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1
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Molecules in coorbit spaces and boundedness of operators

100%
Gröchenig K., Piotrowski M.
Studia Mathematica
|
2009
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tom 192
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nr 1
61-77
EN
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.
2
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Gabor meets Littlewood-Paley: Gabor expansions in $L^{p}(ℝ^{d})$

100%
Gröchenig K., Heil C.
Studia Mathematica
|
2001
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tom 146
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nr 1
15-33
EN
It is known that Gabor expansions do not converge unconditionally in $L^{p}$ and that $L^{p}$ cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that $L^{p}$ can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in $L^{p}$-norm.
3
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On spectrality of the algebra of convolution dominated operators

81%
Fendle G., Gröchenig K., Leinert M.
Banach Center Publications
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2007
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tom 78
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nr 1
145-149
EN
If G is a discrete group, the algebra CD(G) of convolution dominated operators on l²(G) (see Definition 1 below) is canonically isomorphic to a twisted L¹-algebra $l¹(G,l^{∞}(G),T)$. For amenable and rigidly symmetric G we use this to show that any element of this algebra is invertible in the algebra itself if and only if it is invertible as a bounded operator on l²(G), i.e. CD(G) is spectral in the algebra of all bounded operators. For G commutative, this result is known (see [1], [6]), for G noncommutative discrete it appears to be new. This note is about work in progress. Complete details and more will be given in [3].
4
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Convolution-dominated integral operators

81%
Fendler G., Gröchenig K., Leinert M.
Banach Center Publications
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2010
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tom 89
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nr 1
121-127
EN
For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications \cite{fgl08a, fgl08}. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the products are restricted to those given by multiplication with left uniformly continuous functions. This algebra, $CD_{reg}(G)$, is canonically isomorphic to a twisted L¹-algebra. For amenable G that is rigidly symmetric as a discrete group we show the following result: An element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if and only if it is invertible as a bounded operator on L²(G). This report is about work in progress. Complete details and further results will be given in a paper still in preparation.
5
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Compactness criteria in function spaces

81%
Dörfler M., Feichtinger H., Gröchenig K.
Colloquium Mathematicum
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2002
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tom 94
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nr 1
37-50
EN
The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L²(ℝ^{d})$, and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces.
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