Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators Aaφ1,φ2 with symbols a in the wide weighted modulation space Mw∞(R2d), and windows φ1, φ2 in the Gelfand–Shilov space S(1)(Rd). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of Aaφ1,φ2 have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space S(γ)(Rd) , where the parameter γ≥ 1 is related to the growth of the considered weight. An important role is played by τ-pseudodifferential operators Opτ(σ). In that direction we show convenient continuity properties of Opτ(σ) when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Opτ(σ) when the symbol σ belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces

Trace ideals for Fourier integral operators with non-smooth symbols II

We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.Comment: 25 page

Duality covariant quantum field theory on noncommutative Minkowski space

We prove that a scalar quantum field theory defined on noncommutative Minkowski spacetime with noncommuting momentum coordinates is covariant with respect to the UV/IR duality which exchanges coordinates and momenta. The proof is based on suitable resonance expansions of charged noncommutative scalar fields in a background electric field, which yields an effective description of the field theory in terms of a coupled complex two-matrix model. The two independent matrix degrees of freedom ensure unitarity and manifest CT-invariance of the field theory. The formalism describes an analytic continuation of the renormalizable Grosse-Wulkenhaar models to Minkowski signature.Comment: 32 pages; v2: Typos corrected; v3: Further typos corrected - Final version to appear in JHE

Time–frequency localization operators: State of the art

We present localization operators via the short-time Fourier transform. For both modulation and ultra-modulation spaces framework, well-known results about boundedness and Schatten-von Neumann class are reported. Asymptotic eigenvalues’ distribution and decay and smoothness properties for L2-eigenfunctions are exhibited. Eventually, we make a conjecture about smoothness of L2-eigenfunctions for localization operators with Gelfand–Shilov windows and symbols in ultra-modulation spaces