Abstract
We define and analyze Lipschitz spaces ℬα,q associated with a representationx∈g→V(x) of the Lie algebrag by closed operatorsV(x) on the Banach space ℬ together with a heat semigroupS. If the action ofS satisfies certain minimal smoothness hypotheses with respect to the differential structure of (ℬ,g,V) then the Lipschitz spaces support representations ofg for which productsV(x)V(y) are relatively bounded by the Laplacian generatingS. These regularity properties of the ℬα,q can then be exploited to obtain improved smoothness properties ofS on ℬ. In particularC 4-estimates on the action ofS automatically implyC ∞-estimates. Finally we use these results to discuss integrability criteria for (ℬ,g,V).
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References
[BBR] Batty, C.J.K., Bratteli, O., Robinson, D.W.: The heat semigroup, derivations and Reynold's identity. In: Operator algebras and applications. Evans, D.E., Takesaki, M. (eds.). Lond. Math. Soc. Lect. Notes, Vol. 136. Cambridge: Cambridge University Press
[BcF] Berg, C., Forst, G.: Potential theory on locally compact abelian groups. Berlin, Heidelberg, New York: Springer 1975
[BGJR] Bratteli, O., Goodman, F., Jørgensen, P.E.T., Robinson, D.W.: The heat semigroup and integrability of Lie algebras. J. Funct. Anal.79, 351–397 (1988)
[BrJ] Bratteli, O., Jørgensen, P.E.T.: Conservative derivations and dissipative Laplacians. J. Funct. Anal.82, 404–441 (1989)
[BuB] Butzer, P.L., Berens, H.: Semigroups of operators and approximation. Berlin, Heidelberg, New York: Springer 1967
[Orn] Ornstein, D.: A non-inequality for differential operators in theL 1-norm. Arch. Rat. Mech. Anal.11, 40–49 (1962)
[Rob1] Robinson, D.W.: Lie groups and Lipschitz spaces. Duke Math. J.57, 357–395 (1988)
[Rob2] Robinson, D.W.: The differential and integral structure of representations of Lie groups. J. Op. Theor.19, 95–128 (1988)
[Tri] Triebel, H.: Interpolation theory, function spaces, differential operators. Amsterdam: North-Holland 1978
[Yos] Yosida, K.: Functional analysis. Berlin, Heidelberg, New York: Springer 1968
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Communicated by A. Jaffe
Dedicated to Res Jost and Arthur Wightman
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Robinson, D.W. The heat semigroup and integrability of Lie algebras: Lipschitz spaces and smoothness properties. Commun.Math. Phys. 132, 217–243 (1990). https://doi.org/10.1007/BF02278009
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DOI: https://doi.org/10.1007/BF02278009