Avoid common mistakes on your manuscript.
1 Correction to: J Fourier Anal Appl (2019) 25:1134–1146 https://doi.org/10.1007/s00041-018-9612-8
The Paley–Littlewood square function \(g^{*}_{\lambda }\) is not bounded on \(L^{p}\) spaces if \(p\le 1\), as wrongly stated in the paper (see Theorem 2.5). A weaker estimate holds though, namely, \(g^{*}_{\lambda }\) is bounded from the \(H^{p}\) real Hardy space to the usual \(L^{p}\) space if \(p\le 1\). This is proved in references [17, 24]. Thus it is necessary to make two changes in the paper:
- (1)
In Theorem 1.1, after formula (1.5) add the sentence: where the \(L^{p_{j}}\) norms at the right must be replaced by Hardy space norms \(H^{p_{j}}\) if \(p_{j}\le 1\).
- (2)
In Theorem 2.5, at the end of statement (i), add the sentence: where the \(L^{p}\) norm at the right must be replaced by a Hardy space \(H^{p}\) norm if \(p\le 1\).
For completeness, here are the correct statements of Theorems 1.1 and 2.5.
Theorem 1.1
Let \(n\ge 1\). Assume \(s,s_{1},s_{2}\) and \(r,p_{1},p_{2}\) satisfy
Then for all \(u,v\in \mathscr {S}(\mathbb {R}^{n})\) we have
where the \(L^{p_{j}}\) norms at the right must be replaced by Hardy space norms \(H^{p_{j}}\) if \(p_{j}\le 1\).
Moreover, if we define
and we assume in addition \(p_{1},p_{2}>1\) when \(n=1\), then for any weights \(w_{j}\in A_{q_{j}}\) we have
Theorem 2.5
Let \(n\ge 1\), \(\lambda >1\). For any \(u\in \mathscr {S}(\mathbb {R}^{n})\), \(g^{*}_{\lambda }(u)\) satisfies the following estimates, with constants independent of u:
- (i)
\(\Vert g^{*}_{\lambda }(u)\Vert _{L^{p}}\lesssim \Vert u\Vert _{L^{p}}\) for \(\lambda >\max \{1,\frac{2}{p}\}\) and \(0<p<\infty \), where the \(L^{p}\) norm at the right must be replaced by a Hardy space \(H^{p}\) norm if \(p\le 1\).
- (ii)
\(\Vert g^{*}_{\lambda }\Vert _{L^{p}(wdx)}\lesssim \Vert u\Vert _{L^{p}(wdx)}\) for \(\lambda >\max \{1,\frac{2}{p}\}\), \(1<p<\infty \) and \(w\in A_{\min \{p,\frac{p \lambda }{2}\}}\).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fulvio Ricci.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
D’Ancona, P. Correction to: A Short Proof of Commutator Estimates. J Fourier Anal Appl 26, 23 (2020). https://doi.org/10.1007/s00041-019-09724-7
Received:
Published:
DOI: https://doi.org/10.1007/s00041-019-09724-7