Remarks on L2 boundedness of Littlewood–Paley operators

Kôzô Yabuta

doi:10.1515/anly-2017-0044

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Remarks on L2 boundedness of Littlewood–Paley operators

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

Kôzô Yabuta

Published/Copyright: October 25, 2017

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From the journal Analysis Volume 37 Issue 4

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⁢(ℝn) if and only if

supξ′∈Sn-1|∬ℝn×ℝnψ(x)ψ⁢(y)¯log|ξ′⋅(x-y)|dxdy|<∞.”

should be replaced by “under the assumption

(1.0)∬ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯⁢log⁡|ξ′⋅(x-y)||⁢𝑑x⁢𝑑y<∞ for a.e. ⁢ξ′∈𝒮n-1,

gψ is bounded on L2⁢(ℝn) if and only if

supξ′∈Sn-1|∬ℝn×ℝnψ(x)ψ⁢(y)¯log|ξ′⋅(x-y)|dxdy|<∞.”

2. In Remark 1.1, the statement

“⁢Ω∈ℱ1⁢(Sn-1):={Ω∈L1⁢(Sn-1):supξ′∈Sn-1⁡∫Sn-1|Ω⁢(y′)|⁢log⁡1|ξ′⋅y′|⁢d⁢σ⁢(y′)<∞}.”

should be replaced by

“Ω∈ℱ1(Sn-1):={Ω∈L1(Sn-1):supξ′∈Sn-1|∫Sn-1Ω(y′)log1|ξ′⋅y′||dσ(y′)<∞}.”

3. In line 9 on page 216, the statement “Since ψ∈L1⁢(ℝn), this shows the desired assertion.” should be replaced by “Next we check (1.0). In the case n=1, ξ′=1 or =-1 for ξ′∈S0, and so we trivially have

∬ℝ1×ℝ1|ψ⁢(x)⁢ψ⁢(y)¯|⁢log⁡2(ξ′⋅x′)2+(ξ′⋅y′)2⁢d⁢x⁢d⁢y=log⁡2⁢∥ψ∥L1⁢(ℝ)2 for ⁢ξ′∈S0.

In the case n≥2, we have

=∬ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯|⁢∫Sn-1log⁡1|ξ1′|⁢d⁢σ⁢(ξ′)⁢𝑑x⁢𝑑y

=ωn-2⁢∬ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯|⁢∫-11(log⁡1|s|)⁢(1-s2)n-32⁢𝑑s⁢𝑑x⁢𝑑y

=Cn⁢∥ψ∥L1⁢(ℝn)2,

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

∬ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯|⁢log⁡1|ξ′⋅x′|⁢d⁢x⁢d⁢y<∞ for a.e. ⁢ξ′∈Sn-1.

Thus we have

∬ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯|⁢log⁡2(ξ′⋅x′)2+(ξ′⋅y′)2⁢d⁢x⁢d⁢y<∞ for a.e. ⁢ξ′∈Sn-1.

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

∬ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯⁢log⁡|ξ′⋅(x-y)||⁢𝑑x⁢𝑑y

≤∬ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯⁢log⁡(ξ′⋅x′)2+(ξ′⋅y′)2|⁢𝑑x⁢𝑑y

+∫ℝn×ℝn|ψ⁢(x)⁢ψ⁢(y)¯⁢log⁡|x|2+|y|2|⁢𝑑x⁢𝑑y

+∬Sn-1×Sn-1(∫0∞[∫0π2|ψ(rcosθx′)ψ⁢(r⁢sin⁡θ⁢y′)¯|(cosθsinθ)n-1

×|log|cos(θ+tan-1ξ′⋅y′ξ′⋅x′)||dθ]r2⁢n-1dr)dσ(x′)dσ(y′)<∞

for a.e ξ′∈Sn-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. 10.1007/978-0-387-09432-8Search in Google Scholar

[2] K. Yabuta, Remarks on L2 boundedness of Littlewood–Paley operators, Analysis 33 (2003), 209–218. 10.1524/anly.2013.1144Search in Google Scholar

Received: 2017-8-17

Accepted: 2017-9-21

Published Online: 2017-10-25

Published in Print: 2017-11-1

Articles in the same Issue

https://doi.org/10.1515/anly-2017-0044

Keywords for this article

Littlewood–Paley operators