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Analysis

International mathematical journal of analysis and its applications


CiteScore 2018: 0.72

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Volume 37, Issue 4

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Remarks on L2 boundedness of Littlewood–Paley operators

Kôzô Yabuta
  • Corresponding author
  • Research Center for Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1, Sanda 669-1337, Japan
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Published Online: 2017-10-25 | DOI: https://doi.org/10.1515/anly-2017-0044

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

Abstract

In our paper in this journal, entitled “Remarks on L2 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

MSC 2010: 42B25; 42B20

In our paper [2], entitled “Remarks on L2 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 “gψ is bounded on L2(n) if and only if

supξSn-1|n×nψ(x)ψ(y)¯log|ξ(x-y)|dxdy|<.

should be replaced by “under the assumption

n×n|ψ(x)ψ(y)¯log|ξ(x-y)||𝑑x𝑑y<for a.e. ξ𝒮n-1,(1.0)

gψ is bounded on L2(n) if and only if

supξSn-1|n×nψ(x)ψ(y)¯log|ξ(x-y)|dxdy|<.

2. In Remark 1.1, the statement

Ω1(Sn-1):={ΩL1(Sn-1):supξSn-1Sn-1|Ω(y)|log1|ξy|dσ(y)<}.

should be replaced by

Ω1(Sn-1):={ΩL1(Sn-1):supξSn-1|Sn-1Ω(y)log1|ξy||dσ(y)<}.

3. In line 9 on page 216, the statement “Since ψL1(n), this shows the desired assertion.” should be replaced by “Next we check (1.0). In the case n=1, ξ=1 or =-1 for ξS0, and so we trivially have

1×1|ψ(x)ψ(y)¯|log2(ξx)2+(ξy)2dxdy=log2ψL1()2for ξS0.

In the case n2, we have

Sn-1n×n|ψ(x)ψ(y)¯|log1|ξx|dxdydσ(ξ)=n×n|ψ(x)ψ(y)¯|Sn-1log1|ξx|dσ(ξ)𝑑x𝑑y=n×n|ψ(x)ψ(y)¯|Sn-1log1|ξ1|dσ(ξ)𝑑x𝑑y=ωn-2n×n|ψ(x)ψ(y)¯|-11(log1|s|)(1-s2)n-32𝑑s𝑑x𝑑y=CnψL1(n)2,

where ωn-2 is the surface area of the unit sphere in n-1 (see [1, Section 5.2.2]). Hence we get

n×n|ψ(x)ψ(y)¯|log1|ξx|dxdy<for a.e. ξSn-1.

Thus we have

n×n|ψ(x)ψ(y)¯|log2(ξx)2+(ξy)2dxdy<for a.e. ξSn-1.

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

n×n|ψ(x)ψ(y)¯log|ξ(x-y)||𝑑x𝑑yn×n|ψ(x)ψ(y)¯log(ξx)2+(ξy)2|𝑑x𝑑y+n×n|ψ(x)ψ(y)¯log|x|2+|y|2|𝑑x𝑑y+Sn-1×Sn-1(0[0π2|ψ(rcosθx)ψ(rsinθy)¯|(cosθsinθ)n-1×|log|cos(θ+tan-1ξyξx)||dθ]r2n-1dr)dσ(x)dσ(y)<

for a.e ξSn-1. Thus, by (2.12) we obtain the desired assertion.

References

  • [1]

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

  • [2]

    K. Yabuta, Remarks on L2 boundedness of Littlewood–Paley operators, Analysis 33 (2003), 209–218.  Google Scholar

About the article

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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