Global wave-front sets of Banach, Fréchet and modulation space types, and pseudo-differential operators

Coriasco, Sandro; Johansson, Kristina; Toft, Joachim

doi:10.1016/j.jde.2013.01.014

Cited by 27 publications

(61 citation statements)

References 30 publications

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“…On the other hand, at present time, Hörmander's notion of the wave front set attracts a lot of interest among mathematicians and there exists a vast literature related to this basic notion and its important role in the qualitative analysis of PDE and ΨDO. We mention the basic books of Hörmander [5,6] as standard references for classical and Sobolev type wave front sets; the articles [9,7] deal with weighted type wave fronts, while [3,4] study extended wave fronts by considering local and global versions with respect to various Banach and Fréchet spaces of functions over the configuration and the frequency domains.…”

Section: Introductionmentioning

confidence: 99%

Wave fronts via Fourier series coefficients

Maksimović

Pilipović

Sokoloski

et al. 2015

Publ Inst Math (Belgr)

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Section: Introductionmentioning

confidence: 99%

Wave fronts via Fourier series coefficients

Maksimović

Pilipović

Sokoloski

et al. 2015

Publ Inst Math (Belgr)

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“…pSpR n q, SpR n qq, pH r,̺ pR n q, H r´m,̺´µ pR n qq, r, ̺ P R, pS 1 pR n q, S 1 pR n qq are SG-ordered (with respect to any pm, µq P R 2 ). The same holds true for (suitable couples of) modulation spaces, see [15]. is continuous for every a P S m,µ pR 2n q with support such that the projection on the ξ-axis does not intersect R n zΩ.…”

Section: Definition 59 Let B Be a Topological Vector Space Of Distrmentioning

confidence: 77%

“…For any R ą 0 and m P t1, 2, 3u, we set Ω 1,R " t px, ξq P Ω 1 ; |ξ| ě R u, Ω 2,R " t px, ξq P Ω 2 ; |x| ě R u, Ω 3,R " t px, ξq P Ω 3 ; |x|, |ξ| ě R u Evidently, Ω R m is m-conical for every m P t1, 2, 3u. From now on we assume that B in Definition 5.17 is SG-admissible, and recall that Sobolev-Kato spaces and, more generally, modulation spaces, and S pR d q are SG-admissible, see [15,16].…”

Section: Remark 519mentioning

confidence: 99%

Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on $$\mathbb {R}^n$$

2020

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We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R n . We prove wellposedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To this aim, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such expressions. 2010 Mathematics Subject Classification. Primary: 58J40; Secondary: 35S05, 35S30, 47G30, 58J45. Key words and phrases. Fourier integral operator, multi-product, hyperbolic Cauchy problem, involutive characteristics, stochastic PDEs. of operators such that $ & % LEpt, sq " 0 0 ď s ď t ď T, Eps, sq " I, s P r0, Tq, so obtaining by Duhamel's formula the solution Uptq " Ept, 0qG`i ż t 0Ept, sqFpsqdsto the system and then the solution uptq to the corresponding higher order Cauchy problem (1.1) by the reduction procedure.In the present paper we consider the Cauchy problem (1.1) in the SG setting, that is, under the growth condition (1.3), the hyperbolicity condition (1.4) and the involutiveness assumption (1.5) for suitable parameter-dependent, real-valued symbols b jk , d jk P C 8 pr0, Ts; S 0,0 pR 2n qq, j, k P N. The involutiveness of the characteristic roots of the operator, or, equivalently, of the eigenvalues of the (diagonal) principal part of the corresponding first order system is here the main assumption (see Assumption I in Section 3.3 for the precise statement of such condition).We obtain several results concerning the Cauchy problem (1.1): ‚ we prove, in Theorem 5.8, the existence, for every f P C 8 pr0, Ts; H r,ρ pR n qq and g k P H r`m´1´k,ρ`m´1´k pR n q, k " 0, . . . , m´1, of a uniquefor a suitably small 0 ă T 1 ď T, solution of (1.1); we also provide, in (5.15), the explicit expression of u, using SG Fourier integral operators (SG FIOs); ‚ we give a precise description, in Theorem 5.21, of a global wave-front set of the solution, under a (mild) additional condition on the operator L; namely, we prove that, when L is SG-classical, the (smoothness and decay) singularities of the solution of (1.1) with f " 0 are points of unions of arcs of bicharacteristics, generated by the phase functions of the SG FIOs appearing in the expression (5.15), and emanating from (smoothness and decay) singularities of the Cauchy data g k , k " 0, . . . , m´1; ‚ we deal, in Theorem 6.3, with a stochastic version of the Cauchy proble...

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“…Many authors considered various generalisations and characterisations of the Sobolev wave front set and other similar variants; see [18,22,23,24,26,27] and the references there in. This concept was further generalised recently in [5,6] where the wave front set is defined with respect to a general Banach spaces of distributions satisfying appropriate assumptions. In the setting of ultradistributions, the wave front set with respect to Fourier-Lebesgue spaces having sub-exponential weights was considered in [7,16] where the authors also gave a discrete characterisation of it.…”

Section: Introductionmentioning

confidence: 99%