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We first give a short survey on the methods of Microlocal Analysis. In particular we recall some basic facts concerning the theory of pseudodifferential operators. We then present two applications. We first discuss lower bounds for operators with multiple characteristics. Then we give a new formula for the composition of Wick operators.
Ellipticity of operators on manifolds with conical singularities or parabolicity on space-time cylinders are known to be linked to parameter-dependent operators (conormal symbols) on a corresponding base manifold. We introduce the conormal symbolic structure for the case of corner manifolds, where the base itself is a manifold with edges and boundary. The specific nature of parameter-dependence requires...
The aim of this article is to provide a new method to prove global smoothing estimates for dispersive equations such as Schrödinger equations. For the purpose, the Egorov-type theorem via canonical transformation in the form of a class of Fourier integral operators is established, and their weighted L2-boundedness is also proved. The boundedness result is not covered by previous one such as Asada...
We characterize the hypoellipticity inC∞and GevreyGλclasses of 2-variable PDO’s containing powers of anisotropic principal terms. We use an approach based on methods from microlocal analysis. Conditions are imposed on the coefficients of lower order terms. Also a semilinear version is proposed consideringC...
We consider symbolsa(xξ)with a finite number of bounded derivatives with respect to ξ and of weighted Sobolev type inx.Their continuity in weighted Sobolev spaces of sufficiently large order is studied.
Using the symmetric formula for the Wigner transform, we give LP-boundedness results for Weyl transforms with symbols in L1 (ℝn } ℝn). Breaking the symmetry of the Wigner transform, we obtain new results for Weyl transforms with symbols in Lp (ℝn } ℝn),...
For a Hörmander’s symbol classS(mg)it is proved that the weight m is in Lp(ℝ2n, with 1 ⩽ p ⩽ ∞ if and only if all pseudo-differential operators with Weyl symbol inS(mg)are in the Schatten-von Neumann class Sp (L2). Mathematics Subject Classification (2000).Primary 47B10; Secondary 35S05.
A systematic overview of localization operators using a time-frequency approach is given. Sufficient and necessary regularity results for localization operators with symbols and windows living in various function spaces (such as LP or modulation spaces) are discussed. Finally, an exact and an asymptotic product formulae are presented.
Boundedness of localization operators on modulation spaces is studied obtaining results for operators with symbol in Lp(ℝ2n) 1 ≤ p ≤ ∞. In this context the results presented here generalize well-known properties of boundedness and compactness.
Let Mωp,q be the modulation space with parameters p,q ∈ [1,∞] and weight function ω. Also let Mp,q = Mω0p,q , ω0=1. We prove that for certain w, there is a canonical homeomorphism Mω...
Pseudo-differential operators with symbols supported on sectors of dyadic annuli in the Fourier domain are used to perform microlocal analysis of tempered distributions. Microlocal analysis is recalled. The above symbols are made of smooth wavelet frames which are constructed in the Fourier domain by means of modulated smooth tapered functions. The method is used to localize a line of singularities...
A review of system identification based on distribution theory is given. By the Schwartz kernel theorem, to every continuous linear system, there corresponds a unique distribution, calledkernel distribution. Formulae using wavelet transform to access time-frequency information of the kernel distribution are deduced. An application of the formula to system identification of a health...
Partial differential operators are represented in non-standard form in separable two-dimensional orthonormal wavelet bases. A formal pseudodifferential approach is suggested for numerical applications. Applications are to the fifth-order quasilinear thin-film equation on an inclined plane. Mathematics Subject Classification (2000).Primary 65T60; Secondary 35Q51.
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