30 Pages
English
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Nash type Inequalities and

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30 Pages
English

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Nash-type Inequalities and decay of semigroups. Patrick Maheux University of Orleans. Federation Denis Poisson. Workshop in Nanterre . June 4-5-6, 2007.

  • finite measure

  • let ?

  • ∂tt ∂t

  • poincare inequality

  • dimension can

  • ?f ?

  • infinitesimal generator


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Nash-type Inequalities and
decay of semigroups.
Patrick Maheux UniversityofOrl´eans. F´ed´erationDenisPoisson .
Workshop in Nanterre . June 4-5-6,
2007.
SETTING
( X, µ ) measure space with µ a σ -finite measure.
( T t ) t> 0 : a symmetric submarkovian semi-group:
( T t f,h ) = ( f, T t h ) , f,g L 2
• ∀ f L 2 , 0 f 1 0 T t f 1 ( T t ) t> 0 acts as a contraction on L p .
L : infinitesimal generator associated to the semigroup T t = e t L
t = T Tt A t f
E : Dirichlet form associated to ( T t ) :
E ( f ) = ( L f,f ) , f ∈ D ( L )
Important property
f ∈ D ( E ) h = ( f 0) 1 ∈ D ( E )
and
E ( h ) ≤ E ( f )