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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

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
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