Régularité lipschitzienne et solutions de l'équation des ondes sur une variété riemannienne compacte

Verdìère, Yves Colin de; Frisch, M.

doi:10.24033/asens.1318

Cited by 6 publications

(1 citation statement)

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“…The L p properties of solutions to hyperbolic Cauchy problems for the equations of the wave type have been studied in many papers ( [59], [43], [40], [15]). Lipschitz and L p estimates for the wave equation on compact manifolds were derived in [19] and some results for hyperbolic equations are in [63]. General results on L p continuity were obtained in [56].…”

Section: Regularity Of Fourier Integral Operatorsmentioning

confidence: 99%

Singularities of affine fibrations in the regularity theory of Fourier integral operators

Ruzhansky¹

2000

Russ. Math. Surv.

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In the paper the regularity properties of Fourier integral operators are considered in different function spaces. The most interesting case are L p spaces, for which a survey of recent results is made. Thus, the sharp orders are known for operators, satisfying the so-called smooth factorization condition. Further in the paper this condition is analyzed in both real and complex settings. In the last case conditions for the continuity of Fourier integral operators are related to the singularities of affine fibrations in (subsets of) C n , defined by the kernels of Jacobi matrices of holomorphic mappings. Singularities of such fibrations are analyzed in this paper in the general case. In particular, it is shown that for small dimensions n or for small ranks of the Jacobi matrix, all singularities of an affine fibration are removable. Fourier integral operators lead to fibrations, given by the kernels of the Hessian of a phase function of the operator. Based on the analysis of singularities for operators, commuting with translations, in a number of case the factorization condition is shown to be satisfied, which leads to L p estimates for operators. In the other cases, the failure of the factorization condition is exhibited by a number of examples. Results are applied to derive L p estimates for solutions of the Cauchy problem for hyperbolic partial differential operators.

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