##
**The analysis of linear partial differential operators. IV: Fourier integral operators.**
*(English)*
Zbl 0612.35001

Grundlehren der Mathematischen Wissenschaften, 275. Berlin etc.: Springer-Verlag. VII, 352 p. DM 128.00 (1985).

[We continue the review begun with vol. III (1985; Zbl 0601.35001).]

Chapter XXV develops the calculus of Fourier integral operators. § 25.1 introduces Lagrangian distributions as a natural extension of conormal distributions. 25.1.2 gives the localization, 25.1.4 the smoothness properties, and 25.1.5 the important representation by phase functions. The principal symbol, taking into account the Maslov bundle, is constructed in 25.1.9. In § 25.2 Fourier integral operators (FIO’s) are defined as operators with Lagrangian distributions as kernels; adjoints (25.2.2) and compositions (25.2.3) are studied with a refinement (25.2.4) in the case of vanishing principal symbol. \(L^ 2\) continuity is studied in § 25.3: 25.3.1 gives the general result on continuity and compactness, and Egorov’s theorem (25.3.5) is proved after elliptic FIO’s have been defined. This is used to simplify conic symplectic manifolds by homogeneous canonical coordinate changes (25.3.7) and leads to improved \(L^ 2\) continuity (25.3.8) with a partial converse (25.3.9). § 25.4 extends the study to complex Lagrangians defined via Lagrangian ideals (25.4.1). Positivity is defined through positive phase functions which parametrize the complex Lagrangians (25.4.3, 25.4.6). This allows the definition of associated oscillatory integrals (25.4.7) and of the Lagrangian distributions in the complex case (25.4.9). Parallel to § 25.2, the calculus of FIO’s with complex phase is developed in § 25.5 using the results of § 25.4.

Chapter XXVI is devoted to \(\psi\) do’s of principal type. As in the constant coefficient case these are operators for which the existence and regularity theory is determined by the principal symbol. § 26.1 starts the investigation with \(\psi\)do’s possessing a real principal symbol. The general propagation of singularities is proved (26.1.1) by reduction to the operator \(D_ 1\) and explicit calculation. Then the propagation of \(H_{(s)}\) regularity is derived in 26.1.4. This result provides the model for the more general cases treated later since it leads to a semiglobal existence theorem for the adjoint operator (26.1.7), and to a global existence theorem (26.1.9) provided a nontrapping condition for the bicharacteristics is satisfied. From this property then the notion of ’real principal type’ is derived in 26.1.8. The remainder of Section 26.1 is devoted to the construction of twosided parametrices (26.1.14). If \(P\) is a \(\psi\)do of order \(m\) with complex principal symbol \(p\) the situation is much more complicated, since the characteristic set \(p^{-1}(0)\) need not be a manifold and the symplectic form restricted to \(p^{-1}(0)\) need not have constant rank. § 26.2 begins the study of this case assuming that \(\{\operatorname{Re} p, \operatorname{Im} p\}=0\) on the characteristic set which is then an involutive manifold. As such it has a two-dimensional foliation the leaves of which are called bicharacteristics of \(P\). Then \(P\) can be reduced micro-locally to the Cauchy- Riemann operator \(D_ 1+iD_ 2\). It follows that the leaves carry an analytic structure; the main result (Theorem 26.2.1) asserts that the regularity function \(s^*_ u(x,\xi):=\sup \{x\mid u\in H_{(s)}\) at \((x,\xi)\}\) is superharmonic on each leaf if \(Pu\) is \(C^{\infty}\). The opposite extreme case is \(\{\operatorname{Re} p, \operatorname{Im} p\}\neq 0\) on \(p^{-1}(0)\), the characteristic set then being a symplectic manifold; this is studied in § 26.3. Proposition 26.3.1 shows that \(D_ 1+ix_ 1D_ n\) is the model operator in this case. More generally, the operators \(P_ k:=D_ 1+ix^ k_ 1D_ n\) are studied and shown to the micro-hypoelliptic (26.3.4). This implies that P is subelliptic with loss of \(k/(k+1)\) derivatives at a characteristic point \((x^ 0,\xi^ 0)\) if \(H_{\operatorname{Re} p}(x^ 0,\xi^ 0)\neq 0\), \(\{\operatorname{Re} p, \operatorname{Im} p\}>0\), and \(\operatorname{Im} p\) has a zero of exact order \(k\) on each bicharacteristic of \(\operatorname{Re} p\) starting near \((x^ 0,\xi^ 0)\) with sign change from \(-\) to \(+\) or no sign change at all (Theorem 26.3.5). Then 26.3.6 shows that the converse sign change cannot be allowed. The remainder of this section is devoted to explicit parametrix constructions for the model operator \(P_ k\) leading to precise conditions for micro-hypoellipticity (26.3.8 and 26.3.9). § 26.4 explores the relationship between sign changes of \(\operatorname{Im} p\) along bicharacteristics of \(\operatorname{Re} p\) and local solvability. After introducing a suitable notion of local solvability for \(P\) it is shown to be equivalent to an a priori estimate for \(P^*\) (26.4.5). Condition \((\psi)\) is introduced in 26.4.6 to the effect of ruling out the above mentioned sign changes from \(+\) to \(-\), and it is shown that \((\psi)\) is necessary for local solvability (26.4.8). The prove is involved and constructs explicitly solutions violating 26.4.5 if \((\psi)\) or rather an equivalent but more convenient condition (26.4.12) is violated. It is not yet known whether \((\psi)\) is also sufficient for local solvability. In the remaining sections of Chapter XXVI it is proved that a symmetric version of \((\psi)\), called \((P)\), is necessary and sufficient for local solvability of differential operators. This starts in § 25.5 with a classification of the characteristic points and the corresponding microlocal normal forms of \(P\), valid if \((P)\) is assumed. Building on the results obtained in §§ 26.2, 26.3 the propagation of regularity is derived in all cases (Theorems 26.6.2, 26.6.4, 26.9.1, 26.10.1’), and these are combined to give the final result in Theorems 26.11.2 and 26.11.3.

Chapter XXVII gives a microlocal characterization of \(\psi\)do’s subelliptic with loss of \(\delta <1\) derivatives, a property depending on the principal symbol only and already encountered in § 26.3. The condition is stated in terms of repeated Poisson brackets, taking sign changes into account as it must be; the central result is Theorem 27.1.11. The necessity of the condition is proved in § 27.2 after a careful study of the Taylor expansion of the principal symbol. The proof of sufficiency is very complicated and takes up some thirty pages. Basically, the micro-local behavior of \(P\) is split into two alternatives (given in (27.4.12) and (27.4.21)), which necessitates two different kinds of local subelliptic estimates; these are then patched together by a partition of unity in a very delicate way. In the case of differential operators satisfying \((P)\) the proof is much simpler and given separately in § 27.3.

Chapter XXVIII deals with uniqueness for the Cauchy problem. § 28.1 begins with Calderón’s uniqueness theorem (28.1.1) which is proved by factorization in first order operators and establishing the Carleman estimates for these (28.1.2 and 28.1.3). The Carleman estimates are extended to other operators of first or second order in 28.1.4, 28.1.5, and 28.1.6. This allows to conclude uniqueness for operators that can be obtained by composing the more general factors which leads to a substantial generalization of Theorem 28.1.1 (Theorem 28.1.8). Section 28.2 gives microlocal characterizations of Carleman estimates, notably Theorems 28.2.1, 28.2.3, and 28.2.1’. The results motivate the definition of principally normal operators (Definition 28.2.4). The next section uses the material in § 28.1 to prove a unique continuation result for principally normal operators across strongly pseudoconvex hypersurfaces (Theorem 28.3.4). The key is that the convexity condition ensures certain Carleman estimates (28.3.3). Section 28.4 finally gives a generalization of 28.3.4 for second order differential operators with real principal symbol (Theorem 28.4.3) based on yet another subtle Carleman estimate (Proposition 28.4.1).

Chapter XXIX completes the study of spectral asymptotics begun in Chapter XVII in the second order case. Section 29.1 mainly exploits the fact that the calculus of FIO’s gives complete control over the singularities of \(e^{-it P}\) if \(P\) is a first order self-adjoint elliptic \(\psi\)do on a compact manifold without boundary. Thus the restriction of the kernel of \(e^{-it P}B\), B a zero order \(\psi\) do, can be described quite explicitly for small t (Proposition 29.1.2), and combined with the Tauberian argument (Lemma 17.5.6) this leads to a sharp estimate of the spectral function in Theorem 29.1.4. Since the remainder estimate at a point \(x\) is in terms of the smallest period of a bicharacteristic returning to \(x\), this gives by integration sharp estimates of the counting function of the eigenvalues of \(P\) in Theorem 29.1.5 and Corollary 29.1.6. The same ideas are used to obtain a result of ”Szegö type” i.e. to determine the asymptotic behavior of the counting measure of zero order \(\psi\)do’s compressed to the eigenspaces of \(P\) (Theorem 29.1.7). Finally, it is shown that by functional calculus the results extend to self-adjoint elliptic operators of arbitrary positive order (Proposition 29.1.9). § 29.2 is devoted to the case of periodic Hamilton flow for \(P\). If \(\Pi >0\) denotes the common minimal period of all bicharacteristics then the principal symbol of \(e^{-i\Pi P}\) may be assumed to be the identity. It follows from this crucial fact that the eigenvalues cluster around an arithmetic progression. Using the methods of the previous paragraph gives a sharp asymptotic formula for the cardinality of the cluster (Theorem 29.2.2). The distribution of the eigenvalues in each cluster is analyzed further with Szegö type theorems for this case (Theorems 29.2.4, 29.2.5) which lead to very precise but complicated eigenvalue asymptotics (Theorem 29.2.7). The last section of this chapter proves the sharp remainder estimate and the so-called Weyl formula for the Dirichlet problem and operators of second order (Theorem 29.3.3, Corollary 29.3.4). The proof follows the pattern of § 29.1 using Corollary 17.5.11 and the methods of Chapter XXIV as decisive additional tools.

Chapter XXX concludes this impressive treatise with important contributions to long range scattering theory, basically extending the results of Chapter XIV to this much more complicated case. § 30.1 introduces the class of admissible perturbations as sums \(V^ S+V^ L\) of short range and long range perturbations (Definition 30.1.3) and discusses several growth estimates. Section 30.2 describes natural self- adjoint extensions of these operators (Theorem 30.2.1) and proves the limiting absorption principle for the resolvent (Theorem 30.2.10); the essential tool here is the propagation of singularities for operators with a fixed sign for the imaginary part of the principal symbol (§ 26.6). As essential steps in the construction of the modified wave operators the Hamilton-Jacobi equations are solved in § 30.3, with good estimates for the solutions (Theorem 30.3.3), and a Lagrangian submanifold of the critical set is constructed (Theorem 30.3.5). From these facts the existence of the modified wave operators is derived in § 30.4 (Theorem 30.4.1). The last section proves the main result of this Chapter, the asymptotic completeness (Theorem 30.5.10). This is analogous to § 14.6 and based on a suitable notion of distorted Fourier transform, defined in (30.5.1’). The existence of the limit involved is reduced to the asymptotic analysis of a first order operator valued \(\psi\)do.

This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading because of the complexity of the material and the rather high interdependence of the various parts, but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation.

Chapter XXV develops the calculus of Fourier integral operators. § 25.1 introduces Lagrangian distributions as a natural extension of conormal distributions. 25.1.2 gives the localization, 25.1.4 the smoothness properties, and 25.1.5 the important representation by phase functions. The principal symbol, taking into account the Maslov bundle, is constructed in 25.1.9. In § 25.2 Fourier integral operators (FIO’s) are defined as operators with Lagrangian distributions as kernels; adjoints (25.2.2) and compositions (25.2.3) are studied with a refinement (25.2.4) in the case of vanishing principal symbol. \(L^ 2\) continuity is studied in § 25.3: 25.3.1 gives the general result on continuity and compactness, and Egorov’s theorem (25.3.5) is proved after elliptic FIO’s have been defined. This is used to simplify conic symplectic manifolds by homogeneous canonical coordinate changes (25.3.7) and leads to improved \(L^ 2\) continuity (25.3.8) with a partial converse (25.3.9). § 25.4 extends the study to complex Lagrangians defined via Lagrangian ideals (25.4.1). Positivity is defined through positive phase functions which parametrize the complex Lagrangians (25.4.3, 25.4.6). This allows the definition of associated oscillatory integrals (25.4.7) and of the Lagrangian distributions in the complex case (25.4.9). Parallel to § 25.2, the calculus of FIO’s with complex phase is developed in § 25.5 using the results of § 25.4.

Chapter XXVI is devoted to \(\psi\) do’s of principal type. As in the constant coefficient case these are operators for which the existence and regularity theory is determined by the principal symbol. § 26.1 starts the investigation with \(\psi\)do’s possessing a real principal symbol. The general propagation of singularities is proved (26.1.1) by reduction to the operator \(D_ 1\) and explicit calculation. Then the propagation of \(H_{(s)}\) regularity is derived in 26.1.4. This result provides the model for the more general cases treated later since it leads to a semiglobal existence theorem for the adjoint operator (26.1.7), and to a global existence theorem (26.1.9) provided a nontrapping condition for the bicharacteristics is satisfied. From this property then the notion of ’real principal type’ is derived in 26.1.8. The remainder of Section 26.1 is devoted to the construction of twosided parametrices (26.1.14). If \(P\) is a \(\psi\)do of order \(m\) with complex principal symbol \(p\) the situation is much more complicated, since the characteristic set \(p^{-1}(0)\) need not be a manifold and the symplectic form restricted to \(p^{-1}(0)\) need not have constant rank. § 26.2 begins the study of this case assuming that \(\{\operatorname{Re} p, \operatorname{Im} p\}=0\) on the characteristic set which is then an involutive manifold. As such it has a two-dimensional foliation the leaves of which are called bicharacteristics of \(P\). Then \(P\) can be reduced micro-locally to the Cauchy- Riemann operator \(D_ 1+iD_ 2\). It follows that the leaves carry an analytic structure; the main result (Theorem 26.2.1) asserts that the regularity function \(s^*_ u(x,\xi):=\sup \{x\mid u\in H_{(s)}\) at \((x,\xi)\}\) is superharmonic on each leaf if \(Pu\) is \(C^{\infty}\). The opposite extreme case is \(\{\operatorname{Re} p, \operatorname{Im} p\}\neq 0\) on \(p^{-1}(0)\), the characteristic set then being a symplectic manifold; this is studied in § 26.3. Proposition 26.3.1 shows that \(D_ 1+ix_ 1D_ n\) is the model operator in this case. More generally, the operators \(P_ k:=D_ 1+ix^ k_ 1D_ n\) are studied and shown to the micro-hypoelliptic (26.3.4). This implies that P is subelliptic with loss of \(k/(k+1)\) derivatives at a characteristic point \((x^ 0,\xi^ 0)\) if \(H_{\operatorname{Re} p}(x^ 0,\xi^ 0)\neq 0\), \(\{\operatorname{Re} p, \operatorname{Im} p\}>0\), and \(\operatorname{Im} p\) has a zero of exact order \(k\) on each bicharacteristic of \(\operatorname{Re} p\) starting near \((x^ 0,\xi^ 0)\) with sign change from \(-\) to \(+\) or no sign change at all (Theorem 26.3.5). Then 26.3.6 shows that the converse sign change cannot be allowed. The remainder of this section is devoted to explicit parametrix constructions for the model operator \(P_ k\) leading to precise conditions for micro-hypoellipticity (26.3.8 and 26.3.9). § 26.4 explores the relationship between sign changes of \(\operatorname{Im} p\) along bicharacteristics of \(\operatorname{Re} p\) and local solvability. After introducing a suitable notion of local solvability for \(P\) it is shown to be equivalent to an a priori estimate for \(P^*\) (26.4.5). Condition \((\psi)\) is introduced in 26.4.6 to the effect of ruling out the above mentioned sign changes from \(+\) to \(-\), and it is shown that \((\psi)\) is necessary for local solvability (26.4.8). The prove is involved and constructs explicitly solutions violating 26.4.5 if \((\psi)\) or rather an equivalent but more convenient condition (26.4.12) is violated. It is not yet known whether \((\psi)\) is also sufficient for local solvability. In the remaining sections of Chapter XXVI it is proved that a symmetric version of \((\psi)\), called \((P)\), is necessary and sufficient for local solvability of differential operators. This starts in § 25.5 with a classification of the characteristic points and the corresponding microlocal normal forms of \(P\), valid if \((P)\) is assumed. Building on the results obtained in §§ 26.2, 26.3 the propagation of regularity is derived in all cases (Theorems 26.6.2, 26.6.4, 26.9.1, 26.10.1’), and these are combined to give the final result in Theorems 26.11.2 and 26.11.3.

Chapter XXVII gives a microlocal characterization of \(\psi\)do’s subelliptic with loss of \(\delta <1\) derivatives, a property depending on the principal symbol only and already encountered in § 26.3. The condition is stated in terms of repeated Poisson brackets, taking sign changes into account as it must be; the central result is Theorem 27.1.11. The necessity of the condition is proved in § 27.2 after a careful study of the Taylor expansion of the principal symbol. The proof of sufficiency is very complicated and takes up some thirty pages. Basically, the micro-local behavior of \(P\) is split into two alternatives (given in (27.4.12) and (27.4.21)), which necessitates two different kinds of local subelliptic estimates; these are then patched together by a partition of unity in a very delicate way. In the case of differential operators satisfying \((P)\) the proof is much simpler and given separately in § 27.3.

Chapter XXVIII deals with uniqueness for the Cauchy problem. § 28.1 begins with Calderón’s uniqueness theorem (28.1.1) which is proved by factorization in first order operators and establishing the Carleman estimates for these (28.1.2 and 28.1.3). The Carleman estimates are extended to other operators of first or second order in 28.1.4, 28.1.5, and 28.1.6. This allows to conclude uniqueness for operators that can be obtained by composing the more general factors which leads to a substantial generalization of Theorem 28.1.1 (Theorem 28.1.8). Section 28.2 gives microlocal characterizations of Carleman estimates, notably Theorems 28.2.1, 28.2.3, and 28.2.1’. The results motivate the definition of principally normal operators (Definition 28.2.4). The next section uses the material in § 28.1 to prove a unique continuation result for principally normal operators across strongly pseudoconvex hypersurfaces (Theorem 28.3.4). The key is that the convexity condition ensures certain Carleman estimates (28.3.3). Section 28.4 finally gives a generalization of 28.3.4 for second order differential operators with real principal symbol (Theorem 28.4.3) based on yet another subtle Carleman estimate (Proposition 28.4.1).

Chapter XXIX completes the study of spectral asymptotics begun in Chapter XVII in the second order case. Section 29.1 mainly exploits the fact that the calculus of FIO’s gives complete control over the singularities of \(e^{-it P}\) if \(P\) is a first order self-adjoint elliptic \(\psi\)do on a compact manifold without boundary. Thus the restriction of the kernel of \(e^{-it P}B\), B a zero order \(\psi\) do, can be described quite explicitly for small t (Proposition 29.1.2), and combined with the Tauberian argument (Lemma 17.5.6) this leads to a sharp estimate of the spectral function in Theorem 29.1.4. Since the remainder estimate at a point \(x\) is in terms of the smallest period of a bicharacteristic returning to \(x\), this gives by integration sharp estimates of the counting function of the eigenvalues of \(P\) in Theorem 29.1.5 and Corollary 29.1.6. The same ideas are used to obtain a result of ”Szegö type” i.e. to determine the asymptotic behavior of the counting measure of zero order \(\psi\)do’s compressed to the eigenspaces of \(P\) (Theorem 29.1.7). Finally, it is shown that by functional calculus the results extend to self-adjoint elliptic operators of arbitrary positive order (Proposition 29.1.9). § 29.2 is devoted to the case of periodic Hamilton flow for \(P\). If \(\Pi >0\) denotes the common minimal period of all bicharacteristics then the principal symbol of \(e^{-i\Pi P}\) may be assumed to be the identity. It follows from this crucial fact that the eigenvalues cluster around an arithmetic progression. Using the methods of the previous paragraph gives a sharp asymptotic formula for the cardinality of the cluster (Theorem 29.2.2). The distribution of the eigenvalues in each cluster is analyzed further with Szegö type theorems for this case (Theorems 29.2.4, 29.2.5) which lead to very precise but complicated eigenvalue asymptotics (Theorem 29.2.7). The last section of this chapter proves the sharp remainder estimate and the so-called Weyl formula for the Dirichlet problem and operators of second order (Theorem 29.3.3, Corollary 29.3.4). The proof follows the pattern of § 29.1 using Corollary 17.5.11 and the methods of Chapter XXIV as decisive additional tools.

Chapter XXX concludes this impressive treatise with important contributions to long range scattering theory, basically extending the results of Chapter XIV to this much more complicated case. § 30.1 introduces the class of admissible perturbations as sums \(V^ S+V^ L\) of short range and long range perturbations (Definition 30.1.3) and discusses several growth estimates. Section 30.2 describes natural self- adjoint extensions of these operators (Theorem 30.2.1) and proves the limiting absorption principle for the resolvent (Theorem 30.2.10); the essential tool here is the propagation of singularities for operators with a fixed sign for the imaginary part of the principal symbol (§ 26.6). As essential steps in the construction of the modified wave operators the Hamilton-Jacobi equations are solved in § 30.3, with good estimates for the solutions (Theorem 30.3.3), and a Lagrangian submanifold of the critical set is constructed (Theorem 30.3.5). From these facts the existence of the modified wave operators is derived in § 30.4 (Theorem 30.4.1). The last section proves the main result of this Chapter, the asymptotic completeness (Theorem 30.5.10). This is analogous to § 14.6 and based on a suitable notion of distorted Fourier transform, defined in (30.5.1’). The existence of the limit involved is reduced to the asymptotic analysis of a first order operator valued \(\psi\)do.

This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading because of the complexity of the material and the rather high interdependence of the various parts, but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation.

Reviewer: J. Brüning

### MSC:

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

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

47G30 | Pseudodifferential operators |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J32 | Boundary value problems on manifolds |

47F05 | General theory of partial differential operators |

58J47 | Propagation of singularities; initial value problems on manifolds |

35P25 | Scattering theory for PDEs |