Correction to: Left translates of a square integrable function on the Heisenberg group

1 Correction to: Collectanea Mathematica (2020) 71:239–262 https://doi.org/10.1007/s13348-019-00255-4

Theorem 2.3 was proved by Saliani for $n = 1$ in (Proc Am Math Soc 141(3):937–941, 2013), as an application of the following result by Kislyakov.

Theorem 0.1

[1] For every $F \in L^{\infty }(\mathbb {T})$ with $\Vert F\Vert _{\infty } \le 1$ and for every $0<\epsilon \le 1$ there exists a function $G \in U^{\infty }$ with the following properties : $|G|+|F-G|=|F|$, $|\{\xi \in \mathbb {T}: F(\xi ) \ne G(\xi )\}| \le \epsilon \Vert F\Vert _{1}$ and $\Vert G\Vert _{U \infty } \le C\left( 1+\log \left( \epsilon ^{-1}\right) \right)$, where $U^{\infty }$ denotes the space of functions $f \in L^{\infty }(\mathbb {T})$ for which the following norm is finite:

$$ \Vert f\Vert _{U^{\infty }}=\sup \left\{ \left| \sum _{n \le k \le m} \hat{f}(k) \xi ^{k}\right| m, n \in \mathbb {Z}, n \le m, \xi \in \mathbb {T}\right\} $$

Kislyakov proved the above Theorem in a $\sigma$-finite measure space $(\Omega , \mu )$ under the basic assumption that $U^{\infty }$ satisfies both A1 and A2 properties. When the underlying measure space is $(\Omega , \mu )=(\mathbb {T}, m)$, where m is the normalized Lebesgue measure on the unit circle $\mathbb {T}$, it is easy to check that $U^{\infty }$ satisfies A1 property and by a result of Vinogradov [5], $U^{\infty }$ satisfies A2 property as well. Recently it has been proved by Ivana Slamic in [4] that $U^{\infty }$ does not satisfy A2 when the underlying measure space in $\left( \mathbb {T}^{n}, m\right) , n>1$. Therefore, Theorem 0.1 need not hold for $n>1$ and as a consequence, Theorem 2.3 of [2] is not considered proved.

Theorem 2.3 in [2] was used to prove Theorem 4.3 and Theorem 6.3 in the paper. The purpose of Theorem 4.3 was to have converse of Theorem 4.1, which we had already obtained in Theorem 4.2 with an additional assumption that $\left\{ T_{(k, l)}^{t} \varphi :(k, l) \in \right.$ $\left. \mathbb {Z}^{2 n}\right\}$ is a Bessel sequence in $L^{2}\left( \mathbb {R}^{2 n}\right)$. In conclusion, in this erratum we remove Theorem 4.3 from [2] and restate Theorem 6.3 with an additional assumption in the hypothesis that $\left\{ L_{(2 k, l, m)} \varphi :(k, l, m) \in \mathbb {Z}^{2 n+1}\right\}$ is a Bessel sequence in $L^{2}\left( \mathbb {H}^{n}\right)$.