### Abstract

For ψ ∈ C_{0}^{∞} (R^{d}) and m > 0 we consider the maximal operator given byM_{m} f (x, t) = under(sup, r > 0) | under(∫, R^{d}) f (x - y, t - | y |^{m}) frac(1, r^{d}) ψ (frac(y, r)) d y | . It is well known that M_{m} is a L^{p}-bounded operator for 1 < p ≤ ∞. Also A. Seeger and T. Tao proved that M_{m} is of weak-type L log log L if m ≠ 1. In this paper we consider the case m = 1 and prove M_{1} maps the standard Hardy space H^{1} to weak L^{1} if d ≥ 4.

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

Pages (from-to) | 152-162 |

Number of pages | 11 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 351 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Mar 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Maximal functions
- Singular integrals

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Endpoint estimates for some maximal operators associated to the circular conical surface.** / Heo, Ya-Ryong.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Endpoint estimates for some maximal operators associated to the circular conical surface

AU - Heo, Ya-Ryong

PY - 2009/3/1

Y1 - 2009/3/1

N2 - For ψ ∈ C0∞ (Rd) and m > 0 we consider the maximal operator given byMm f (x, t) = under(sup, r > 0) | under(∫, Rd) f (x - y, t - | y |m) frac(1, rd) ψ (frac(y, r)) d y | . It is well known that Mm is a Lp-bounded operator for 1 < p ≤ ∞. Also A. Seeger and T. Tao proved that Mm is of weak-type L log log L if m ≠ 1. In this paper we consider the case m = 1 and prove M1 maps the standard Hardy space H1 to weak L1 if d ≥ 4.

AB - For ψ ∈ C0∞ (Rd) and m > 0 we consider the maximal operator given byMm f (x, t) = under(sup, r > 0) | under(∫, Rd) f (x - y, t - | y |m) frac(1, rd) ψ (frac(y, r)) d y | . It is well known that Mm is a Lp-bounded operator for 1 < p ≤ ∞. Also A. Seeger and T. Tao proved that Mm is of weak-type L log log L if m ≠ 1. In this paper we consider the case m = 1 and prove M1 maps the standard Hardy space H1 to weak L1 if d ≥ 4.

KW - Maximal functions

KW - Singular integrals

UR - http://www.scopus.com/inward/record.url?scp=56549114279&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56549114279&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2008.10.011

DO - 10.1016/j.jmaa.2008.10.011

M3 - Article

AN - SCOPUS:56549114279

VL - 351

SP - 152

EP - 162

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -