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

Bendikov, Alexander; Maheux, Patrick

doi:10.1090/S0002-9947-07-04020-2

Nash type inequalities for fractional powers of non-negative self-adjoint operators
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by Alexander Bendikov and Patrick Maheux PDF

Trans. Amer. Math. Soc. 359 (2007), 3085-3097 Request permission

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 $-A^{\alpha }$.

References

D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), no. 4, 1033–1074. MR 1386760, DOI 10.1512/iumj.1995.44.2019
A. D. Bendikov, Symmetric stable semigroups on the infinite-dimensional torus, Exposition. Math. 13 (1995), no. 1, 39–79. MR 1339813
A. Bendikov and L. Saloff-Coste, On- and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces, Amer. J. Math. 122 (2000), no. 6, 1205–1263. MR 1797661, DOI 10.1353/ajm.2000.0043
Christian Berg and Gunnar Forst, Potential theory on locally compact abelian groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87, Springer-Verlag, New York-Heidelberg, 1975. MR 0481057, DOI 10.1007/978-3-642-66128-0
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
M. Biroli and P. Maheux, Inégalités de Sobolev logarithmiques et de type Nash pour des semi-groupes sous-markoviens symétriques. Preprint.
Thierry Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141 (1996), no. 2, 510–539. MR 1418518, DOI 10.1006/jfan.1996.0140
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. MR 569058
Leonard Gross, Logarithmic Sobolev inequalities and contractivity properties of semigroups, Dirichlet forms (Varenna, 1992) Lecture Notes in Math., vol. 1563, Springer, Berlin, 1993, pp. 54–88. MR 1292277, DOI 10.1007/BFb0074091
Herbert Heyer, Probability measures on locally compact groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 94, Springer-Verlag, Berlin-New York, 1977. MR 0501241, DOI 10.1007/978-3-642-66706-0
N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
V. M. Zolotarev, One-dimensional stable distributions, Translations of Mathematical Monographs, vol. 65, American Mathematical Society, Providence, RI, 1986. Translated from the Russian by H. H. McFaden; Translation edited by Ben Silver. MR 854867, DOI 10.1090/mmono/065