Luigi Rodino

Linear partial differential operators in Gevrey spaces

Differential operators with constant coefficients Gevrey pseudo-differential operators of infinite order canonical transformations and classical analytic Fourier integral operators propagation of Gevrey singularities Gevrey hypoellipticity the Cauchy problem in the Gevrey classes local solvability in Gevrey classes.

Global Pseudo-differential calculus on Euclidean spaces

Background meterial.- Global Pseudo-Differential Calculus.- ?-Pseudo-Differential Operators and H-Polynomials.- G-Pseudo-Differential Operators.- Spectral Theory.- Non-Commutative Residue and Dixmier Trace.- Exponential Decay and Holomorphic Extension of Solutions.

Classes of degenerate elliptic operators in Gelfand-Shilov spaces

We propose a novel approach for the study of the uniform regularity and the decay at infinity for Shubin type pseudo-differential operators which are globally hypoelliptic but not necessarily globally and even locally elliptic. The basic idea is to use the special role of the Hermite functions for the characterization of inductive and projective Gelfand-Shilov spaces. In this way we transform the problem to infinite dimensional linear systems on S Banach spaces of sequences by using Fourier series expansion with respect to the Hermite functions. As applications of our general results we obtain new theorems for global hypoellipticity for classes of degenerate operators in tensorized generalizations of Shubin spaces and in inductive and projective Gelfand-Shilov spaces.

Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential operators in a Sjostrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sjostrand class consists of generalized metaplectic operators. As a consequence, the Schrodinger equation preserves the phase-space concentration, as measured by modulation space norms.

Wick calculus: A time-frequency approach

The purpose of this paper is to present a formula for the product of two Wick operators, defined in terms of different pairs of windows φ1, φ2. In principle, Wick operators can be converted to Weyl operators, and hence one may apply to them the standard symbolic calculus [14, 21]. It is natural, however, to consider the product in the Wick form, and try to compute directly the symbol in terms of the symbols of the factors; see in this direction [3, 5, 15, 16, 19]. Recently Ando and Morimoto [1] have given a full expansion for the Wick symbol of the product in the case when all the windows coincide with the Gaussian function. We propose here a general formula. The expression is somewhat non-standard, because we write the product as a sum of anti-Wick operators corresponding to a sequence of different pairs of windows, with decreasing order. This seems to us the only possible expression of reasonable simplicity in the generic case. In the remaining part of this Introduction we recall the definition of Wick operators and state the composition result. In Section 2 we summarize some concepts of time-frequency methods used in the proof. In Section 3 we introduce the classes of symbols we are arguing on. They are, essentially, those of Shubin [18], as generalized in [3]. Let us emphasize that other classes of symbols, under weaker assumptions on derivative estimates, would work as well. In Section 4 we prove the result. In Section 5 we give a composition formula for the particular case of Gaussian functions as a pair of windows and we recapture the results of Lerner [15, 16], Ando and Morimoto [1]. Section 6 is devoted to miscellaneous comments. Namely we show how to pass from a pair of windows to another and, finally, we construct a parametrix for the elliptic Wick operators by using our formula; a natural application, which we hope to detail in future papers, concerns regularity results in the frame of the modulation spaces [9, 12]. Before stating the precise definition, let us observe that Wick operators have been considered in the past under rather different points of view, and different names. They were introduced by Berezin [2] as a quantization procedure, and as an approximation of pseudodifferential operators (“wave packets”) by Cordoba and Fefferman [7, 11]. From the point of view of the time-frequency analysis, which we shall adopt in the following, they have been studied by Daubechies [8] and

Gabor representations of evolution operators

We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodierential operators with symbols of Gevrey, analytic and ultra- analytic type. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrodinger-type propagators (14), surprisingly reveal to be an equally ecient tool for representing solutions to hyperbolic and parabolic-type dierential equations with constant coecients. In fact, the Gabor matrix represen- tation of the corresponding propagator displays super-exponential decay away from the diagonal.

Wave packet analysis of Schrödinger equations in analytic function spaces

Abstract We consider a class of linear Schrodinger equations in R d , with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic singularities.

Hypoellipticity and Local Solvability in Gevrey Classes

Let P be a linear partial differential operator with coefficients in the Gevrey class Gs. We prove first that if P is s-hypoelliptic then its transposed operator tP is s-locally solvable, thus extending to the Gevrey classes the well-known analogous result in the C∞class. We prove also that if P is s-hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s-hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.

Fourier integral operators and inhomogeneous Gevrey classes

SummaryFourier integral operators with inhomogeneous amplitude and phase junction are studied in the frame of Gevrey classes. Applications are given to propagation of singularities for a pseudodifferential equation.

Sub-exponential decay and uniform holomorphic extensions for semilinear pseudodifferential equations

The goal of the present paper is to derive a simultaneous description of the decay and the regularity properties for elliptic equations in ℝ n with coefficients admitting irregular decay at infinity of the type O(|x|−σ), σ > 0, filling the gap between the case of Cordes globally elliptic operators and the case of regular/Fuchs behavior at infinity. Representative examples in ℝ n are the equations where 0 < σ <2, ⟨x⟩ = (1 + |x|2)1/2, ω(x) a bounded smooth function, f given and F[u] a polynomial in u, and similar Schrödinger equations at the endpoint of the spectrum. Other relevant examples are given by linear and nonlinear ordinary differential equations with irregular type of singularity for x → ∞, admitting solutions y(x) with holomorphic extension in a strip and sub-exponential decay of type |y(x)| ≤Ce −ϵ|x| r ; 0 < r < 1. Sobolev estimates for the linear case are proved in the frame of a suitable pseudodifferential calculus; decay and uniform holomorphic extensions are then obtained in terms of Gelfand–Shilov spaces by an inductive technique. The same technique allows to extend the results to the semilinear case.