The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators

Abstract

We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are well-known when the weights are moderated. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators in the context of these modulation spaces. 9

Bilinear Localization Operators on Modulation Spaces

“…We refer to [20,21,[29][30][31] for the proof and more details on STFT in other spaces of Gelfand-Shilov type.…”

“…We restrict ourselves to ( , ) = ⟨ ⟩ ⟨ ⟩ , , ∈ R, since the convolution and multiplication estimates which will be used later on are formulated in terms of weighted spaces with such polynomial weights. As already mentioned, weights of exponential type growth are used in the study of GelfandShilov spaces and their duals in [16,[29][30][31]. We refer to [36] for a survey on the most important types of weights commonly used in time-frequency analysis.…”

On the Inverse to the Harmonic Oscillator

“…The definition extends uniquely to any a ∈ S ′ 1/2 (R 2d ), and then [18].) In the case t = 0, then Op 0 (a) agrees with the Kohn-Nirenberg representation a(x, D), and if t = 1/2, then Op 1/2 (a) is equal to the Weyl quantization Op w (a).…”

“…To obtain (0.8) for |X| ≥ 1, we finally use the fact that by (2.2) we have 18) for |X| = 0. Here it follows by induction on |α| that…”

“…It follows that H is an invertible and globally elliptic operator on S and S s , and their dual spaces, for every s ≥ 1/2 (cf. e. g. [17,18] ′ , and their proofs that these results remain valid after the standard harmonic oscillator has been replaced by the operator in (2.29).…”