##
**F.B.I. transformation. Second microlocalization and semilinear caustics.**
*(English)*
Zbl 0760.35004

Lecture Notes in Mathematics. 1522. Berlin: Springer-Verlag. 100 p. (1992).

The text provides a self-contained introduction into the theory of second microlocalization along a Lagrangian submanifold, and its application to the study of conormal singularities for solutions of semilinear hyperbolic partial differential equations. The technical tools are rather advanced (generalized Fourier-Bros-Iagolnitzer (FBI) transformations, symplectic geometry, plurisubharmonical functions, subanalytic sets and functions, stratifications) but the exposition is unusually clear and precise.

The first chapter deals with FBI transformation \[ Tu(x,\lambda)=\int e^{-\lambda/2(x-t)^ 2}u(t)dt \] \((x\in\mathbb{C}^ n\), \(0<\lambda<\infty\), \(t\in\mathbb{R}^ n\), \(u(x)\) is a compactly supported distribution) and the generalization \[ T_ gu(x,\lambda)=\int e^{i\lambda g(x,t)}u(t)dt \] with FBI phases \(g\) satisfying \({(\partial g/\partial t)}(x_ 0,t_ 0)\in\mathbb{R}^ n-\{0\}\), \(\text{Im}\partial^ 2g/\partial t^ 2>\varepsilon>0\), \(\text{det}(\partial^ 2g/\partial x\partial t)\neq 0\) near the point \((x_ 0,t_ 0)\) under consideration. Then the microlocal \(H^ s\)-regularity is expressed in terms of FBI transformations.

The next chapter studies the second microlocalizations. For the special Lagrange submanifold \(T^*_{(0)}\mathbb{R}^ n\subset T^*\mathbb{R}^ n\), the conormal regularity \(u\in H^{s,+\infty}_{(0)}\) means that \(X_ 1\dots X_ ku\in H^ s_{loc}\) for every vector fields \(X_ 1,\dots,X_ k\) vanishing at 0, and \(u\) is 2-microlocally regular if \(u\in H^{s,+\infty}_{(0)}\) for appropriate \(s\in\mathbb{R}\). This is expressed in terms of FBI transforms (even for the case of the regularity in a cone) which leads to the concept of the conormal regularity along a Lagrangian submanifold by a special choice of the phase \(g\). A trace formula relating the wave front of the restriction \(u|_ N\) to a submanifold to the wave front of \(u\) and the second microlocalization concludes the second chapter.

The third chapter recalls the subanalytic sets, maps, and stratifications. Then the critical points of locally Lipschitzian subanalytic functions are studied to derive the geometric upper bound for the singular spectrum, and for the second microsupport of a distribution defined as boundary values of a certain ramified function. The results cannot be easily explained owing to the technical complexity.

The fourth chapter applies the preceding methods to the Cauchy problem \[ \partial^ 2u/\partial t^ 2-\sum^ d\partial^ 2u/\partial x^ 2_ j=\sum p_ k(t,x)u^ k\quad (u\in C^ 0(\mathbb{R}_ t,H^ s), \]

\[ u|_{t=0}=u_ 0\in H^ s_{loc},\quad\partial u/\partial t|_{t=0}=u_ 1\in H^{s-1}_{loc} \] (where \(s>d/2)\). The Lebeu theorem concerning the geometric upper bound for the wave front if the Cauchy data are conormal along an analytic submanifold \(V\) of the hyperplane \(t=0\) is proved. The result is valid in large time, after the formation of caustics. Close to \(t=0\), the singularities behave like in the linear case but on a longer interval of time, a nonlinear interaction appears. At last, an improvement of the result for the case \(d=2\) if the union of bicharacteristics issued from \(V\) develops as a swallow-tail singularity concludes the book.

The first chapter deals with FBI transformation \[ Tu(x,\lambda)=\int e^{-\lambda/2(x-t)^ 2}u(t)dt \] \((x\in\mathbb{C}^ n\), \(0<\lambda<\infty\), \(t\in\mathbb{R}^ n\), \(u(x)\) is a compactly supported distribution) and the generalization \[ T_ gu(x,\lambda)=\int e^{i\lambda g(x,t)}u(t)dt \] with FBI phases \(g\) satisfying \({(\partial g/\partial t)}(x_ 0,t_ 0)\in\mathbb{R}^ n-\{0\}\), \(\text{Im}\partial^ 2g/\partial t^ 2>\varepsilon>0\), \(\text{det}(\partial^ 2g/\partial x\partial t)\neq 0\) near the point \((x_ 0,t_ 0)\) under consideration. Then the microlocal \(H^ s\)-regularity is expressed in terms of FBI transformations.

The next chapter studies the second microlocalizations. For the special Lagrange submanifold \(T^*_{(0)}\mathbb{R}^ n\subset T^*\mathbb{R}^ n\), the conormal regularity \(u\in H^{s,+\infty}_{(0)}\) means that \(X_ 1\dots X_ ku\in H^ s_{loc}\) for every vector fields \(X_ 1,\dots,X_ k\) vanishing at 0, and \(u\) is 2-microlocally regular if \(u\in H^{s,+\infty}_{(0)}\) for appropriate \(s\in\mathbb{R}\). This is expressed in terms of FBI transforms (even for the case of the regularity in a cone) which leads to the concept of the conormal regularity along a Lagrangian submanifold by a special choice of the phase \(g\). A trace formula relating the wave front of the restriction \(u|_ N\) to a submanifold to the wave front of \(u\) and the second microlocalization concludes the second chapter.

The third chapter recalls the subanalytic sets, maps, and stratifications. Then the critical points of locally Lipschitzian subanalytic functions are studied to derive the geometric upper bound for the singular spectrum, and for the second microsupport of a distribution defined as boundary values of a certain ramified function. The results cannot be easily explained owing to the technical complexity.

The fourth chapter applies the preceding methods to the Cauchy problem \[ \partial^ 2u/\partial t^ 2-\sum^ d\partial^ 2u/\partial x^ 2_ j=\sum p_ k(t,x)u^ k\quad (u\in C^ 0(\mathbb{R}_ t,H^ s), \]

\[ u|_{t=0}=u_ 0\in H^ s_{loc},\quad\partial u/\partial t|_{t=0}=u_ 1\in H^{s-1}_{loc} \] (where \(s>d/2)\). The Lebeu theorem concerning the geometric upper bound for the wave front if the Cauchy data are conormal along an analytic submanifold \(V\) of the hyperplane \(t=0\) is proved. The result is valid in large time, after the formation of caustics. Close to \(t=0\), the singularities behave like in the linear case but on a longer interval of time, a nonlinear interaction appears. At last, an improvement of the result for the case \(d=2\) if the union of bicharacteristics issued from \(V\) develops as a swallow-tail singularity concludes the book.

Reviewer: J.Chrastina (Brno)

### MSC:

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |

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

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35L70 | Second-order nonlinear hyperbolic equations |

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

35L05 | Wave equation |

35L67 | Shocks and singularities for hyperbolic equations |