Maximal operators associated with some singular submanifolds

Ya-Ryong Heo, Sunggeum Hong, Chan Woo Yang

Research output: Contribution to journalArticle

Abstract

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|) 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: ℝdsym → 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.

Original languageEnglish
Pages (from-to)4597-4629
Number of pages33
JournalTransactions of the American Mathematical Society
Volume369
Issue number7
DOIs
Publication statusPublished - 2017

Fingerprint

Maximal Operator
Submanifolds
Henri Léon Lebésgue
Subset
Singularity
Decay Estimates
Borel Measure
Measurable function
Smooth function
Hypersurface
n-dimensional
Boundedness
Corollary
Class

Keywords

  • Maximal operators

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 369, No. 7, 2017, p. 4597-4629.

Research output: Contribution to journalArticle

@article{68b73410a57a4b2faad38159fd45625e,
title = "Maximal operators associated with some singular submanifolds",
abstract = "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.",
keywords = "Maximal operators",
author = "Ya-Ryong Heo and Sunggeum Hong and Yang, {Chan Woo}",
year = "2017",
doi = "10.1090/tran/6785",
language = "English",
volume = "369",
pages = "4597--4629",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "7",

}

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 -