Densities and differentiability properties of Gauss semigroups on a Lie group

Siebert, Eberhard

doi:10.1090/S0002-9939-1984-0740190-5

Densities and differentiability properties of Gauss semigroups on a Lie group

Author: Eberhard Siebert
Journal: Proc. Amer. Math. Soc. 91 (1984), 298-305
MSC: Primary 60B15; Secondary 43A05, 47D05
DOI: https://doi.org/10.1090/S0002-9939-1984-0740190-5
MathSciNet review: 740190
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider an absolutely continuous Gauss semigroup on a connected Lie group. Integrability and boundedness properties for the corresponding densities are established. Moreover it is shown that the Gauss measures transform integrable functions into infinitely differentiable solutions of certain partial differential equations. Finally, we prove that the semigroup acts on many Banach spaces as a differentiable operator semigroup.

[1] E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, Springer-Verlag, Berlin and New York, 1963.
[2] H. Heyer, Probability measures on locally compact groups, Springer-Verlag, Berlin and New York, 1977. MR 0501241 (58:18648)
[3] A. Hulanicki, Subalgebra of ${L_1}(G)$ associated with Laplacian on a Lie group, Colloq. Math. 31 (1974), 259-287. MR 0372536 (51:8743)
[4] P. E. T. Jørgensen, Representations of differential operators on a Lie group, J. Funct. Anal. 20 (1975), 105-135. MR 0383469 (52:4350)
[5] J. Kisyński, Holomorphicity of semigroups of operators generated by sublaplacians on Lie groups, Function Theoretic Methods for Partial Differential Equations (Proc. Internat. Sympos., Darmstadt, 1976), Lecture Notes in Math., Vol. 561, Springer-Verlag, Berlin and New York, 1976, pp. 283-297. MR 0477879 (57:17381)
[6] -, On semigroups generated by differential operators on Lie groups, J. Funct. Anal. 31 (1979), 234-244. MR 525954 (81e:22010)
[7] E. Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572-615. MR 0107176 (21:5901)
[8] A. Pazy, On the differentiability and compactness of semigroups of linear operators, J. Math. Mech. 17 (1968), 1131-1141. MR 0231242 (37:6797)
[9] H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York, 1974. MR 0423039 (54:11023)
[10] E. Siebert, Supports of holomorphic convolution semigroups and densities of symmetric convolution semigroups on a locally compact group, Arch. Math. 36 (1981), 423-433. MR 629273 (82j:43001)
[11] -, Absolute continuity, singularity, and supports of Gauss semigroups on a Lie group, Monatsh. Math. 93 (1982), 239-253. MR 661571 (83m:60018)
[12] -, Holomorphy of convolution semigroups, Proc. 7th Conf. on Probability Measures on Groups, Oberwolfach, 1983 (to appear).