Localized Frames and Compactness

Abstract

Abstract. We introduce the concept of weak localization for continuous frames and use this concept to define a class of weakly localized operators. This class contains many important classes of operators, including: Short Time Fourier Transform multipliers, CalderonToeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, some pseudodifferential operators, some Calderon-Zygmund operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transforms.

Some notes about distribution frame multipliers

“…For instance, it is known that in L(D, D × ) a partial multiplication is defined (see [2,46]), thus a symbolic calculus may be developed for multipliers in L(D, D × ). The idea behind Definition 4.3 are connected to localization frames which were introduced in [27] and further studied (sometimes with variations) in [6,7,11,21,22,28]. More precisely, some result in Section 4 can be extended considering a certain decay of x → f |ω x instead of assume that f |ω x is null outside a bounded set.…”

On dual molecules and convolution-dominated operators

“…for a well-localized envelope function Φ ∈ L 1 (G); see for instance [6,35,86]. Although we do not use any specific results from the literature, we formulate many of the technical lemmas in terms of such operators.…”

“…In this section we introduce the class of integral kernels that will be considered in the remainder. See[6,35,37,82,86] for related notions.…”