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

Keywords

  • Maximal operators

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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