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, 1974) that, for any \(f\in L^1(G)\), the Bochner–Riesz mean \(S_R^\delta (f)\) converges almost everywhere to f, provided \(\delta >(n-1)/2\). In this paper, we show that, at the critical index \(\delta =(n-1)/2\), there exists an \(f\in L^1(G)\) such that
This is an analogue of a well-known result of Kolmogoroff (Fund Math 4(1):324–328, 1923) for Fourier series on the circle, and a result of Stein (Ann Math 2(74):140–170, 1961) for Bochner–Riesz means on the tori \(\mathbb {T}^{n}, n\ge 2\). We also study localization properties of the Bochner–Riesz mean \(S_{R}^{(n-1)/2}(f)\) for \(f\in L^1(G)\).
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Notes
Alternatively, one can take Q to be any fundamental domain centered at 0, and \(Q_0\) a sufficiently small ball centered at 0.
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Dashan Fan was partially supported by the National Natural Science Foundation of China (Grant Nos. 11471288, 11601456).
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Chen, X., Fan, D. On almost everywhere divergence of Bochner–Riesz means on compact Lie groups. Math. Z. 289, 961–981 (2018). https://doi.org/10.1007/s00209-017-1983-z
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DOI: https://doi.org/10.1007/s00209-017-1983-z