Paolo Boggiatto

Localization Operators with LP Symbols on Modulation Spaces

Boundedness of localization operators on modulation spaces is studied obtaining results for operators with symbol in L p (ℝ2n ) 1 ≤ p ≤ ∞. In this context the results presented here generalize well-known properties of boundedness and compactness.

Advances in pseudo-differential operators

Microlocal Analysis and Applications.- The Conormal Symbolic Structure of Corner Boundary Value Problems..- A New Proof of Global Smoothing Estimates for Dispersive Equations.- Gevrey Hypoellipticity of p-Powers of Non-Hypoelliptic Operators.- Continuity in Weighted Sobolev Spaces of LP Type for Pseudo-Differential Operators with Completely Nonsmooth Symbols.- Symmetry-Breaking for Wigner Transforms and LP-Boundedness of Weyl Transforms.- Pseudo-Differential Operators and Schatten-von Neumann Classes.- Localization Operators via Time-Frequency Analysis.- Localization Operators with LP Symbols on Modulation Spaces.- Convolutions and Embeddings for Weighted Modulation Spaces.- Pseudo-Differential Operators, Microlocal Analysis and Image Restoration.- Applications of Wavelet Transforms to System Identification.- Two-Dimensional Wavelet Bases for Partial Differential Operators and Applications.

Anti-Wick quantization with symbols in ^{} spaces

We give a classification of pseudo-differential operators with anti-Wick symbols belonging to L p spaces: if p = 1 the corresponding operator belongs to trace classes; if 1 < p < 2 we get Hilbert-Schmidt operators; finally, if p < oo, the operator is compact. This classification cannot be improved, as shown by some examples.

A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform

Motivated by problems in signal analysis, we define a class of time-frequency representations which is based on the short-time Fourier transform and depends on two fixed windows. We show that this class can be viewed as a link between the classical Rihaczek representation and the spectrogram. Correspondingly we formulate for this class a suitable general form of the uncertainty principle which have, as limit case, the uncertainty principles for the Rihaczek representation and for the spectrogram. We finally consider the questions of marginal distributions. We compute them in terms of convolutions with the windows and prove simple conditions for which average and standard deviation of the distributions in our class coincide with that of their marginals.

Non-commutative residues for anisotropic pseudo-differential operators in Rn

Abstract In this paper we define a non-commutative residue for the algebra of classical anisotropic pseudo-differential operators on R n modulo regularizing operators. Furthermore, by means of the Weyl formula, we show that this trace coincides with the Dixmier trace on operators of order −| M |.

Pseudo-differential operators and related topics

Strongly Elliptic Second Order Systems with Spectral Parameter in Transmission Conditions on a Nonclosed Surface.- Well-Posedness of the Cauchy Problem for Some Degenerate Hyperbolic Operators.- Quasilinear Hyperbolic Equations with SG-Coefficients.- Representation of Solutions and Regularity Properties for Weakly Hyperbolic Systems.- Global Calculus of Fourier Integral Operators, Weighted Estimates, and Applications to Global Analysis of Hyperbolic Equations.- Lp-Continuity for Pseudo-Differential Operators.- Fredholm Property of Pseudo-Differential Operators on Weighted Holder-Zygmund Spaces.- Weyl Transforms and Convolution Operators on the Heisenberg Group.- Uncertainty Principle, Phase Space Ellipsoids and Weyl Calculus.- Pseudo-Differential Operator and Reproducing Kernels Arising in Geometric Quantization.- Hudsons Theorem and Rank One Operators in Weyl Calculus.- Distributions and Pseudo-Differential Operators on Infinite-Dimensional Spaces with Applications in Quantum Physics.- Ultradistributions and Time-Frequency Analysis.- Frames and Generalized Shift-Invariant Systems.- The Wigner Distribution of Gaussian Weakly Harmonizable Stochastic Processes.- Reproducing Groups for the Metaplectic Representation.

Two-Window Spectrograms and Their Integrals

We analyze in this paper some basic properties of two-window spectrograms, introduced in a previous work. This is achieved by the analysis of their kernel, in view of their immersion in the Cohen class of time-frequency representations. Further we introduce weighted averages of two-window spectrograms depending on varying window functions. We show that these new integrated representations improve some features of both the classical Rihaczek representation and the two-window spectrogram which in turns can be viewed as limit cases of them.

Partial Differential Equations of Multi-Quasi-Elliptic Type.

The theory of multi-quasi-elliptic operators and associated Sobolev spaces is revised in this article and possible developments of the research are indicated concerning operators of principal type and non-linear equations. Moreover some new results concerning operators with anti-Wick symbols are presented.

Wigner Representations Associated with Linear Transformations of the Time-Frequency Plane

In this paper we define a variation of theWigner form depending on a linear transformation of the time-frequency plane and study the corresponding properties. This construction yields a natural geometric interpretation of the so-called “ghost frequencies” showed, among others, by the Wigner quadratic representation. In particular we prove that the representations in this new class which satisfy the support properties are just the so-called t-Wigner forms. On the other hand for these new forms an “essential” re-adjustement of the supports is showed to be possible, whereas their features of moving almost arbitrarily the ghost frequencies is used to define representations with no interferences at all for a certain class of signals.

Multi-quasi-elliptic operators and anti-Wick symbols

The Theory of multi-quasi-elliptic operators and associated Sobolev spaces is presented here making use of operators with anti-Wick symbols. This leads to remarkable simplifications in the definitions and the proves of the theorems. On the other side, it gives a positive answer to the problem of order reduction for this type of Sobolev spaces.