Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups

Dungey, Nick

Canadian journal of mathematics, Tome 57 (2005) no. 6, pp. 1193-1214

Résumé

Let $K$ be a function on a unimodular locally compact group $G$ , and denote by ${{K}_{n}}\,=\,K\,*\,K\,*\cdots *\,K$ the $n$ -th convolution power of $K$ . Assuming that $K$ satisfies certain operator estimates in ${{L}^{2}}\left( G \right)$ , we give estimates of the norms ${{\left\| {{K}_{n}} \right\|}_{2}}$ and ${{\left\| {{K}_{n}} \right\|}_{\infty }}$ for large $n$ . In contrast to previous methods for estimating ${{\left\| {{K}_{n}} \right\|}_{\infty }}$ , we do not need to assume that the function $K$ is a probability density or nonnegative. Our results also adapt for continuous time semigroups on $G$ . Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.

Détail

DOI : 10.4153/CJM-2005-047-1

Mots-clés : Primary: 22E30, secondary: 35B40, 43A99

Bibliographie

[1] [1] Auscher, P., ter Elst, A. F. M. and Robinson, D.W., On positive Rockland operators. Colloq. Math. 67(1994), 197–216. Google Scholar

[2] [2] Coulhon, T., Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141(1996), 510–539. Google Scholar

[3] [3] Davies, E B., Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, 1989. Google Scholar

[4] [4] Davies, E B., Long time asymptotics of fourth order parabolic equations. J. Anal.Math. 67(1995), 323–345. Google Scholar

[5] [5] Davies, E B., Lp spectral theory of higher-order elliptic differential operators. Bull. LondonMath. Soc. 29(1997), 513–546. Google Scholar

[6] [6] Dungey, N., Higher order operators and Gaussian bounds on Lie groups of polynomial growth. J. Operator Theory 46(2001), 45–61. Google Scholar

[7] [7] Dungey, N., ter Elst, A. F. M. and Robinson, D.W., On anomalous asymptotics of heat kernels on groups of polynomial growth. Mathematics Research Report 01-006, The Australian National University, 2001. Google Scholar

[8] [8] Dungey, N., ter Elst, A. F. M. and Robinson, D.W., Analysis on Lie GroupsWith Polynomial Growth. Progress in Mathematics 214, Birkhäuser, Boston,MA, 2003. Google Scholar

[9] [9] Dungey, N., ter Elst, A. F. M., Robinson, D.W. and Adam, Sikora, Asymptotics of subcoercive semigroups on nilpotent Lie groups. J. Operator Theory 45(2001), 81–110. Google Scholar

[10] [10] Dziubanski, J., Hebisch, W. and Zienkiewicz, J., Note on semigroups generated by positive Rockland operators on graded homogeneous groups. Studia Math. 110(1994), 115–126. Google Scholar

[11] [11] ter Elst, A. F. M. and Robinson, D.W., Weighted subcoercive operators on Lie groups. J. Funct. Anal. 157(1998), 88–163. Google Scholar

[12] [12] ter Elst, A. F. M. and Robinson, D.W., On anomalous asymptotics of heat kernels. In: Evolution Equations and Their Applications To Physical and Life Sciences, Lecture Notes in Pure and Appl. Math. 215, Dekker, New York, 2001, pp. 89–103. Google Scholar

[13] [13] ter Elst, A. F. M. and Robinson, D.W. Gaussian bounds for complex subelliptic operators on Lie groups of polynomial growth. Bull. Austral. Math. Soc. 67(2003), 201–218. Google Scholar

[14] [14] ter Elst, A. F. M., Robinson, D.W. and Sikora, A., Heat kernels and Riesz transforms on nilpotent Lie groups. Colloq. Math. 74(1997), 191–218. Google Scholar

[15] [15] Guivarc’h, Y., Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101(1973), 333–379. Google Scholar

[16] [16] Hebisch, W., Estimates on the semigroups generated by left invariant operators on Lie groups. J. Reine Angew.Math. 423(1992), 1–45. Google Scholar

[17] [17] Hebisch, W. and Saloff-Coste, L., Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21(1993), 673–709. Google Scholar

[18] [18] Kato, T., Perturbation theory for linear operators. Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, Berlin, 1984. Google Scholar

[19] [19] Nagel, A., Ricci, F., and Stein, E. M., Harmonic analysis and fundamental solutions on nilpotent Lie groups. In: Analysis and Partial Differential Equations, Lecture Notes in Pure and Applied Mathematics 122, Dekker, New York, 1990, 249–275. Google Scholar

[20] [20] Nagel, A., Stein, E. M., and S.Wainger, Balls and metrics defined by vector fields. I: basic properties. Acta Math. 155(1985), 103–147. Google Scholar

[21] [21] Robinson, D.W., Elliptic operators and Lie groups. Oxford Mathematical Monographs, Oxford University Press, Oxford, 1991. Google Scholar

[22] [22] Saloff-Coste, L., Probability on groups: random walks and invariant diffusions. Notices Amer.Math. Soc. 48(2001), 968–977. Google Scholar

[23] [23] Varopoulos, N. T., Convolution powers on locally compact groups. Bull. Sci. Math. 111(1987), 333–342. Google Scholar

[24] [24] Varopoulos, N. T., Analysis on Lie groups. Rev.Mat. Iberoamericana 12(1996), 791–917. Google Scholar

[25] [25] Varopoulos, N. T., Saloff-Coste, L., and Coulhon, T., Analysis and Geometry on Groups. Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992. Google Scholar