\widetilde p(\eta)}\text{ in Pol}^0(m,n),\eta\to\infty\right\} \] where \(\text{Pol}^0(m,n)\) is the set of polynomials in \(n\) variables, with degree at most \(m\), with the zero polynomial removed, and \(L_\theta(P)\) defined analogously when \((\eta/|\eta|)\to(\theta/|\theta|)\) with \(\theta\neq 0\). Then (th. 10.2.4 and 10.2.5) \(\Lambda'(P)\) is the smaller subspace that contains supp\(E(P)\), and, if \(F=\{(x,\theta)\in\mathbb{R}^n\times(\mathbb{R}^n \setminus 0)\), \(x\in\Lambda'(Q)\) for some \(Q\in L_\theta(P)\}\), \(F_0= pr_{\mathbb{R}^n}F\) then (th. 10.2.11), \(WF(E(P))\subset F\), sing supp\( E(P)\subset F_0\). Mention is given also of a nice example for which the unique temperate solution is not regular. §10.3 studies the equation \(P(D)u=f\) with \(f\in{\mathcal E}'\), and introduces the concept of comparison of operators, a concept which will be systematically exaiained in §10.4, and which leads to the definition of operators of principal type. Sections 10.5 and 10.6 contain the now classical results of Malgrange on approximation of solutions of homogeneous equations and the relations between \(P\)-convexity and existence theorems; the proofs follow the lines of Malgrange’s proofs with simplifications resulting from the existence of regular fundamental solutions. The existence results for the equation \(P(D)u=f\) with \(f\in{\mathcal D}'(X)\), connected with \(P\)-convexity with regard to singular support (which in the “classical” terminology, was called strong \(P\)-convexity) make use of the splitting into existence theory in \({\mathcal D}'/C^\infty\) and in \(C^\infty\), and the exposition follows closely that of the author’s paper [Enseign. Math., II. Sér. 17, 99–163 (1971; Zbl 0224.35084)], and occupies §10.7. Very interesting is the geometrical meaning of these convexity conditions whose study is done in §10.8. We find here a complete characterization of \(P\)-convexity for supports when \(P\) is the wave operator.

Chapter XI discusses the interior regularity of solutions of differential equations. §11.1 gives a detailed account of hypoellipticity (where essential use is made of the theorems of the preceding chapter concerning localizations), §11.2 studies partially hyperelliptic operators. The next section §11.3 considers analogues for singularities to the non-uniqueness theorems of §8.6, results which play for \(P\)-convexity with respect to singular supports the same role as these of §8.6 with respect to \(P\)-convexity with respect to supports (th. 11.3.1, Coroll. 11.3.2, Coroll. 11.3.3). A very strong converse of th. 11.3.1 is the following beautiful result (th. 11.36): If \(X\) is an open set in \(\mathbb{R}^n\), \(x_0\in X\) and \(\varphi_1,\dots,\varphi_k \in C^1(X)\) real valued functions such that \(d \varphi_1(x_0),\dots,d \varphi_k(x_0)\) are linearly independent, denoting by \(W\) the space spanned by these differentials and assuming that \(\sigma_p(W)=0\), then if \(X=\{x\in X,\varphi_j(x)<\varphi_j(x_0)j=1,\dots,k\}\) and if \(u\in {\mathcal D}' (X)\), \(P(D)u\in C^\infty(\overline X)\) and \(u\in C^\infty (\underline X)\) it follows that, \(u\in C^\infty\) in a neighbourhood of \(x_0\), independent of \(u\). Here \(\sigma_p(W)=\inf_{t>1} \varliminf_{\xi\to\infty}P_w(\xi,t)/\widetilde P (\xi,t)\) where \(\widetilde P_w(\xi,t)=\sup\{|P(\xi+\theta)|,\theta\in W,| \theta|\leq t\}\). This \(\sigma_p(W)\) is an appropriate substitute for the characteristic hyperplanes and plays an importent role. The last section discusses optimal estimates for the growth of the derivative of the solutions of a hypoelliptic equation.

Chapter XII “The Cauchy and mixed problems” has 86 pages, and is really marvelous. It begins with a study of the Cauchy problem for the wave equation in \(\mathbb{R}^{3+1}\) (a variant of Riesz’s formula), continues with the oscillatory Cauchy problem for the wave equation, the asymptotics being given (th. 12.2.3). In §12.3 the author proves that a differential operator \(P\) must satisfy a restrictive algebraic condition in order that a weak existence theorem be valid (12.3.1), and so in a natural way, the hyperbolic polynomials are introduced. The next section studies in detail the properties of hyperbolic polynomials; the use of the Tarski-Seidenberg theorem simplifies the proofs. In §12.5 the Cauchy problem for a hyperbolic equation is shown to have a solution, constructing first a regular fundamental solution, whose singularities are studied in §12.6 (th. 12.6.2). One finds also Herglotz-Petrovskii formula, and an interesting discussion leading to the “Petrovskii condition”; of course, for further results one has to consult the great papers of M. F. Atiyah, R. Bott and L. Gårding [Acta Math. 124, 109–184 (1970; Zbl 0191.11203); ibid. 131, 145–206 (1973; Zbl 0266.55045)]. Section 12.7 contains sees precise results on the support of the solution in the hyperbolic case and on the non-existence of solutions in the nonhyperbolic case. It begins with a theorem (th. 12.7.1) originally due to F. John and then with a nice theorem (th. 12.7.5) asserting the existence of solution of the Cauchy problem in certain Gevrey classes when the principal part of the operator is hyperbolic. The characteristic Cauchy problem is examined in §12.8. The main result here is th. 12.8.1, which is really a difficult result. This proof (in fact the implication (iii)=(iv)) is quite involved; first one proves an intermediate result (prop. 12.6.4), then some lemmas about the behaviour of algebraic functions when away of their branching points), after that an a-priori estimation (th. 12.8.12) which enables immediately to prove a local existence theorem, then an approximation result concerning the solutions of an operator \(P\) which satisfies condition (iii) namely that if \(\tau(\xi)\) is a solution of \(P(\xi+\tau N)=0\), analytic and single valued in a ball \(B\) with real center and radius \(A_1>0\), there exists \(A_2\) such that \(\sup_8\text{Im}\,\tau(\xi)\geq A_2\). (This condition is to be compared with Petrovskii condition: \(\text{Im}\,\tau> C\) if \(P(\xi+\tau N)=0\), \(\xi\in\mathbb{R}^r\) and if the highest power of \(\tau\) in \(P(\xi+\tau N)\) is independent of \(\xi)\) and after all these preparations the proof is along the same lines as the proof of existence of solutions for open sets being \(P\)-convex with respect to supports (th. 10.6.7); an operator \(P(D)\) having the equivalent properties of th. 12.8.1 with respect to the hyperplane \(H=\{x|\langle x,N\rangle\geq 0\}\) is called an evolution operator. Mixed problems are described in §12.9, only in the case of a half space, the general case being left to volume III. The mixed problem to be studied will be the following: \(P(D)u=f\) in \(H\cap H'\), \(D^2(u-\varphi)=0\) in \(H'\cap\partial H\), \(|\alpha|<m\) and \(B_j (D)=\varphi_j\); \(j=1,\dots,\mu\) on \(H\cap\partial H'\) where \(H=\{x \mid\langle x,N\rangle\geq 0\}\), \(H'=\{x\mid\langle x,\theta\rangle\geq 0\}\), \(N,\theta\) linearly independent vector in \(\mathbb{R}^n\). After Some remarks to be used in the sequel relative to boundary value problems for ordinary differential operators, one finds an existence theorem (th. 12.9.2) for the homogeneous boundary problem (assuming Lopatinsky’s condition). Finally th. 12.9.3 and 12.9.3 give the necessary conditions for the existence and uniqueness of solutions of the mixed problem considered. For “hyperbolic mixed problems”, one constructs “fundamental solutions” (th. 12.9.12). As an interesting example one finds a very nice discussion showing when the oblique derivative boundary operator gives a hyperbolic mixed problem for the wave-operator.

Chapter XIII deals with operators which are not with constant coefficients, but which can be considered as bounded (in the spaces \(B_{p,k})\) perturbations of p.d.o. with constant coefficients. These are the operators of constant strength. After deriving the basic properties of these operators in §13.1, giving existence results when the coefficients are mereley continuous in §13.2 and again existence results when the coefficients are \(C^\infty\) (the main result is th. 13.3.3 (these are of local nature), the author proves for this class hypoellipticity results (but, as it is well known and also pointed out by him, there are many other hypoelliptic operators (with \(C^\infty\) coefficients) not belonging to this class). Using what he calls localization at infinity of an operator \(P(x,D)\), the author proves some global existence results. But the most significant section of this chapter is §13.6 concerning the non-uniqueness for the Cauchy problem, where it is shown that the uniqueness results for the non-characteristic Cauchy problem break-down completely to case of \(C^\infty\) coefficients. We mention here th. 13.6.1 (with its corollaries) and th. 13.6.15. This last theorem states that there exists a fourth-order elliptic differential operator \(P\) to \(\mathbb{R}^3\), with \(C^\infty\) coefficients such that the equation \(Pu =0\) has a non-trivial solution \(u\in C_0^\infty (\mathbb{R}^3)\)! Some very interesting examples are given in this section. Although vary technical, it is a very good thing to have these results gathered together (for recent development one should nevertheless see the recent monograph by Ch. Zuily [“Uniqueness and non-uniqueness in the Cauchy-problem”, Prog. Math. 33 (1983; Zbl 0521.35003)]).

Chapter XIV is an introduction to scattering theory, for which the stationary method is used. Considering the importance of this topic it is a very good idea to have a chapter on it incorporated in a general treatise on p.d.o. I will list only the sections of this chapter; §14.1 introduces some subspaces of \(L^2 (\mathbb{R}^n)\) which appear naturally in these problems, §14.2. Division by functions with simple zeros, §14.3. The resolvent of the unperturbed operator, §14.4 short range perturbations, §14.5. The boundary values of the resolvent and the point spectrum, §14.6. The distorted Fourier transforms and the continuous spectrum, §14.7. Absence of embedded eigenvalues.

Chapter XV “Analytic function theory and differential equations” begins with a study of the inhomogeneous Cauchy-Riemann equations in \(\mathbb C^n\) by means of the celebrated method of \(L^2\)-estimates with weights (due to the author). After proving the existence results (th. 15.1.1 and th. 15.1.2) one gives an extension result (due also to him) to illustrate how th. 15.1.2 is used. As a consequence, the Martineau-Ehrenpreis characterization of the Fourier-Laplace transforms of analytic functionals with support in a compact convex set is given. Another beautiful application is th. 15.15 which roughly says that all plurisubharmonic functions in \(\mathbb{C}^n\) are limits (in \(L^1_{\text{loc}} (\mathbb{C}^n))\) of functions of the form \(\frac 1N\log|f(z)|\) with \(f\) entire.

The next 3 sections contain the analytic part of the proof of the fundamental principle, i.e., representation of solutions by means of Fourier-Laplace representations. To simplify things (i.e., in order to avoid local algebra arguments, as well as cohomology with bounds for forms of weight degrees) the case of a single operator is considered. One can understand perfectly this point of view, but nevertheless regret that a complete proof along this line in the general case is not to be found here. As far as I know, sections 15.2 and 15.4 contain a discussion which is published here for the first time on how one can still use \(L^2\) estimates, even if the weight that appears in the Fourier transform of \(C_0^\infty (K)\) is not pluri-subharmonic, by constructing equivalent strictly plurisubharmonic functions. In §15.2, the topology of \(B^c_{3,k}= B_{2,k}\cap{\mathcal E}'(X)\), where as usual \(X\) is an open set in \(\mathbb{R}^n\) and is a weight function, is described in a suitable way to make possible the use of the results of §15.1 (th. 15.2.1). In §15.4, the author gives a description of neighborhood of 0 in terms of Fourier transforms for \(C^\infty_0(X)\) when \(X\) is convex.

The last chapter XVI is devoted to the study of “Convolution equations”; it contains some very beautiful results, and its presence is justified in a volume about operators with constant coefficient by the fact that a convolution operator is translation invariant, and the Fourier-Laplace transformation is still the main tool. The chapter begins with two sections about, respectively, subharmonic and plurisubharmonic functions. The novel feature here is th. 16.1.8 and its extension to push functions (th. 16.2.4) whose consequences play an important role in the sequel. §16.3, “The support and singular support of a convolution” introduces Ehrenpreis’ invertible distributions, and gives some results about the convex hull of the support at a distribution \(u\in\delta'\) in connexion with \({\mathcal H}(u)\), where \({\mathcal H}(u)\) is defined to be the set of those supporting functions \(h\) of compact convex sets such that \((\exists)\) a sequence \(\xi_v\to \infty\) such that \(L_u(z,\xi_v)\) is converging to a psh function with precisely the supporting function \(h\). Here \(L_u(z,\xi)\) is defined to be \[ |\log| \widehat u(\xi+z\log|\xi|)H/\log\xi(|\xi|>2) \] for every \(u\in\delta'\). We mention another proof given here of the theorem of supports (already proved in Chapter IV). Note also th. 16.3.9, the very useful th. 16.3.9 (the negation of the conditions in th. 16.3.9) and some relevant examples. §16.4 contains a proof of the approximation theorem (th. 16.4.1), a proof simplified by the use of analytic functionals. Note also the existence theorem for the equation \(\mu*u=f\) where \(f\) is real analytic in \(X\), \(0\neq\mu\in\delta'\) and \(X\) is convex. §16.5 contains the main results on inhomogeneous convolution equations. The role of invertibility is put in evidence. Th. 16.3.22 and th. 16.5.26 justify the introduction of very slowly decreasing \(\widehat\mu\); precisely for these distributions we have solutions in \({\mathcal D}_F'\). The last two sections consider hypoelliptic, respectively hyperbolic convolution equations. The volume ends with an appendix “some algebraic lemmas” which contains, among other results, a proof due to P. Cohen, of the Tarski-Seidenberg theorem and a result of P. Cohen stating that the graph of an analytic continuation of an algebraic function is semi-algebraic.

This brief sketch of the contest of this second volume cannot give an idea good enough of how rich and interesting it really is. Even if the domain p.d.o. with constant coefficients is not so active now – the knowledge of it is very important for those interested mainly in operators with variable coefficients. One sees at his best not only the extraordinary strength of the author in some really difficult and deep results, but also his care for the reader. Certainly, also this volume will be a classic for many years to come.

Chapter X (the first of this volume), “Existence and approximation of solutions of differential equations”, introduces the spaces \(B_{p,k}\), where \(k\) is a “temperate weight function”, i.e., a positive function defined on \(\mathbb{R}^n\) such that \(k(\xi+\eta)\leq(1+C|\xi|)^Nk(\eta)\), \(\forall\xi\), \(\eta\in\mathbb{R}^n\) and studies their properties (without references to restrictions to subspaces of \(\mathbb{R}^n)\); the next section starts with an existence theorem (th. 10.8.1) of fundamental solutions belonging to \(B^{\text{loc}}_{\omega,\widetilde p}\) for any not zero differential operator with constant coefficient \(P(D)\). Such a fundamental solution is called regular. The new feature is the study of singularities of the fundamental solution \(E(P)\) constructed precedently. For this the author considers the sets \[ \Lambda(P)= \{\eta\in\mathbb{R}^n,P(\xi+i\eta)= P (\xi)\},\quad \Lambda'(P)=\{x\in\mathbb{R}^n,\langle x,\eta\rangle=0,\eta\in \Lambda(P)\}, \]

\[ L(P)=\left\{\text{set of limits of }\xi-\frac{P(\xi+ \eta)} {\widetilde p(\eta)}\text{ in Pol}^0(m,n),\eta\to\infty\right\} \] where \(\text{Pol}^0(m,n)\) is the set of polynomials in \(n\) variables, with degree at most \(m\), with the zero polynomial removed, and \(L_\theta(P)\) defined analogously when \((\eta/|\eta|)\to(\theta/|\theta|)\) with \(\theta\neq 0\). Then (th. 10.2.4 and 10.2.5) \(\Lambda'(P)\) is the smaller subspace that contains supp\(E(P)\), and, if \(F=\{(x,\theta)\in\mathbb{R}^n\times(\mathbb{R}^n \setminus 0)\), \(x\in\Lambda'(Q)\) for some \(Q\in L_\theta(P)\}\), \(F_0= pr_{\mathbb{R}^n}F\) then (th. 10.2.11), \(WF(E(P))\subset F\), sing supp\( E(P)\subset F_0\). Mention is given also of a nice example for which the unique temperate solution is not regular. §10.3 studies the equation \(P(D)u=f\) with \(f\in{\mathcal E}'\), and introduces the concept of comparison of operators, a concept which will be systematically exaiained in §10.4, and which leads to the definition of operators of principal type. Sections 10.5 and 10.6 contain the now classical results of Malgrange on approximation of solutions of homogeneous equations and the relations between \(P\)-convexity and existence theorems; the proofs follow the lines of Malgrange’s proofs with simplifications resulting from the existence of regular fundamental solutions. The existence results for the equation \(P(D)u=f\) with \(f\in{\mathcal D}'(X)\), connected with \(P\)-convexity with regard to singular support (which in the “classical” terminology, was called strong \(P\)-convexity) make use of the splitting into existence theory in \({\mathcal D}'/C^\infty\) and in \(C^\infty\), and the exposition follows closely that of the author’s paper [Enseign. Math., II. Sér. 17, 99–163 (1971; Zbl 0224.35084)], and occupies §10.7. Very interesting is the geometrical meaning of these convexity conditions whose study is done in §10.8. We find here a complete characterization of \(P\)-convexity for supports when \(P\) is the wave operator.

Chapter XI discusses the interior regularity of solutions of differential equations. §11.1 gives a detailed account of hypoellipticity (where essential use is made of the theorems of the preceding chapter concerning localizations), §11.2 studies partially hyperelliptic operators. The next section §11.3 considers analogues for singularities to the non-uniqueness theorems of §8.6, results which play for \(P\)-convexity with respect to singular supports the same role as these of §8.6 with respect to \(P\)-convexity with respect to supports (th. 11.3.1, Coroll. 11.3.2, Coroll. 11.3.3). A very strong converse of th. 11.3.1 is the following beautiful result (th. 11.36): If \(X\) is an open set in \(\mathbb{R}^n\), \(x_0\in X\) and \(\varphi_1,\dots,\varphi_k \in C^1(X)\) real valued functions such that \(d \varphi_1(x_0),\dots,d \varphi_k(x_0)\) are linearly independent, denoting by \(W\) the space spanned by these differentials and assuming that \(\sigma_p(W)=0\), then if \(X=\{x\in X,\varphi_j(x)<\varphi_j(x_0)j=1,\dots,k\}\) and if \(u\in {\mathcal D}' (X)\), \(P(D)u\in C^\infty(\overline X)\) and \(u\in C^\infty (\underline X)\) it follows that, \(u\in C^\infty\) in a neighbourhood of \(x_0\), independent of \(u\). Here \(\sigma_p(W)=\inf_{t>1} \varliminf_{\xi\to\infty}P_w(\xi,t)/\widetilde P (\xi,t)\) where \(\widetilde P_w(\xi,t)=\sup\{|P(\xi+\theta)|,\theta\in W,| \theta|\leq t\}\). This \(\sigma_p(W)\) is an appropriate substitute for the characteristic hyperplanes and plays an importent role. The last section discusses optimal estimates for the growth of the derivative of the solutions of a hypoelliptic equation.

Chapter XII “The Cauchy and mixed problems” has 86 pages, and is really marvelous. It begins with a study of the Cauchy problem for the wave equation in \(\mathbb{R}^{3+1}\) (a variant of Riesz’s formula), continues with the oscillatory Cauchy problem for the wave equation, the asymptotics being given (th. 12.2.3). In §12.3 the author proves that a differential operator \(P\) must satisfy a restrictive algebraic condition in order that a weak existence theorem be valid (12.3.1), and so in a natural way, the hyperbolic polynomials are introduced. The next section studies in detail the properties of hyperbolic polynomials; the use of the Tarski-Seidenberg theorem simplifies the proofs. In §12.5 the Cauchy problem for a hyperbolic equation is shown to have a solution, constructing first a regular fundamental solution, whose singularities are studied in §12.6 (th. 12.6.2). One finds also Herglotz-Petrovskii formula, and an interesting discussion leading to the “Petrovskii condition”; of course, for further results one has to consult the great papers of M. F. Atiyah, R. Bott and L. Gårding [Acta Math. 124, 109–184 (1970; Zbl 0191.11203); ibid. 131, 145–206 (1973; Zbl 0266.55045)]. Section 12.7 contains sees precise results on the support of the solution in the hyperbolic case and on the non-existence of solutions in the nonhyperbolic case. It begins with a theorem (th. 12.7.1) originally due to F. John and then with a nice theorem (th. 12.7.5) asserting the existence of solution of the Cauchy problem in certain Gevrey classes when the principal part of the operator is hyperbolic. The characteristic Cauchy problem is examined in §12.8. The main result here is th. 12.8.1, which is really a difficult result. This proof (in fact the implication (iii)=(iv)) is quite involved; first one proves an intermediate result (prop. 12.6.4), then some lemmas about the behaviour of algebraic functions when away of their branching points), after that an a-priori estimation (th. 12.8.12) which enables immediately to prove a local existence theorem, then an approximation result concerning the solutions of an operator \(P\) which satisfies condition (iii) namely that if \(\tau(\xi)\) is a solution of \(P(\xi+\tau N)=0\), analytic and single valued in a ball \(B\) with real center and radius \(A_1>0\), there exists \(A_2\) such that \(\sup_8\text{Im}\,\tau(\xi)\geq A_2\). (This condition is to be compared with Petrovskii condition: \(\text{Im}\,\tau> C\) if \(P(\xi+\tau N)=0\), \(\xi\in\mathbb{R}^r\) and if the highest power of \(\tau\) in \(P(\xi+\tau N)\) is independent of \(\xi)\) and after all these preparations the proof is along the same lines as the proof of existence of solutions for open sets being \(P\)-convex with respect to supports (th. 10.6.7); an operator \(P(D)\) having the equivalent properties of th. 12.8.1 with respect to the hyperplane \(H=\{x|\langle x,N\rangle\geq 0\}\) is called an evolution operator. Mixed problems are described in §12.9, only in the case of a half space, the general case being left to volume III. The mixed problem to be studied will be the following: \(P(D)u=f\) in \(H\cap H'\), \(D^2(u-\varphi)=0\) in \(H'\cap\partial H\), \(|\alpha|<m\) and \(B_j (D)=\varphi_j\); \(j=1,\dots,\mu\) on \(H\cap\partial H'\) where \(H=\{x \mid\langle x,N\rangle\geq 0\}\), \(H'=\{x\mid\langle x,\theta\rangle\geq 0\}\), \(N,\theta\) linearly independent vector in \(\mathbb{R}^n\). After Some remarks to be used in the sequel relative to boundary value problems for ordinary differential operators, one finds an existence theorem (th. 12.9.2) for the homogeneous boundary problem (assuming Lopatinsky’s condition). Finally th. 12.9.3 and 12.9.3 give the necessary conditions for the existence and uniqueness of solutions of the mixed problem considered. For “hyperbolic mixed problems”, one constructs “fundamental solutions” (th. 12.9.12). As an interesting example one finds a very nice discussion showing when the oblique derivative boundary operator gives a hyperbolic mixed problem for the wave-operator.

Chapter XIII deals with operators which are not with constant coefficients, but which can be considered as bounded (in the spaces \(B_{p,k})\) perturbations of p.d.o. with constant coefficients. These are the operators of constant strength. After deriving the basic properties of these operators in §13.1, giving existence results when the coefficients are mereley continuous in §13.2 and again existence results when the coefficients are \(C^\infty\) (the main result is th. 13.3.3 (these are of local nature), the author proves for this class hypoellipticity results (but, as it is well known and also pointed out by him, there are many other hypoelliptic operators (with \(C^\infty\) coefficients) not belonging to this class). Using what he calls localization at infinity of an operator \(P(x,D)\), the author proves some global existence results. But the most significant section of this chapter is §13.6 concerning the non-uniqueness for the Cauchy problem, where it is shown that the uniqueness results for the non-characteristic Cauchy problem break-down completely to case of \(C^\infty\) coefficients. We mention here th. 13.6.1 (with its corollaries) and th. 13.6.15. This last theorem states that there exists a fourth-order elliptic differential operator \(P\) to \(\mathbb{R}^3\), with \(C^\infty\) coefficients such that the equation \(Pu =0\) has a non-trivial solution \(u\in C_0^\infty (\mathbb{R}^3)\)! Some very interesting examples are given in this section. Although vary technical, it is a very good thing to have these results gathered together (for recent development one should nevertheless see the recent monograph by Ch. Zuily [“Uniqueness and non-uniqueness in the Cauchy-problem”, Prog. Math. 33 (1983; Zbl 0521.35003)]).

Chapter XIV is an introduction to scattering theory, for which the stationary method is used. Considering the importance of this topic it is a very good idea to have a chapter on it incorporated in a general treatise on p.d.o. I will list only the sections of this chapter; §14.1 introduces some subspaces of \(L^2 (\mathbb{R}^n)\) which appear naturally in these problems, §14.2. Division by functions with simple zeros, §14.3. The resolvent of the unperturbed operator, §14.4 short range perturbations, §14.5. The boundary values of the resolvent and the point spectrum, §14.6. The distorted Fourier transforms and the continuous spectrum, §14.7. Absence of embedded eigenvalues.

Chapter XV “Analytic function theory and differential equations” begins with a study of the inhomogeneous Cauchy-Riemann equations in \(\mathbb C^n\) by means of the celebrated method of \(L^2\)-estimates with weights (due to the author). After proving the existence results (th. 15.1.1 and th. 15.1.2) one gives an extension result (due also to him) to illustrate how th. 15.1.2 is used. As a consequence, the Martineau-Ehrenpreis characterization of the Fourier-Laplace transforms of analytic functionals with support in a compact convex set is given. Another beautiful application is th. 15.15 which roughly says that all plurisubharmonic functions in \(\mathbb{C}^n\) are limits (in \(L^1_{\text{loc}} (\mathbb{C}^n))\) of functions of the form \(\frac 1N\log|f(z)|\) with \(f\) entire.

The next 3 sections contain the analytic part of the proof of the fundamental principle, i.e., representation of solutions by means of Fourier-Laplace representations. To simplify things (i.e., in order to avoid local algebra arguments, as well as cohomology with bounds for forms of weight degrees) the case of a single operator is considered. One can understand perfectly this point of view, but nevertheless regret that a complete proof along this line in the general case is not to be found here. As far as I know, sections 15.2 and 15.4 contain a discussion which is published here for the first time on how one can still use \(L^2\) estimates, even if the weight that appears in the Fourier transform of \(C_0^\infty (K)\) is not pluri-subharmonic, by constructing equivalent strictly plurisubharmonic functions. In §15.2, the topology of \(B^c_{3,k}= B_{2,k}\cap{\mathcal E}'(X)\), where as usual \(X\) is an open set in \(\mathbb{R}^n\) and is a weight function, is described in a suitable way to make possible the use of the results of §15.1 (th. 15.2.1). In §15.4, the author gives a description of neighborhood of 0 in terms of Fourier transforms for \(C^\infty_0(X)\) when \(X\) is convex.

The last chapter XVI is devoted to the study of “Convolution equations”; it contains some very beautiful results, and its presence is justified in a volume about operators with constant coefficient by the fact that a convolution operator is translation invariant, and the Fourier-Laplace transformation is still the main tool. The chapter begins with two sections about, respectively, subharmonic and plurisubharmonic functions. The novel feature here is th. 16.1.8 and its extension to push functions (th. 16.2.4) whose consequences play an important role in the sequel. §16.3, “The support and singular support of a convolution” introduces Ehrenpreis’ invertible distributions, and gives some results about the convex hull of the support at a distribution \(u\in\delta'\) in connexion with \({\mathcal H}(u)\), where \({\mathcal H}(u)\) is defined to be the set of those supporting functions \(h\) of compact convex sets such that \((\exists)\) a sequence \(\xi_v\to \infty\) such that \(L_u(z,\xi_v)\) is converging to a psh function with precisely the supporting function \(h\). Here \(L_u(z,\xi)\) is defined to be \[ |\log| \widehat u(\xi+z\log|\xi|)H/\log\xi(|\xi|>2) \] for every \(u\in\delta'\). We mention another proof given here of the theorem of supports (already proved in Chapter IV). Note also th. 16.3.9, the very useful th. 16.3.9 (the negation of the conditions in th. 16.3.9) and some relevant examples. §16.4 contains a proof of the approximation theorem (th. 16.4.1), a proof simplified by the use of analytic functionals. Note also the existence theorem for the equation \(\mu*u=f\) where \(f\) is real analytic in \(X\), \(0\neq\mu\in\delta'\) and \(X\) is convex. §16.5 contains the main results on inhomogeneous convolution equations. The role of invertibility is put in evidence. Th. 16.3.22 and th. 16.5.26 justify the introduction of very slowly decreasing \(\widehat\mu\); precisely for these distributions we have solutions in \({\mathcal D}_F'\). The last two sections consider hypoelliptic, respectively hyperbolic convolution equations. The volume ends with an appendix “some algebraic lemmas” which contains, among other results, a proof due to P. Cohen, of the Tarski-Seidenberg theorem and a result of P. Cohen stating that the graph of an analytic continuation of an algebraic function is semi-algebraic.

This brief sketch of the contest of this second volume cannot give an idea good enough of how rich and interesting it really is. Even if the domain p.d.o. with constant coefficients is not so active now – the knowledge of it is very important for those interested mainly in operators with variable coefficients. One sees at his best not only the extraordinary strength of the author in some really difficult and deep results, but also his care for the reader. Certainly, also this volume will be a classic for many years to come.

Reviewer: G. Gussi

### MSC:

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

35G05 | Linear higher-order PDEs |

35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |

47A40 | Scattering theory of linear operators |

35P25 | Scattering theory for PDEs |

35G10 | Initial value problems for linear higher-order PDEs |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

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

32T99 | Pseudoconvex domains |

35B65 | Smoothness and regularity of solutions to PDEs |

65H10 | Numerical computation of solutions to systems of equations |

35L05 | Wave equation |

35B45 | A priori estimates in context of PDEs |

35P05 | General topics in linear spectral theory for PDEs |