### Abstract

Let U be a bounded open subset of ℝ^{d} and let Ω be a Lebesgue measurable subset of U. Let γ = (γ_{1}, · · ·, γ_{n}): U \ Ω → ℝ^{n} be a Lebesgue measurable function, and let µ be a Borel measure on ℝ^{d+n} defined by where ψ is a smooth function supported in U. In this paper we give some conditions under which the Fourier decay estimates |µ(ξ)| ≤ C(1+|ξ|)^{-ε} hold for some ε > 0. As a corollary we obtain the L^{p}-boundedness properties of the maximal operators MS associated with a certain class of possibly non-smooth n-dimensional submanifolds of ℝ^{d+n}, i.e., where Ω_{sym} is a radially symmetric Lebesgue measurable subset of ℝ^{d}, γ(y) = (γ_{1}(y), · · ·, γ_{n}(y)), γ_{i}(ty) = tai^{γi}(y) for each t > 0 where ai ∈ ℝ, and the function γ_{i}: ℝ^{d} \Ω_{sym} → ℝ satisfies some singularity conditions over a certain subset of ℝ^{d}. Also we investigate the endpoint (parabolic H^{1}, L^{1,∞}) mapping properties of the maximal operators M_{H} associated with a certain class of possibly non-smooth hypersurfaces, i.e., where the function γ: ℝ^{d} → ℝ satisfies some singularity conditions over a certain subset of ℝ^{d} and γ(ty) = t^{m}γ(y) for each t > 0 where m > 0.

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

Pages (from-to) | 4597-4629 |

Number of pages | 33 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2017 |

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### Keywords

- Maximal operators

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*369*(7), 4597-4629. https://doi.org/10.1090/tran/6785

**Maximal operators associated with some singular submanifolds.** / Heo, Ya-Ryong; Hong, Sunggeum; Yang, Chan Woo.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 369, no. 7, pp. 4597-4629. https://doi.org/10.1090/tran/6785

}

TY - JOUR

T1 - Maximal operators associated with some singular submanifolds

AU - Heo, Ya-Ryong

AU - Hong, Sunggeum

AU - Yang, Chan Woo

PY - 2017

Y1 - 2017

N2 - Let U be a bounded open subset of ℝd and let Ω be a Lebesgue measurable subset of U. Let γ = (γ1, · · ·, γn): U \ Ω → ℝn be a Lebesgue measurable function, and let µ be a Borel measure on ℝd+n defined by where ψ is a smooth function supported in U. In this paper we give some conditions under which the Fourier decay estimates |µ(ξ)| ≤ C(1+|ξ|)-ε hold for some ε > 0. As a corollary we obtain the Lp-boundedness properties of the maximal operators MS associated with a certain class of possibly non-smooth n-dimensional submanifolds of ℝd+n, i.e., where Ωsym is a radially symmetric Lebesgue measurable subset of ℝd, γ(y) = (γ1(y), · · ·, γn(y)), γi(ty) = taiγi(y) for each t > 0 where ai ∈ ℝ, and the function γi: ℝd \Ωsym → ℝ satisfies some singularity conditions over a certain subset of ℝd. Also we investigate the endpoint (parabolic H1, L1,∞) mapping properties of the maximal operators MH associated with a certain class of possibly non-smooth hypersurfaces, i.e., where the function γ: ℝd → ℝ satisfies some singularity conditions over a certain subset of ℝd and γ(ty) = tmγ(y) for each t > 0 where m > 0.

AB - Let U be a bounded open subset of ℝd and let Ω be a Lebesgue measurable subset of U. Let γ = (γ1, · · ·, γn): U \ Ω → ℝn be a Lebesgue measurable function, and let µ be a Borel measure on ℝd+n defined by where ψ is a smooth function supported in U. In this paper we give some conditions under which the Fourier decay estimates |µ(ξ)| ≤ C(1+|ξ|)-ε hold for some ε > 0. As a corollary we obtain the Lp-boundedness properties of the maximal operators MS associated with a certain class of possibly non-smooth n-dimensional submanifolds of ℝd+n, i.e., where Ωsym is a radially symmetric Lebesgue measurable subset of ℝd, γ(y) = (γ1(y), · · ·, γn(y)), γi(ty) = taiγi(y) for each t > 0 where ai ∈ ℝ, and the function γi: ℝd \Ωsym → ℝ satisfies some singularity conditions over a certain subset of ℝd. Also we investigate the endpoint (parabolic H1, L1,∞) mapping properties of the maximal operators MH associated with a certain class of possibly non-smooth hypersurfaces, i.e., where the function γ: ℝd → ℝ satisfies some singularity conditions over a certain subset of ℝd and γ(ty) = tmγ(y) for each t > 0 where m > 0.

KW - Maximal operators

UR - http://www.scopus.com/inward/record.url?scp=85017301567&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017301567&partnerID=8YFLogxK

U2 - 10.1090/tran/6785

DO - 10.1090/tran/6785

M3 - Article

AN - SCOPUS:85017301567

VL - 369

SP - 4597

EP - 4629

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -