## Abstract

In our paper in this journal, entitled “Remarks on ${L}^{2}$ boundedness of Littlewood–Paley operators”, there are two incomplete statements and incompleteness in the proof of the main theorem. In this short note we will correct them.

Show Summary Details# Remarks on ${L}^{2}$ boundedness of Littlewood–Paley operators

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### International mathematical journal of analysis and its applications

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This Erratum corrects the original online version which can be found here: https://doi.org/10.1515/anly.2013.1144

In our paper in this journal, entitled “Remarks on ${L}^{2}$ boundedness of Littlewood–Paley operators”, there are two incomplete statements and incompleteness in the proof of the main theorem. In this short note we will correct them.

Keywords: Littlewood–Paley operators

In our paper [2], entitled “Remarks on ${L}^{2}$ boundedness of Littlewood–Paley operators”, there are two incomplete statements and incompleteness in the proof of the main theorem.

1. From line 11 to line 12 in the Introduction, the statement “${g}_{\psi}$ is bounded on ${L}^{2}({\mathbb{R}}^{n})$ if and only if

$\underset{{\xi}^{\prime}\in {S}^{n-1}}{sup}|{\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}\psi (x)\overline{\psi (y)}\mathrm{log}|{\xi}^{\prime}\cdot (x-y)|dxdy|<\mathrm{\infty}.\text{\u201d}$

should be replaced by “under the assumption

${\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}\left|\psi (x)\overline{\psi (y)}\mathrm{log}|{\xi}^{\prime}\cdot (x-y)|\right|\mathit{d}x\mathit{d}y<\mathrm{\infty}\mathit{\hspace{1em}}\text{for a.e.}{\xi}^{\prime}\in {\mathcal{\mathcal{S}}}^{n-1},$(1.0)

${g}_{\psi}$ is bounded on ${L}^{2}({\mathbb{R}}^{n})$ if and only if

$\underset{{\xi}^{\prime}\in {S}^{n-1}}{sup}|{\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}\psi (x)\overline{\psi (y)}\mathrm{log}|{\xi}^{\prime}\cdot (x-y)|dxdy|<\mathrm{\infty}.\text{\u201d}$

2. In Remark 1.1, the statement

$\text{\u201c}\mathrm{\Omega}\in {\mathcal{\mathcal{F}}}_{1}({S}^{n-1}):=\{\mathrm{\Omega}\in {L}^{1}({S}^{n-1}):\underset{{\xi}^{\prime}\in {S}^{n-1}}{sup}{\int}_{{S}^{n-1}}|\mathrm{\Omega}({y}^{\prime})|\mathrm{log}\frac{1}{|{\xi}^{\prime}\cdot {y}^{\prime}|}d\sigma ({y}^{\prime})<\mathrm{\infty}\}.\text{\u201d}$

should be replaced by

$\text{\u201c}\mathrm{\Omega}\in {\mathcal{\mathcal{F}}}_{1}({S}^{n-1}):=\{\mathrm{\Omega}\in {L}^{1}({S}^{n-1}):\underset{{\xi}^{\prime}\in {S}^{n-1}}{sup}|{\int}_{{S}^{n-1}}\mathrm{\Omega}({y}^{\prime})\mathrm{log}\frac{1}{|{\xi}^{\prime}\cdot {y}^{\prime}|}|d\sigma ({y}^{\prime})<\mathrm{\infty}\}.\text{\u201d}$

3. In line 9 on page 216, the statement “Since $\psi \in {L}^{1}({\mathbb{R}}^{n})$, this shows the desired assertion.” should be replaced by “Next we check (1.0). In the case $n=1$, ${\xi}^{\prime}=1$ or $=-1$ for ${\xi}^{\prime}\in {S}^{0}$, and so we trivially have

${\iint}_{{\mathbb{R}}^{1}\times {\mathbb{R}}^{1}}|\psi (x)\overline{\psi (y)}|\mathrm{log}\frac{2}{\sqrt{{({\xi}^{\prime}\cdot {x}^{\prime})}^{2}+{({\xi}^{\prime}\cdot {y}^{\prime})}^{2}}}dxdy=\mathrm{log}\sqrt{2}{\parallel \psi \parallel}_{{L}^{1}(\mathbb{R})}^{2}\mathit{\hspace{1em}}\text{for}{\xi}^{\prime}\in {S}^{0}.$

In the case $n\ge 2$, we have

${\int}_{{S}^{n-1}}}{\displaystyle {\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}}|\psi (x)\overline{\psi (y)}|\mathrm{log}{\displaystyle \frac{1}{|{\xi}^{\prime}\cdot {x}^{\prime}|}}dxdyd\sigma ({\xi}^{\prime})={\displaystyle {\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}}|\psi (x)\overline{\psi (y)}|{\displaystyle {\int}_{{S}^{n-1}}}\mathrm{log}{\displaystyle \frac{1}{|{\xi}^{\prime}\cdot {x}^{\prime}|}}d\sigma ({\xi}^{\prime})\mathit{d}x\mathit{d}y$$={\displaystyle {\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}}|\psi (x)\overline{\psi (y)}|{\displaystyle {\int}_{{S}^{n-1}}}\mathrm{log}{\displaystyle \frac{1}{|{\xi}_{1}^{\prime}|}}d\sigma ({\xi}^{\prime})\mathit{d}x\mathit{d}y$$={\omega}_{n-2}{\displaystyle {\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}}|\psi (x)\overline{\psi (y)}|{\displaystyle {\int}_{-1}^{1}}\left(\mathrm{log}{\displaystyle \frac{1}{|s|}}\right){(1-{s}^{2})}^{\frac{n-3}{2}}\mathit{d}s\mathit{d}x\mathit{d}y$$={C}_{n}{\parallel \psi \parallel}_{{L}^{1}({\mathbb{R}}^{n})}^{2},$

where ${\omega}_{n-2}$ is the surface area of the unit sphere in ${\mathbb{R}}^{n-1}$ (see [1, Section 5.2.2]). Hence we get

${\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}|\psi (x)\overline{\psi (y)}|\mathrm{log}\frac{1}{|{\xi}^{\prime}\cdot {x}^{\prime}|}dxdy<\mathrm{\infty}\mathit{\hspace{1em}}\text{for a.e.}{\xi}^{\prime}\in {S}^{n-1}.$

Thus we have

${\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}|\psi (x)\overline{\psi (y)}|\mathrm{log}\frac{2}{\sqrt{{({\xi}^{\prime}\cdot {x}^{\prime})}^{2}+{({\xi}^{\prime}\cdot {y}^{\prime})}^{2}}}dxdy<\mathrm{\infty}\mathit{\hspace{1em}}\text{for a.e.}{\xi}^{\prime}\in {S}^{n-1}.$

Using the above estimate, and observing the proof of estimates (2.4)–(2.11), we see that

${\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}}\left|\psi (x)\overline{\psi (y)}\mathrm{log}|{\xi}^{\prime}\cdot (x-y)|\right|\mathit{d}x\mathit{d}y$$\le {\displaystyle {\iint}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}}\left|\psi (x)\overline{\psi (y)}\mathrm{log}\sqrt{{({\xi}^{\prime}\cdot {x}^{\prime})}^{2}+{({\xi}^{\prime}\cdot {y}^{\prime})}^{2}}\right|\mathit{d}x\mathit{d}y$$+{\displaystyle {\int}_{{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}}}\left|\psi (x)\overline{\psi (y)}\mathrm{log}\sqrt{{|x|}^{2}+{|y|}^{2}}\right|\mathit{d}x\mathit{d}y$$+{\displaystyle {\iint}_{{S}^{n-1}\times {S}^{n-1}}}({\displaystyle {\int}_{0}^{\mathrm{\infty}}}[{\displaystyle {\int}_{0}^{\frac{\pi}{2}}}\left|\psi (r\mathrm{cos}\theta {x}^{\prime})\overline{\psi (r\mathrm{sin}\theta {y}^{\prime})}\right|{(\mathrm{cos}\theta \mathrm{sin}\theta )}^{n-1}$$\times \left|\mathrm{log}\left|\mathrm{cos}(\theta +{\mathrm{tan}}^{-1}{\displaystyle \frac{{\xi}^{\prime}\cdot {y}^{\prime}}{{\xi}^{\prime}\cdot {x}^{\prime}}})\right|\right|d\theta ]{r}^{2n-1}dr)d\sigma ({x}^{\prime})d\sigma ({y}^{\prime})<\mathrm{\infty}$

for a.e ${\xi}^{\prime}\in {S}^{n-1}$. Thus, by (2.12) we obtain the desired assertion.

- [1]
L. Grafakos, Classical Fourier Analysis, 2nd ed., Grad. Texts in Math. 249, Springer, New York, 2008. Google Scholar

- [2]
K. Yabuta, Remarks on ${L}^{2}$ boundedness of Littlewood–Paley operators, Analysis 33 (2003), 209–218. Google Scholar

**Received**: 2017-08-17

**Accepted**: 2017-09-21

**Published Online**: 2017-10-25

**Published in Print**: 2017-11-01

**Citation Information: **Analysis, Volume 37, Issue 4, Pages 243–244, ISSN (Online) 2196-6753, ISSN (Print) 0174-4747, DOI: https://doi.org/10.1515/anly-2017-0044.

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