Multiparameter singular integrals and maximal operators along flat surfaces

Yong Kum Cho; Sunggeum Hong; Joonil Kim; Chan Woo Yang

doi:10.4171/RMI/566

Multiparameter singular integrals and maximal operators along flat surfaces

Yong Kum Cho, Sunggeum Hong, Joonil Kim, Chan Woo Yang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We study double Hilbert transforms and maximal functions along surfaces of the form (t₁, t₂, γ₁(t ₁)γ₂(t₂)). The L^P(ℝ ³) boundedness of the maximal operator is obtained if each γ_i is a convex increasing and γ_i(0) = 0. The double Hilbert transform is bounded in L^P(ℝ³) if both γ_i's above are extended as even functions. If γ₁ is odd, then we need an additional comparability condition on γ₂. This result is extended to higher dimensions and the general hyper-surfaces of the form (t₁,..., t_n, Γ(t₁,..., t_n)) on ℝⁿ⁺¹.

Original language	English
Pages (from-to)	1047-1073
Number of pages	27
Journal	Revista Matematica Iberoamericana
Volume	24
Issue number	3
DOIs	https://doi.org/10.4171/RMI/566
Publication status	Published - 2008

Keywords

Flat surface
Multiple hubert transform
Singular radon transform

ASJC Scopus subject areas

Mathematics(all)

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10.4171/RMI/566

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title = "Multiparameter singular integrals and maximal operators along flat surfaces",

abstract = "We study double Hilbert transforms and maximal functions along surfaces of the form (t1, t2, γ1(t 1)γ2(t2)). The LP(ℝ 3) boundedness of the maximal operator is obtained if each γi is a convex increasing and γi(0) = 0. The double Hilbert transform is bounded in LP(ℝ3) if both γi's above are extended as even functions. If γ1 is odd, then we need an additional comparability condition on γ2. This result is extended to higher dimensions and the general hyper-surfaces of the form (t1,..., tn, Γ(t1,..., tn)) on ℝn+1.",

keywords = "Flat surface, Multiple hubert transform, Singular radon transform",

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T1 - Multiparameter singular integrals and maximal operators along flat surfaces

AU - Cho, Yong Kum

AU - Hong, Sunggeum

AU - Kim, Joonil

AU - Yang, Chan Woo

PY - 2008

Y1 - 2008

N2 - We study double Hilbert transforms and maximal functions along surfaces of the form (t1, t2, γ1(t 1)γ2(t2)). The LP(ℝ 3) boundedness of the maximal operator is obtained if each γi is a convex increasing and γi(0) = 0. The double Hilbert transform is bounded in LP(ℝ3) if both γi's above are extended as even functions. If γ1 is odd, then we need an additional comparability condition on γ2. This result is extended to higher dimensions and the general hyper-surfaces of the form (t1,..., tn, Γ(t1,..., tn)) on ℝn+1.

AB - We study double Hilbert transforms and maximal functions along surfaces of the form (t1, t2, γ1(t 1)γ2(t2)). The LP(ℝ 3) boundedness of the maximal operator is obtained if each γi is a convex increasing and γi(0) = 0. The double Hilbert transform is bounded in LP(ℝ3) if both γi's above are extended as even functions. If γ1 is odd, then we need an additional comparability condition on γ2. This result is extended to higher dimensions and the general hyper-surfaces of the form (t1,..., tn, Γ(t1,..., tn)) on ℝn+1.

KW - Flat surface

KW - Multiple hubert transform

KW - Singular radon transform

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DO - 10.4171/RMI/566

M3 - Article

AN - SCOPUS:58549112781

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VL - 24

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EP - 1073

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

IS - 3

ER -

Multiparameter singular integrals and maximal operators along flat surfaces

Abstract

Keywords

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