TY - JOUR

T1 - Maximal operators associated with some singular submanifolds

AU - Heo, Yaryong

AU - Hong, Sunggeum

AU - Yang, Chan Woo

N1 - 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.

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 -