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Harmonic analysis and pseudodifferential analysis of the light cone. (Analyse harmonique et analyse pseudo-différentielle du cône de lumière.) (French) Zbl 0643.35118

Centre National de la Recherche Scientifique (France). Astérisque, 156 (1987). Paris: Société Mathématique de France. 201 p. FF 135.00; $ 24.00 (1988).
The author considers the cone in \({\mathbb R}_ t\times {\mathbb R}^ n_ x\), \(C=\{(t,x)\), \(r(t,x)>0\}\), where \(r(t,x)=t^2-x^2_1-\dots-x^2_n\). The domain \(C\) is a symmetric space for a Riemannian structure invariant under the Lorentz group and the transformation \((t,x)\to (at,ax)\), \(a>0.\)
Pseudodifferential operators \(A\) are introduced on \(C\), with symbol \(\sigma\) defined on the cotangent space \(T^*(C)\), which admits a group of affine transformations containing the Poincaré group. In particular, when \(n=0\) and \(C\) reduces to \({\mathbb R}_+\), the operator \(A\) associated to the symbol \(\sigma(t,\eta)\) turns out to be \[ Af(s)=\iint_{t>0}\exp (2\pi i(s-t)\eta)(s/t)^{1/2} \sigma ((st)^{1/2},\eta)f(t)\,dt\, d\eta. \] The main result of the book is the boundedness of the operators with classical symbol on the space \(L^ 2(C)\) of the functions \(u\) such that \[ \int_{C}| u(t,x)|^ 2 r(t,x)^{-(n+1)/2}\, dt\, dx<\infty. \] A symbolic calculus is also developed modelling the standard pseudo-differential calculus on \({\mathbb R}^ n,\) with covariance under the above group action.
Some applications are given to partial differential equations in \(C\), with emphasis on problems from relativistic mechanics.
Reviewer: L.Rodino

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
58J40 Pseudodifferential and Fourier integral operators on manifolds
81S10 Geometry and quantization, symplectic methods
22E43 Structure and representation of the Lorentz group
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C35 Differential geometry of symmetric spaces
58J70 Invariance and symmetry properties for PDEs on manifolds
81R30 Coherent states
81T20 Quantum field theory on curved space or space-time backgrounds