Dual ultracontractivity and its applications

Abstract

The ultracontractivity is well studied and several equivalent conditions are known. In this paper, we introduce the dual notion of the ultracontractivity, which we call the dual ultracontractivity. We give necessary and sufficient conditions for the dual ultracontractivity. As an application, we discuss one-dimensional diffusion processes. We can write the conditions for the dual ultracontractivity in terms of speed measures.

Keywords

Dual ultracontractivity / one-dimensional diffusion processes / Nash inequality / interpolation

References

[1] Bakry D, Coulhon T, Ledoux M, Saloff-Coste L. Sobolev inequalities in disguise. Indiana Univ Math J, 1995, 44(4): 1033-1074

[2] Bobkov S, Götze F. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J Funct Anal, 1999, 163(1): 1-28

[3] Coulhon T. Dimension à l’infini d’un semi-groupe analytique. Bull Sci Math, 1990, 114(4): 485-500

[4] Coulhon T. Dimensions of continuous and discrete semigroups on the L^p-spaces. In: Semigroup Theory and Evolution Equations (Delft, 1989). Lecture Notes in Pure and Appl Math, 135. New York: Dekker, 1991, 93-99

[5] Coulhon T. Itération de Moser et estimation gaussienne du noyau de la chaleur. J Operator Theory, 1993, 29(1): 157-165

[6] Davies E B. Heat Kernels and Spectral Theory. Cambridge: <PublisherName Language="chs">Cambrid<?Pub Caret?>ge University Press</PublisherName>, 1989

[7] Mao Y H. Nash inequalities for Markov processes in dimension one. Acta Math Sin (Engl Ser), 2002, 18(1): 147-156

[8] Varopoulos N Th, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge: Cambridge University Press, 1992

[9] Wang F-Y. Functional inequalities for the decay of sub-Markov semigroups. Potential Anal, 2003, 18(1): 1-23

[10] Wang F-Y. Functional Inequalities, Markov Semigroups and Spectral Theory. Beijing: Science Press, 2005

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg