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

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

• Viorel Catana
• University Politehnica of Bucharest
• Department of Mathematics I
• catana_viorel_at_yahoo.co.uk

The 10th French-Romanian Conference on Applied
Mathematics Poitiers (Futuroscope ) ,France ,
August 26-31 , 2010
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Topics

• Introduction
• Basic definitions and notations
• Preliminary results
• The generalized Hermite semigroup
• The inverse of the generalized Hermite operator
• The heat kernel of the generalized Hermite
  operator
• The Green function of the generalized Hermite
  operator
• Global hypoellipticity
• Ultracontractivity and Hypercontractivity
• The abstract Hermite semigroup
• The abstract Cauchy problem (ACP) for the
  abstract Hermite operator

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Introduction
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References
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The twisted Laplacian
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Wongs research about twisted Laplacian
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Basic definitions notations
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The generalized Hermite operator
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Basic transformations
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The generalized Hermite functions
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Preliminary results
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Main results concerning the Hermite semigroup
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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 book 16 by
Wong).
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The inverse of the generalized Hermite operator
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The heat kernel
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Preliminary result
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The Green function of the generalized Hermite
operator
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Global hypoellipticity
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Ultracontractivity and Hypercontractivity
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The abstract Cauchy problem (ACP) for the
abstract Hermite operator
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The abstract Hermite operator
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The semigroup of the abstract Hermite operator
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An estimate of the abstract Hermite semigroup
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The (ACP) for the abstract Hermite operator
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Merci pour votre aimable attention!Thank you
for your kind attention!