## A heat semigroup version of Bernstein’s theorem on Lie groups.(English)Zbl 0719.43008

Let f be a continuous function on the one-dimensional torus $${\mathbb{T}}$$. A classical theorem of S. N. Bernstein states that if the modulus of continuity $$\omega_{\infty}(f)$$ of f satisfies $$\omega_{\infty}(f)(t)=O(t^{\epsilon +})$$ for $$\epsilon >0$$, then the Fourier series of f is absolutely convergent. Versions of Bernstein’s theorem have been established for many classes of Lie groups. See for instance G. I. Gaudry and R. Pini [Math. Proc. Camb. Philos. Soc. 99, 297-305 (1986; Zbl 0612.43011); J. Aust. Math. Soc., Ser. A 43, 385-397 (1987; Zbl 0633.43002)]; C. S. Herz [J. Math. Mech. 18, 283-323 (1968; Zbl 0177.157)]; I. R. Inglis [Boll. Unione Mat. Ital., VI. Ser., A 2, 39-46 (1983; Zbl 0528.43008)].
The purpose of the present paper is to establish a heat semigroup version of Bernstein’s theorem, applicable to any unimodular group, and to underline the essential geometric content of the result. Estimates for the norms of the heat kernels for small time and large time by N. Th. Varopoulos [J. Funct. Anal. 76, 346-410 (1988; Zbl 0634.22008)] reflect the dimension, and the measure-theoretic growth of the group, respectively. The authors’ main theorem is stated in terms of certain Lipschitz spaces whose definition incorporates these two geometric features of the group in question. In this respect, it harks back to an earlier paper of G. I. Gaudry and R. Pini [loc. cit.]. The geometric content is further underlined by showing that, in a certain sense, the theorem is best possible. The results are stated in terms of semigroup analogous of the $$A_ p$$ spaces of A. Figà-Talamanca [Duke Math. J. 32, 495-501 (1965; Zbl 0142.104)].
Reviewer: G.I.Gaudry

### MSC:

 43A80 Analysis on other specific Lie groups 22E30 Analysis on real and complex Lie groups 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 35K05 Heat equation
Full Text:

### References:

 [1] Bergh, J., Löfström, J.: Interpolation Spaces. Berlin-Heidelberg-New York: Springer. 1976. · Zbl 0344.46071 [2] Butzer, P. L., Berens, H.: Semigroups of Operators and Approximation. Berlin-Heidelberg-New York: Springer. 1967. · Zbl 0164.43702 [3] Cowling, M. G.: Harmonic analysis on semigroups. Annals of Math.117, 267-283 (1983). · Zbl 0528.42006 [4] Cowling, M. G., Gaudry, G. I., Giulini, S., Mauceri, G.: Weak type (1, 1) estimates for heat kernel maximal functions on Lie groups. Trans. Amer. Math. Soc. To appear. · Zbl 0722.22006 [5] Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France92, 181-236 (1964). · Zbl 0169.46403 [6] Folland, G. B., Stein, E. M.: Hardy Spaces on Homogeneous Groups. Math. Notes No. 28. Princeton Univ. Press. 1982. · Zbl 0508.42025 [7] Gaudry, G. I., Pini, R.: Bernstein’s theorem for compact connected Lie groups. Math. Proc. Camb. Phil. Soc.99, 297-305 (1986). · Zbl 0612.43011 [8] Gaudry, G. I., Pini, R.: Motion groups and absolutely convergent Fourier transforms. J. Austral. Math. Soc.43, 385-397 (1987). · Zbl 0633.43002 [9] Giulini, S.: Approximation and Besov spaces on stratified groups. Proc. Amer. Math. Soc.96, 569-578 (1986). · Zbl 0605.41013 [10] Herz, C. S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech.18, 283-323 (1986). · Zbl 0177.15701 [11] Hörmander, L.: Hypoelliptic second-order differential equations. Acta Math.119, 147-171 (1967). · Zbl 0156.10701 [12] Hulanicki, A.: Subalgebra ofL 1 (G) associated with Laplacian on a Lie group. Colloq. Math.31, 259-287 (1974). · Zbl 0316.43005 [13] Inglis, I. R.: Bernstein’s theorem and the Fourier algebra of the Heisenberg group. Boll. Un. Mat. Ital. (6)2-A, 39-46 (1983). · Zbl 0528.43008 [14] Meda, S., Pini, R.: Lipschitz spaces on compact Lie groups. Mh. Math.105, 177-191 (1988). · Zbl 0639.43003 [15] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II, New York-San Francisco-London: Academic Press. 1975. · Zbl 0308.47002 [16] Robinson, D. W.: Lie groups and Lipschitz spaces. Duke Math. J.57, 357-395 (1988). · Zbl 0687.46024 [17] Stein, E. M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton Univ. Press. 1970. · Zbl 0193.10502 [18] Varopoulos, N. Th.: Analysis on Lie groups. J. Funct. Anal.76, 346-410 (1988). · Zbl 0634.22008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.