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- Title
On almost everywhere divergence of Bochner-Riesz means on compact Lie groups.
- Authors
Chen, Xianghong; Fan, Dashan
- Abstract
Let G be a connected, simply connected, compact semisimple Lie group of dimension n. It has been shown by Clerc (Ann Inst Fourier Grenoble 24(1):149-172, <xref>1974</xref>) that, for any f∈L1(G)<inline-graphic></inline-graphic>, the Bochner-Riesz mean SRδ(f)<inline-graphic></inline-graphic> converges almost everywhere to f, provided δ>(n-1)/2<inline-graphic></inline-graphic>. In this paper, we show that, at the critical index δ=(n-1)/2<inline-graphic></inline-graphic>, there exists an f∈L1(G)<inline-graphic></inline-graphic> such that lim supR→∞|SR(n-1)/2(f)(x)|=∞,a.e.x∈G.<graphic></graphic>This is an analogue of a well-known result of Kolmogoroff (Fund Math 4(1):324-328, <xref>1923</xref>) for Fourier series on the circle, and a result of Stein (Ann Math 2(74):140-170, <xref>1961</xref>) for Bochner-Riesz means on the tori Tn,n≥2<inline-graphic></inline-graphic>. We also study localization properties of the Bochner-Riesz mean SR(n-1)/2(f)<inline-graphic></inline-graphic> for f∈L1(G)<inline-graphic></inline-graphic>.
- Publication
Mathematische Zeitschrift, 2018, Vol 289, Issue 3/4, p961
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-017-1983-z