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

Authors: Alexander Bendikov and Patrick Maheux
Journal: Trans. Amer. Math. Soc. 359 (2007), 3085-3097
MSC (2000): Primary 39B62, 47A60, 26A12, 26A33, 81Q10
Published electronically: January 25, 2007
MathSciNet review: 2299447
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Abstract | References | Similar Articles | Additional Information

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

Patrick Maheux
Affiliation: Département de Mathématiques, MAPMO-Fédération Denis Poisson, Université d’Or- léans, BP 6759, F 45 067 Orleans Cedex 2, France

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
Published electronically: 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)
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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