L^2-bounded singular integrals on a purely unrectifiable set in R^d

Författare

  • Joan Mateu Universitat Autònoma de Barcelona, Departament de Matemàtiques and Centre de Reserca Matemàtica
  • Laura Prat Universitat Autònoma de Barcelona, Departament de Matemàtiques and Centre de Reserca Matemàtica

Nyckelord:

Purely unrectifiable set, singular integral operator, Cantor type set, T(1)-theorem

Abstract

We construct an example of a purely unrectifiable measure \(\mu\) in \(\mathbf{R}^d\) for which the singular integrals associated to the kernels \(K(x)=P_{2k+1}(x)/|x|^{2k+d}\), with \(k\geq 1\) and \(P_{2k+1}\) a homogeneous harmonic polynomial of degree \(2k+1\), are bounded in \(L^2(\mu)\). This contrasts starkly with the results concerning the Riesz kernel \(x/|x|^d\) in \(\mathbf{R}^d\).
Sektion
Articles

Publicerad

2021-06-24

Referera så här

Mateu, J., & Prat, L. (2021). L^2-bounded singular integrals on a purely unrectifiable set in R^d. Annales Fennici Mathematici, 46(1), 187–200. Hämtad från https://afm.journal.fi/article/view/109766