Microhyperbolic Operators in Gevrey Classes

Abstract

. Introduction Kashiwara and Kawai [16] defined microhyperbolicity and proved that the microlocal Cauchy problem for microhyperbolic pseudodifferential operators is well-posed in the framework of microfunctions, which is a microlocalization of the results obtained by Bony and Schapira [3].In the microlocal studies of pseudo-differential operators, the concept of microhyperbolicity is very useful.From their results one can obtain results on propagation of analytic singularities (propagation of micro-analyticities) of solutions for microhyperbolic operators (see [28]). On the other hand, Bronshtein [5] proved that the hyperbolic Cauchy problem is well-posed in some Gevrey classes which are intermediate spaces between the space of real analytic functions and C°° (see, also, [14], [15]). So we can generalize the definition of microhyperbolicity in the framework of some Gevrey classes, to say the least of it. In doing so, we expect to get a clue to a generalization of microhyperbolicity and microlocal studies of microhyperbolic operators in the framework of C°°.In this paper we shall consider microhyperbolic operators in Gevrey classes and prove microlocal well-posedness of the microlocal Cauchy problem and theorems on propagation of singularities for microhyperbolic operators. Our aims are to show how one can obtain microlocal results (microlocal well-posedness and, therefore, a microlocal version of Holmgren's uniqueness theorem) from methods to prove well-posedness of the Cauchy problem and to show that theorems on propagation of singularities are immediate consequences of a microlocal version of Holmgren's uniqueness theorem, using

Local solvability for partial differential equations with multiple characteristics in mixed Gevrey-C ∞ spaces

“…Spaces of the kind of H s,ψ φ,r have been first introduced by Kajitani-Wakabayashi [16] and then used in various situations in the study of partial differential equations in Gevrey classes, cf. for example Gramchev-Rodino [10] for the isotropic case, Marcolongo-Oliaro [20], De Donno-Oliaro [6] in the anisotropic frame, Bourdaud-Reissig-Sickel [1] where the composition is investigated, Jornet-Oliaro [15] where spaces related to H s,ψ φ,r are considered in the frame of ultradifferentiable functions.…”

“…The conjugation allows us to transfer on P(w, D) the local solvability in mixed Gevrey-C ∞ classes. This technique has already been used in some papers, starting from the work of Kajitani-Wakabayashi [16]; we refer also to Gramchev-Rodino [10] for the isotropic case, Marcolongo-Oliaro [20] in the anisotropic frame, and to De Donno-Oliaro [6,7], in which the influence of the lower order terms is also taken into account. One of the novelties here is that we propose in the conjugation, a microlocalisation near the anisotropic characteristic set {(w, ζ ) ∈ × (R p+q \{0}) : p 1 (w, ξ ) − p 2 (w, η) = 0}, which allows us to obtain solvability in mixed Gevrey-C ∞ classes, whereas the results in the above quoted papers are only in Gevrey.…”

Quantized Contact Transformations and Pseudodifferential Operators of Infinite Order

“…Suppose that (1.13) for PeS Xip^ and QeS^. If we let cP| operate on the both members of (1.13), we have (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) AoQ(x'^f) = 0^2(0 1 (P))(x f^f )…”

The <i>C</i>∞-well posed Cauchy problem for hyperbolic operators dominated by time functions

“…(ii) We can also define 'time functions' for microhyperbolic functions (sym bols) (see [8] and [20]). We note that the ƒÉj(x, ƒÌ') are locally Lipschitz continuous (see [2], [21]).…”