## 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 |
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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 |

### Bibliographical note

Funding Information:This research was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology NRF-2015R1A1A1A05001304, NRF-2014R1A1A3049983, and NRF-2013R1A1A2013659.

Publisher Copyright:

© 2017 American Mathematical Society.

## Keywords

- Maximal operators

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics