Regularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods

In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity properties.

Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending previous work by Langenbruch. As a consequence we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.

Global regularity of second order twisted differential operators

In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension $2$. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree $2$. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.

Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

We use techniques from time-frequency analysis to show that the space $mathcal S_omega$ of rapidly decreasing $omega$-ultradifferentiable functions is nuclear for every weight function $omega(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $mathcal S_{(M_p)}$ when defined with $L^{2},$norms is nuclear exactly when condition $(M2)$ of Komatsu holds.

About the nuclearity of ${mathcal S}_{(M_{p})}$ and ${mathcal S}_{omega}$

We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space ${mathcal S}_{(M_p)}$ to be nuclear. As a consequence, we obtain that for a weight function $omega$ satisfying the mild condition: $2omega(t)leq omega(Ht)+H$ for some $H>1$ and for all $tgeq0$, the space ${mathcal S}_omega$ in the sense of Bjorck is also nuclear.

Real Paley-Wiener theorems in spaces of ultradifferentiable functions

We develop real Paley-Wiener theorems for classes ${mathcal S}_omega$ of ultradifferentiable functions and related $L^{p}$-spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.

Cohen class of time-frequency representations and operators: boundedness and uncertainty principles

This paper presents a proof of an uncertainty principle of Donoho-Stark type involving $varepsilon$-concentration of localization operators. More general operators associated with time-frequency representations in the Cohen class are then considered. For these operators, which include all usual quantizations, we prove a boundedness result in the $L^p$ functional setting and a form of uncertainty principle analogous to that for localization operators.

The Gabor wave front set in spaces of ultradifferentiable functions

Given a non-quasianalytic subadditive weight function $omega$ we consider the weighted Schwartz space $mathcal{S}_omega$ and the short-time Fourier transform on $mathcal{S}_omega$, $mathcal{S}_omega$ and on the related modulation spaces with exponential weights. In this setting we define the $omega$-wave front set $WF_omega(u)$ and the Gabor $omega$-wave front set $WF^G_omega(u)$ of $uinmathcal{S}_{omega}$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.

Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms

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 $mathcal S$ and in the space ${mathcal S}_omega$ as introduced by Bjorck for weight functions $omega$. Several examples are discussed in this new setting.

Two Aspects of the Donoho-Stark Uncertainty Principle

We present some forms of uncertainty principle which involve in a new way localization operators, the concept of $varepsilon$-concentration and the standard deviation of $L^2$ functions. We show how our results improve the classical Donoho-Stark estimate in two different aspects: a better general lower bound and a lower bound in dependence on the signal itself.