Abstract
We introduce a class of bilinear localization operators and show how to interpret them as bilinear Weyl pseudodifferential operators. Such interpretation is well known in linear case whereas in bilinear case it has not been considered so far. Then we study continuity properties of both bilinear Weyl pseudodifferential operators and bilinear localization operators which are formulated in terms of a modified version of modulation spaces.
Highlights
Localization operators were introduced by Berezin in the study of general Hamiltonians related to quantization problem in quantum mechanics [1]
Thereafter, a more detailed study of localization in phase space together with basic facts on localization operators and their applications in optics and signal analysis was given by Daubechies in [3]
To define localization operators we start with the short-time Fourier transform, a time-frequency representation related to Feichtinger’s modulation spaces [25, 26]
Summary
Localization operators were introduced by Berezin in the study of general Hamiltonians related to quantization problem in quantum mechanics [1]. In contrast to [15] in this paper we do not observe multilinear version of localization operators, since we use sharp convolution estimates for modulation spaces given in [24]; see Theorems 8 and 9. These results are well suited for the study of bilinear operators, but their extension to multilinear case is a challenging problem. These results are boundedness of bilinear ΨDOs on Mps,,tq (Theorem 14) and sufficient and necessary conditions for boundedness of bilinear localization operators on Mps,,tq (Theorems 15 and 16, resp.). We use the notation A ≲ B to indicate that A ≤ cB for a suitable constant c > 0, whereas A ≍ B means that c−1A ≤ B ≤ cA for some c ≥ 1
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