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Correction to: A Generalized Fejér’s Theorem for Locally Compact Groups

  • Huichi HuangEmail author
Correction
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1 Correction to: J Geom Anal (2018) 28:909–920  https://doi.org/10.1007/s12220-017-9847-7

As pointed out by Hanfeng Li, we fix a gap in Theorem 1.2.

There are three issues in the paper.
  1. (1)

    Let G be a locally compact group with the unit \(e_G\) and a fixed left Haar measure \(\mu \). Let \(\nu \) be the right Haar measure given by \(\nu (A)=\mu (A^{-1})\) for every Borel subset A of G.

     
Theorem 1.2 is stated as follows.

Theorem

(A generalized Fejér’s theorem)

Consider a locally compact group G with a fixed left Haar measure \(\mu \) and the corresponding right Haar measure \(\nu \). Let \(\{F_\theta \}_{\theta \in \Theta }\) be an approximate identity of \(L^1(G)\). Assume that there exists a local partition \(\{A_1,A_2,\ldots , A_k\}\) of G such that \(\displaystyle \lim _{\theta }\int _{A_j}F_\theta (y)\,d\mu (y)=\lambda _j\) for every \(1\le j\le k\).

For an f in \(L^\infty (G)\), if there exists x in G such that \(\displaystyle \lim _{\begin{array}{c} y\rightarrow e_G \\ y\in A_j \end{array}} f(y^{-1}x)\) (denoted by \(f(x,A_j)\)) exists for every \(1\le j\le k\), then
$$\begin{aligned} \lim _{\theta }F_\theta *f(x)=\sum _{j=1}^k \lambda _j f(x,A_j). \end{aligned}$$
Moreover if \(\displaystyle \lim _{\theta }\sup _{y\in \mathcal {N}^c}|F_\theta (y)|=0\) for any neighborhood \(\mathcal {N}\) of \(e_G\), then for every f in \(L^1(G,\nu )\) (or \(L^\infty (G)\)) such that each \(f(x,A_j)\) exists for some x in G, we have
$$\begin{aligned} \lim _{\theta }F_\theta *f(x)=\sum _{j=1}^k \lambda _j f(x,A_j). \end{aligned}$$
In the proof of Theorem 1.2, there are two corrections:
  1. (a)

    The sentence “Now assume that \(\displaystyle \lim _{\theta }\sup _{y\in \mathcal {N}^c}|F_\theta (y)|=0\) for any neighborhood \(\mathcal {N}\) of \(e_G\) and f is in \(L^1(G)\) such that each \(f(x,A_j)\) exists for some x in G ” in the line -5 on page 5 is changed to “Now assume that \(\displaystyle \lim _{\theta }\sup _{y\in \mathcal {N}^c}|F_\theta (y)|=0\) for any neighborhood \(\mathcal {N}\) of \(e_G\) and f is in \(L^1(G,\nu )\) such that each \(f(x,A_j)\) exists for some x in G”.

     
  2. (b)
    the last identities are changed to:
    $$\begin{aligned} \limsup _{\theta }\Big |\int _{\mathcal {N}^c} F_\theta (y)f(y^{-1}x)\,d\mu (y)\Big |\le \Big [\lim _{\theta }\sup _{y\in \mathcal {N}^c}|F_\theta (y)|\Big ] \Vert f\Vert _{{L^1(G,\nu )}}=0. \end{aligned}$$
     
  1. (2)

    Statement of Corollary 2.2 is changed to the following:

     

Corollary

Given any approximate identity \(\{F_\theta \}_{\theta \in \Theta }\) of \(L^1(G)\) and local partition \(\{A_1,\ldots , A_k\}\) of G, if for an f in \(L^\infty (G)\), every \(f(x, A_j)\) exists, then there exists a subnet \(\Theta _1\) of \(\Theta \) such that every \(\displaystyle \lim _{\theta \in \Theta _1}\int _{A_j}F_\theta (y)\,d\mu (y)\) exists (denoted by \(\lambda _j(\Theta _1)\)) and
$$\begin{aligned} \displaystyle \lim _{\theta \in \Theta _1} F_\theta *f(x)=\sum _{j=1}^k \lambda _j(\Theta _1)f(x,A_j). \end{aligned}$$
Moreover if \(\displaystyle \lim _{\theta \in \Theta _1}\sup _{y\in \mathcal {N}^c}|F_\theta (y)|=0\) for every neighborhood \(\mathcal {N}\) of \(e_G\), then for every f in \(L^1(G,\nu )\) (or \(L^\infty (G)\)) such that every \(f(x,A_j)\) exists for some x in G, we have
$$\begin{aligned} \displaystyle \lim _{\theta \in \Theta _1} F_\theta *f(x)=\sum _{j=1}^k \lambda _j(\Theta _1)f(x,A_j). \end{aligned}$$
  1. (3)

    Before Corollary 3.5, add a definition.

     
For \(k=(k_1,\ldots ,k_d)\) in \(\{0,1\}^d\), define
$$\begin{aligned} J_k=\displaystyle \prod _{l=1}^d J_{k_l} \end{aligned}$$
with \(J_0=(-\infty ,0)\) and \(J_1=[0,\infty )\).

Notes

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China

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