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Nash type inequalities for fractional powers of non-negative self-adjoint operators

Author(s): Alexander Bendikov; Patrick Maheux
Journal: Trans. Amer. Math. Soc. 359 (2007), 3085-3097.
MSC (2000): Primary 39B62, 47A60, 26A12, 26A33, 81Q10
Posted: January 25, 2007
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Abstract: Assuming that a Nash type inequality is satisfied by a non-negative self-adjoint operator $ A$, we prove a Nash type inequality for the fractional powers $ A^{\alpha}$ of $ A$. Under some assumptions, we give ultracontractivity bounds for the semigroup $ (T_{t,{\alpha}})$ generated by $ -A^{\alpha}$.

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Additional Information:

Alexander Bendikov
Affiliation: Mathematical Institute of the Wroclaw University, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Email: bendikov@math.uni.wroc.pl

Patrick Maheux
Affiliation: Département de Mathématiques, MAPMO-Fédération Denis Poisson, Université d'Or- léans, BP 6759, F 45 067 Orleans Cedex 2, France
Email: pmaheux@univ-orleans.fr

DOI: 10.1090/S0002-9947-07-04020-2
PII: S 0002-9947(07)04020-2
Keywords: Nash inequality, fractional powers of operators, semigroup of operators, logarithmic Sobolev inequality, ultracontractivity property, Dirichlet form
Received by editor(s): March 11, 2002
Received by editor(s) in revised form: April 11, 2005
Posted: January 25, 2007
Additional Notes: This research was partially supported by the European Commission (IHP Network ``Harmonic Analysis and Related Problems'' 2002-2006, contract HPRN-CT-2001-00273-HARP)
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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A.Bendikov.T.Coulhon.L.Saloff-Coste, Ultracontractivity and embedding into L^{\infty}, Math.Ann. (4) 337 (2007), 817-853. (English)

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