Abstract. We introduce a τ -dependent Wigner representation, Wig τ , τ ∈ [0, 1], which permits us to define a general theory connecting time-frequency representations on one side and pseudo-differential operators on the other. The scheme includes various types of time-frequency representations, among the others the classical Wigner and Rihaczek representations and the most common classes of pseudo-differential operators. We show further that the integral over τ of Wig τ yields a new representation Q possessing features in signal analysis which considerably improve those of the Wigner representation, especially for what concerns the so-called "ghost frequencies". The relations of all these representations with respect to the generalized spectrogram and the Cohen class are then studied. Furthermore, a characterization of the L p -boundedness of both τ -pseudo-differential operators and τ -Wigner representations are obtained.

In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, W F * , of classical distributions. In particular, we have the following inclusionwhere Ω is an open subset of R n , P is a linear partial differential operator with coefficients in a suitable ultradifferentiable class, and Σ is the characteristic set of P . Some applications are also investigated.Mathematics Subject Classification (2010). Primary 46F05; Secondary 35A18, 35A21.

Given a non-quasianalytic subadditive weight function ω we consider the weighted Schwartz space S ω and the short-time Fourier transform on S ω , S ′ ω and on the related modulation spaces with exponential weights. In this setting we define the ω-wave front set WF ′ ω (u) and the Gabor ω-wave front set WF G ω (u) of u ∈ S ′ ω , and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.2010 Mathematics Subject Classification. Primary 35A18; Secondary 46F05, 42C15, 35S05.

Abstract. We study the behaviour of linear partial differential operators with polynomial coefficients via a Wigner type transform. In particular, we obtain some results of regularity in the Schwartz space S and in the space S ω as introduced by Björck for weight functions ω. Several examples are discussed in this new setting.

We study boundedness and compactness properties for the Weyl quantization with symbols in L q (R 2d ) acting on L p (R d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L q * (R 2d ) and L q (R 2d ), respectively smaller and larger than the L q (R 2d ), and showing that the Weyl correspondence is bounded on L q * (R 2d ) (and yields compact operators), whereas it is not on L q (R 2d ). We conclude with a remark on weak-type L p boundedness.