Coorbit space theory for quasi-Banach spaces

Rauhut, Holger

doi:10.4064/sm180-3-4

Cited by 57 publications

(85 citation statements)

References 17 publications

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“…If we ask for more regularity on the window g, then the previous result can be extended to suitable (quasi-)Banach spaces, as shown below ( [14,18] and [28,Theorem 8.3]). …”

Section: Gabor Frames [14]mentioning

confidence: 98%

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Sparsity of Gabor representation of Schrödinger propagators

Cordero

Nicola

Rodino

2009

Applied and Computational Harmonic Analysis

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Recent papers show how tight frames of curvelets and shearlets provide optimally sparse representation of hyperbolic-type Fourier integral operators (FIOs) [E.J. Candés, L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris 336 (5) (2003) 395-398; E.J. Candés, L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. In this paper we address to another class of FIOs, employed by Helffer and Robert to study spectral properties of globally elliptic operators of quantum mechanics [B. Helffer, Théorie spectrale pour des operateurs globalement elliptiques, Astérisque, Société Mathématique de France, 1984; B. Helffer, D. Robert, Comportement asymptotique precise du spectre d'operateurs globalement elliptiques dans R d , Sem. Goulaouic-Meyer-Schwartz 1980-81, École Polytechnique, 1980, Exposé II], and hence studied by many other authors, see, e.g., [A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett. 4 (1997) 53-67; F. Concetti, J. Toft, Schatten-von Neumann properties for Fourier integral operators with non-smooth symbols I, Ark. Mat., in press]. An example is provided by the resolvent of the Cauchy problem for the Schrödinger equation with a quadratic Hamiltonian. We show that Gabor frames provide optimally sparse representations of such operators. Numerical examples for the Schrödinger case demonstrate the fast computation of these operators.

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“…If we ask for more regularity on the window g, then the previous result can be extended to suitable (quasi-)Banach spaces, as shown below ( [14,18] and [28,Theorem 8.3]). …”

Section: Gabor Frames [14]mentioning

confidence: 98%

“…The quasi-Banach cases were first introduced in [16], then studied in [25,28]. In particular, these spaces are examples of the general coorbit space theory for quasi-Banach spaces studied in [28].…”

Section: Modulation Spacesmentioning

confidence: 98%

Sparsity of Gabor representation of Schrödinger propagators

Cordero

Nicola

Rodino

2009

Applied and Computational Harmonic Analysis

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“…We also recall the definition of the so-called Wiener amalgam spaces [2,3,10]. First of all we denote by L p s , s ∈ R, 0 < p ∞, the weighted L p space of functions f in R d such that x s f (x) is in L p , with the obvious quasi-norm (norm if p 1).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Remarks on Fourier multipliers and applications to the wave equation

Cordero

Nicola

2009

Journal of Mathematical Analysis and Applications

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Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [A. Bényi, K.A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, preprint, April 2007 (available at ArXiv:0704.0833v1)]. The same arguments may apply to obtain local wellposedness for the NLKG equation.

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“…Theorem 3.5 can probably be formulated for quasi-Banach spaces as well by using Rauhut's extension of coorbit space theory [24].…”

Section: The Last Inequality Above Follows Immediately From Propositimentioning

confidence: 99%

Molecules in coorbit spaces and boundedness of operators

Gröchenig

Piotrowski

2009

Studia Math.

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Abstract. We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For time-frequency molecules we recover some boundedness results on modulation spaces, for time-scale molecules we obtain the boundedness on homogeneous Besov spaces.1. Introduction. A remarkable principle of classical analysis states that an operator that maps atoms to molecules is bounded. Here an "atom" is a function on R d satisfying certain support and moment conditions, and norm bounds. Atoms arose first in the study of atomic decompositions of real Hardy spaces [2] and singular integral operators on Hardy spaces (see [13,25]. The notion of an atom was later diversified to adapt to the BesovTriebel-Lizorkin spaces [10], and then generalized to "molecules", which are functions satisfying norm bounds, moment and decay conditions (instead of support conditions); see [3,10]. The resulting molecular decompositions of function spaces have been successfully applied to study the boundedness properties of Calderón-Zygmund operators on Besov-Triebel-Lizorkin spaces (see [8][9][10][11]27] for some of the main contributions). The technical part of the proofs is to show that the operator under consideration maps smooth atoms into smooth molecules. Using norm estimates for atomic and molecular decompositions, one then obtains the boundedness of the operator. A similar strategy has been used in [14] to study a class of pseudodifferential operators on Besov-Triebel-Lizorkin spaces.In this paper we study atoms, molecules, and the boundedness of operators in the context of coorbit theory. In coorbit theory one can attach to every irreducible, unitary, integrable representation π of a locally compact group G on a Hilbert space H a class of π-invariant Banach spaces Co Y that is parametrized by function spaces Y on the group G. These

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Coorbit space theory for quasi-Banach spaces

Cited by 57 publications

References 17 publications

Sparsity of Gabor representation of Schrödinger propagators

Sparsity of Gabor representation of Schrödinger propagators

Remarks on Fourier multipliers and applications to the wave equation

Molecules in coorbit spaces and boundedness of operators

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