Results 1  10
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24
PSEUDODIFFERENTIAL OPERATORS WITH GENERALIZED SYMBOLS AND REGULARITY THEORY
, 2005
"... We study pseudodifferential operators with amplitudes aε(x, ξ) depending on a singular parameter ε → 0 with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for ε → 0, refined versions of estimates for classical pseudodifferential operators. W ..."
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We study pseudodifferential operators with amplitudes aε(x, ξ) depending on a singular parameter ε → 0 with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for ε → 0, refined versions of estimates for classical pseudodifferential operators. We apply these estimates to nets of regularizations of exotic operators as well as operators with amplitudes of low regularity, providing a unified method for treating both classes. Further, we develop a full symbolic calculus for pseudodifferential operators acting on algebras of Colombeau generalized functions. As an application, we formulate a sufficient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodifferential equations.
Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity
, 2006
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On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets
, 2005
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REAL ANALYTIC GENERALIZED FUNCTIONS
, 2007
"... Abstract. Real analytic generalized functions are investigated as well as the analytic singular support and analytic wave front of a generalized function in G(Ω) are introduced and described. ..."
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Abstract. Real analytic generalized functions are investigated as well as the analytic singular support and analytic wave front of a generalized function in G(Ω) are introduced and described.
Generalized Oscillatory Integrals and Fourier Integral Operators
, 2008
"... In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase functions as well as amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by th ..."
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Cited by 2 (1 self)
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In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase functions as well as amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need of a general framework for partial differential operators with nonsmooth coefficients and distribution data. The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets.
GEOMETRICAL EMBEDDINGS OF DISTRIBUTIONS INTO ALGEBRAS OF GENERALIZED FUNCTIONS
, 709
"... Abstract. We use spectral theory to produce embeddings of distributions into algebras of generalized functions on a closed Riemannian manifold. These embeddings are invariant under isometries and preserve the singularity structure of the distributions. 1. ..."
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Abstract. We use spectral theory to produce embeddings of distributions into algebras of generalized functions on a closed Riemannian manifold. These embeddings are invariant under isometries and preserve the singularity structure of the distributions. 1.
Banach ˜ Calgebras
, 811
"... We study Banach ˜ Calgebras, i.e., complete ultrapseudonormed algebras over the ring ˜ C of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach ˜ C ..."
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We study Banach ˜ Calgebras, i.e., complete ultrapseudonormed algebras over the ring ˜ C of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach ˜ Calgebras and indicate a particular class of Banach ˜ Calgebras that overcomes these limitations to a large extent. We also investigate C ∗algebras over ˜ C. Key words: algebras of generalized functions, Banach algebras, C ∗algebras.
Generalized analytic functions on generalized domains
"... We define the algebra ˜ G(A) of Colombeau generalized functions on a subset A of the space of generalized points ˜ R d. If A is an open subset of ˜ R d, such generalized functions can be identified with pointwise maps from A into the ring of generalized numbers ˜ C. We study analyticity in ˜ G(A), w ..."
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We define the algebra ˜ G(A) of Colombeau generalized functions on a subset A of the space of generalized points ˜ R d. If A is an open subset of ˜ R d, such generalized functions can be identified with pointwise maps from A into the ring of generalized numbers ˜ C. We study analyticity in ˜ G(A), where A is an open subset of ˜ C. In particular, if the domain is an open ball for the sharp norm on ˜C, we characterize analyticity and give a unicity theorem involving the values at generalized points. Key words: algebras of generalized functions. 2000 Mathematics subject classification: 46F30. 1
SPHERICAL COMPLETENESS OF THE NONARCHIMEDEAN RING OF COLOMBEAU GENERALIZED NUMBERS
, 2007
"... Abstract. We show spherical completeness of the ring of Colombeau generalized real (or complex) numbers endowed with the sharp norm. As an application, we establish a HahnBanach extension theorem for ultrapseudonormed modules (over the ring of generalized numbers) of generalized functions in the ..."
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Abstract. We show spherical completeness of the ring of Colombeau generalized real (or complex) numbers endowed with the sharp norm. As an application, we establish a HahnBanach extension theorem for ultrapseudonormed modules (over the ring of generalized numbers) of generalized functions in the sense of Colombeau. 1.
Regularity, Local and Microlocal Analysis in Theories of Generalized Functions
, 2008
"... We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the Blocal analysis of sections of A. But the microlocal ana ..."
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We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the Blocal analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a ”frequential microlocal analysis” and into a ”microlocal asymptotic analysis”. The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations.