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

The continuity envelope for the Besov and Triebel-Lizorkin spaces of generalized smoothness B (s, ) pq (R n ) and F (s, ) pq (R n ), respectively, are computed in the critical case s = n/p, provided that satisfies an appropriate critical condition. Surprisingly, in this critical situation, the corresponding optimal index is ∞, when compared with all the known results. Moreover, in the particular case of the classical spaces, we solve an open problem posed by Haroske in Envelopes and Sharp Embeddings of Function Spaces, Research Notes in Mathematics, vol. 437, Chapman & Hall, Boca Raton, 2007. As an immediate application of our results we give an upper estimate for approximation numbers of related embeddings.

The paper investigates the asymptotic behaviour of entropy and approximation numbers of compact embeddings between weighted modulation spaces.

Abstract. The main purpose of the present paper is to extend the theory of non-smooth atomic decompositions to anisotropic function spaces of Besov and TriebelLizorkin type. Moreover, the detailed analysis of the anisotropic homogeneity property is carried out. We also present some results on pointwise multipliers in special anisotropic function spaces.

A linear and bounded operator T between Banach spaces X and Y has Fourier type 2 with respect to a locally compact abelian group G if there exists a constant c > 0 such that Tf 2 ≤ c f 2 holds for all X-valued functions f ∈ L X 2 (G) wheref is the Fourier transform of f . We show that any Fourier type 2 operator with respect to the classical groups has Fourier type 2 with respect to any locally compact abelian group. This generalizes previous special results for the Cantor group and for closed subgroups of R n .