Correction to: A Short Proof of Commutator Estimates

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

$$\begin{aligned} s=s_{1}+s_{2}\in (0,2), \quad s_{j}\in (0,1), \qquad \frac{1}{r}=\frac{1}{p_{1}}+\frac{1}{p_{2}}, \quad \frac{2n}{n+2s_{j}}<p_{j}<\infty . \end{aligned}$$

Then for all $u,v\in \mathscr {S}(\mathbb {R}^{n})$ we have

$$\begin{aligned} \Vert D ^{s}(uv)-uD ^{s}v-vD ^{s}u\Vert _{L^{r}} \lesssim \Vert D ^{s_{1}}u\Vert _{L^{p_{1}}} \Vert D ^{s_{2}}v\Vert _{L^{p_{2}}} \end{aligned}$$

(1.1)

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

$$\begin{aligned} \textstyle q_{j}=p_{j}\left( \frac{1}{2}+\frac{s_{j}}{n}\right) \quad \text {if } n\ge 2, \qquad q_{j}=\min \left\{ p_{j},p_{j}\left( \frac{1}{2}+s_{j}\right) \right\} \quad \text {if } n=1, \end{aligned}$$

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

$$\begin{aligned} \Vert D ^{s}(uv)-uD ^{s}v-vD ^{s}u\Vert _{L^{r}(w_{1}^{r/p_{1}}w_{2}^{r/p_{2}}dx)} \lesssim \Vert D ^{s_{1}}u\Vert _{L^{p_{1}}(w_{1}dx)} \Vert D ^{s_{2}}v\Vert _{L^{p_{2}}(w_{2}dx)}. \end{aligned}$$

(1.2)

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 $w\in A_{\min \{p,\frac{p \lambda }{2}\}}$.