Displaying similar documents to “A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators”

On microlocal analyticity of solutions of first-order nonlinear PDE

Shif Berhanu (2009)

Annales de l’institut Fourier

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We study the microlocal analyticity of solutions u of the nonlinear equation u t = f ( x , t , u , u x ) where f ( x , t , ζ 0 , ζ ) is complex-valued, real analytic in all its arguments and holomorphic in ( ζ 0 , ζ ) . We show that if the function u is a C 2 solution, σ ∈ Char L u and 1 i σ ( [ L u , L u ¯ ] ) < 0 or if u is a C 3 solution, σ ∈ Char L u , σ ( [ L u , L u ¯ ] ) = 0 , and σ ( [ L u , [ L u , L u ¯ ] ] ) ≠ 0 , then σ ∉ W F a u . Here W F a u denotes the analytic wave-front set of u and Char L u is the characteristic set of the linearized operator. When m = 1 , we prove a more general result involving the repeated brackets of L u and L u ¯ of any...

Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Kenichi Ito, Shu Nakamura (2012)

Annales de l’institut Fourier

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We consider Schrödinger operators H on ℝ n with variable coefficients. Let H 0 = - 1 2 ▵ be the free Schrödinger operator and we suppose H is a “short-range” perturbation of H 0 . Then, under the nontrapping condition, we show that the time evolution operator: e - i t H can be written as a product of the free evolution operator e - i t H 0 and a Fourier integral operator W ( t ) which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators....

On the Fefferman-Phong inequality

Abdesslam Boulkhemair (2008)

Annales de l’institut Fourier

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We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by n 2 + 4 + ϵ improving thus the bound 2 n + 4 + ϵ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type S 0 , 0 0 , we show that this number is bounded by n + 4 + ϵ ; more precisely, for a non negative symbol a , the Fefferman-Phong inequality holds if ∂ x α ∂ ξ β a ( x , ξ ) are bounded for, roughly, 4 ≤ | α | + | β | ≤ n + 4 + ϵ . To obtain such results and others, we first prove an abstract result which...

Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle

Christiane Rousseau, Colin Christopher (2007)

Annales de l’institut Fourier

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We consider germs of one-parameter generic families of resonant analytic diffeomorphims and we give a complete modulus of analytic classification by means of the unfolding of the Écalle modulus. We describe the parametric resurgence phenomenon. We apply this to give a complete modulus of orbital analytic classification for the unfolding of a generic resonant saddle of a 2-dimensional vector field by means of the unfolding of its holonomy map. Here again the modulus is an unfolding of...

An o-minimal structure which does not admit C ∞ cellular decomposition

Olivier Le Gal, Jean-Philippe Rolin (2009)

Annales de l’institut Fourier

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We present an example of an o-minimal structure which does not admit C ∞ cellular decomposition. To this end, we construct a function H whose germ at the origin admits a C k representative for each integer k , but no C ∞ representative. A number theoretic condition on the coefficients of the Taylor series of H then insures the quasianalyticity of some differential algebras 𝒜 n ( H ) induced by H . The o-minimality of the structure generated by H is deduced from this quasianalyticity property. ...