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

• Huichi Huang
Correction

## 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 $$J_1=[0,\infty )$$.