Weyl transforms and the generalized Weyl transforms and the generalized Hermite operator.The abstract Hermite operator.The abstract Cauchy problem (ACP) for the Cauchy problem (ACP) for the

abstract Hermite operator.abstract Hermite operator.

Viorel Catana Viorel Catana University Politehnica of Bucharest University Politehnica of Bucharest

Department of Mathematics IDepartment of Mathematics Icatana_viorel@yahoo.co.ukcatana_viorel@yahoo.co.uk

The 10th French-Romanian Conference on Applied Mathematics

Poitiers (Futuroscope ) ,France , August 26-31 , 2010

TopicsTopics

IntroductionIntroduction Basic definitions and notationsBasic definitions and notations Preliminary results Preliminary results The generalized Hermite semigroupThe generalized Hermite semigroup The inverse of the generalized Hermite operatorThe inverse of the generalized Hermite operator The heat kernel of the generalized Hermite operatorThe heat kernel of the generalized Hermite operator The Green function of the generalized Hermite The Green function of the generalized Hermite

operator operator Global hypoellipticityGlobal hypoellipticity Ultracontractivity and HypercontractivityUltracontractivity and Hypercontractivity The abstract Hermite semigroupThe abstract Hermite semigroup The abstract Cauchy problem (ACP) for the abstract The abstract Cauchy problem (ACP) for the abstract

Hermite operatorHermite operator

IntroductionIntroduction

ReferencesReferences

The twisted LaplacianThe twisted Laplacian

Wong’s research about Wong’s research about twisted Laplaciantwisted Laplacian

Basic definitions & notationsBasic definitions & notations

The generalized Hermite The generalized Hermite operatoroperator

Basic transformationsBasic transformations

The generalized Hermite The generalized Hermite functionsfunctions

Preliminary resultsPreliminary results

Main results concerning the Main results concerning the Hermite semigroupHermite semigroup

For the proof of Theorem 6 we need the following two results (see For the proof of Theorem 6 we need the following two results (see Chapter 17 of the book [17] and Theorem 15.4 and Theorem 17.1 in the Chapter 17 of the book [17] and Theorem 15.4 and Theorem 17.1 in the

book [16] by Wong).book [16] by Wong).

The inverse of the generalized The inverse of the generalized Hermite operatorHermite operator

The heat kernelThe heat kernel

Preliminary resultPreliminary result

The Green function of the The Green function of the generalized Hermite generalized Hermite

operatoroperator

Global hypoellipticityGlobal hypoellipticity

Ultracontractivity and Ultracontractivity and HypercontractivityHypercontractivity

The abstract Cauchy problem The abstract Cauchy problem (ACP) for the abstract Hermite (ACP) for the abstract Hermite

operatoroperator

The abstract Hermite operatorThe abstract Hermite operator

The semigroup of the abstract The semigroup of the abstract Hermite operatorHermite operator

An estimate of the abstract An estimate of the abstract Hermite semigroupHermite semigroup

The (ACP) for the abstract The (ACP) for the abstract Hermite operatorHermite operator

Merci pour votre aimable Merci pour votre aimable attention!attention!

Thank you for your kind Thank you for your kind attention!attention!