### Abstract

We study double Hilbert transforms and maximal functions along surfaces of the form (t_{1}, t_{2}, γ_{1}(t _{1})γ_{2}(t_{2})). 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}(ℝ^{3}) if both γ_{i}'s above are extended as even functions. If γ_{1} is odd, then we need an additional comparability condition on γ_{2}. This result is extended to higher dimensions and the general hyper-surfaces of the form (t_{1},..., t_{n}, Γ(t_{1},..., t_{n})) on ℝ^{n+1}.

Original language | English |
---|---|

Pages (from-to) | 1047-1073 |

Number of pages | 27 |

Journal | Revista Matematica Iberoamericana |

Volume | 24 |

Issue number | 3 |

Publication status | Published - 2008 Dec 1 |

### Fingerprint

### Keywords

- Flat surface
- Multiple hubert transform
- Singular radon transform

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Revista Matematica Iberoamericana*,

*24*(3), 1047-1073.

**Multiparameter singular integrals and maximal operators along flat surfaces.** / Cho, Yong Kum; Hong, Sunggeum; Kim, Joonil; Yang, Chan Woo.

Research output: Contribution to journal › Article

*Revista Matematica Iberoamericana*, vol. 24, no. 3, pp. 1047-1073.

}

TY - JOUR

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 - 2008/12/1

Y1 - 2008/12/1

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

UR - http://www.scopus.com/inward/record.url?scp=58549112781&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58549112781&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:58549112781

VL - 24

SP - 1047

EP - 1073

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 3

ER -