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An accuracy improvement in Egorov's theorem.

Jorge Drumond Silva (2007)

Publicacions Matemàtiques

We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in Sm/Sm-1 for the pseudodifferential operators, one uses refined principal symbols in Sm/Sm-2, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally...

Analytic index formulas for elliptic corner operators

Boris Fedosov, Bert-Wolfgang Schulze, Nikolai Tarkhanov (2002)

Annales de l’institut Fourier

Spaces with corner singularities, locally modelled by cones with base spaces having conical singularities, belong to the hierarchy of (pseudo-) manifolds with piecewise smooth geometry. We consider a typical case of a manifold with corners, the so-called "edged spindle", and a natural algebra of pseudodifferential operators on it with special degeneracy in the symbols, the "corner algebra". There are three levels of principal symbols in the corner algebra, namely the interior,...

Anisotropic classes of homogeneous pseudodifferential symbols

Árpád Bényi, Marcin Bownik (2010)

Studia Mathematica

We define homogeneous classes of x-dependent anisotropic symbols γ , δ m ( A ) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent symbols in...

Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one.

Alexander V. Sobolev (2006)

Revista Matemática Iberoamericana

We consider a periodic pseudo-differential operator on the real line, which is a lower-order perturbation of an elliptic operator with a homogeneous symbol and constant coefficients. It is proved that the density of states of such an operator admits a complete asymptotic expansion at large energies. A few first terms of this expansion are found in a closed form.

Boundary value problems and layer potentials on manifolds with cylindrical ends

Marius Mitrea, Victor Nistor (2007)

Czechoslovak Mathematical Journal

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis and Kral-Wedland . We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the...

Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

Fabio Nicola (2010)

Studia Mathematica

We study Fourier integral operators of Hörmander’s type acting on the spaces L p ( d ) c o m p , 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in L p . We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on L p ( d ) c o m p if the mapping x x Φ ( x , η ) is constant on the fibres, of codimension r, of an affine...

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