Gianluca Garello

L p -microlocal regularity for pseudodifferential operators of quasi-homogeneous type†

The authors consider pseudodifferential operators whose symbols have decay at infinity of quasi-homogeneous type and study their behaviour on the wave front set of distributions in weighted Zygmund–Hölder spaces and weighted Sobolev spaces in Lp -framework. Then microlocal properties for solutions to linear partial differential equations with coefficients in weighted Zygmund–Hölder spaces are obtained. †This article is devoted to the special issue ‘Växjö Conference 2008’.

Continuity in Weighted Sobolev Spaces Of Lp Type for Pseudo-Differential Operators with Completely Nonsmooth Symbols

We consider symbolsa(xξ)with a finite number of bounded derivatives with respect to ξ and of weighted Sobolev type inx.Their continuity in weighted Sobolev spaces of sufficiently large order is studied.

Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols

In this paper a result of continuity for pseudo-differential operators with non-regular symbols on spaces of quasi-homogeneous type is given. More precisely, the symbols a(x, ξ) take their values in a quasi-homogeneous Besov space with respect to the x variable; moreover a finite number of derivatives with respect to the second variable satisfies, in Besov norm, decay estimates of quasi-homogeneous type.

Continuity in Weighted Besov Spaces for Pseudodifferential Operators with Non-regular Symbols

The authors state and prove a result of continuity in weighted Besov spaces for a class of pseudodifferential operators whose symbol a(x, ξ) admits a finite number of bounded derivatives with respect to ξ and is of weighted Besov type in the x variable.

Microlocal properties for pseudodifferential operators of type 1,1

The author studies the action of a new selfadjoint algebra of pseudodifferential operators of type (1,1) on Sobolev wave front sets for distributions.

Regularity for Quasi-Elliptic Pseudo-Differential Operators with Symbols in Hölder Classes

After proving a result of continuity for pseudo-differential operators whose symbols belong to Holder spaces and satisfy a decay of quasihomogeneous type, the authors construct a suitable symbolic calculus and a parametrrx for quasi-elliptic operators; these tools are applied to the study of quasi-elliptic linear partial differential equations with H:older coefficients.

Pseudo-Differential Operators and Generalized Functions

At the ninth congress of the International Society for Analysis Applications and Computations (ISAAC), ISAAC Group in Pseudo-Differential Operators (IGPDO) and the ISAAC Group in Generalized Functions (IGGF) agreed to continue with the publication of a joint volume, as for the eight ISAAC Congress in Moscow 2013, of selecting papers from their two special sessions. Generalized functions as a general framework for almost all fields in analysis, and pseudo-differential operators as a basis of microlocal analysis in combination with harmonic and complex analysis with many applications, fit well and offer rich synergies for the further development of analysis in general.Moreover, the participants of both sessions agreed to dedicate this volume to Professor Michael Oberguggenberger at Insbruck university, Austria on his 60th birthday. Professor Oberguggenberger is one of the founder of the algebraic approach to generalized function theory with many contributions to the qualitative analysis of partial differential equations and a leader of the International Asso- ciation for Generalized Functions based in Vienna. Professor Oberguggenberger is highly appreciated as a scientist and a strong expert on generalized functions, especially Columbeau algebras. He is also very appreciated as a modest and encouraging person who supervised several PhD student to their examination. It is a pleasure for us to dedicate the volume to him.This joint volume is titledPseudo-Differential Operators and Generalized Functionsand consists of invited papers, mainly based on the scientific activities of the groups IGPDO and IGGF at the ninth ISAAC congress in Krakow, Poland, during August 2013. The volume is intended to be an independent sequel to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators”, “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” and “Pseudo-Differential Operators, Generalized Functions and Asymptotics”. These volumes were based on, respectively, the 4th ISAAC congress in Toronto in 2003, conference in V axjo 2004, 5th ISAAC congress in Catana in 2005, 6th ISAAC congress in Ankara in 2007, workshop in Toronto in 2008, 7th ISAAC congress in London in 2009, and 8th ISAAC congress in Moscow ISAAC in 2011.The volume consists of 19 peer-reviewed contributions representing modern trends in the theory of generalized functions and pseudo-differential operators. Topics include algebras of generalized functions, ultra-distributions, partial differential equations, micro-local analysis, harmonic analysis, global analysis, geometry, quantization, mathematical physics, and time-frequency analysis. Variety of applications especially in the framework of manifolds with singular metrics and general relativity, microlocal analysis and the analysis of equations with singularities will be interested for a wide audience including graduate students and researchers in partial differential equations, mathematical physics, various fields of analysis, stochastic analysis and geometry.The papers can be sorted roughly into two groups. The first group of papers is related to generalized functions and deals with various problems of equations with singular coefficients and data within algebras of generalized functions where the classical method of regularizations is well established. Moreover in this setting, local analysis is well adapted and analyzed towards Hoder type spaces and in the direction of generalized manifolds and applications in general relativity.In the second group of papers, various kinds of Fourier analysis are more present, involving micro-local analysis, harmonic analysis, theory of ultra-distributions, time-frequency analysis, etc. For example, Wiener type Tauberian theorems related to generalized integral transforms stochastic equations are adapted to classical distribution theory. Ultradistribution spaces are analyzed in connection with global type operators and wave fronts. Gabor analysis via modulation spaces or Hermite expansions is developed for various Gelfand-Shilov classes. Time-frequency methods are applied on evolution operators and on random MIMO systems.

L^P microlocal properties for vector weighted pseudodifferential operators with smooth symbols

The authors introduce a class of pseudodifferential operators, whose symbols satisfy completely inhomogeneous estimates at infinity for the derivatives.

Lp-Continuity for Pseudo-Differential Operators

The authors give a short survey about the Lp-continuity of pseudodifferential operators both with smooth and non-smooth symbols