Symmetric and zygmund measures in several variables

E. Doubtsov, A. Nicolau

Research output: Contribution to journalArticleResearchpeer-review

10 Citations (Scopus)

Abstract

Let ω : (0, ∞) → (0, ∞) be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure μ ∈ ℝn is called ω-Zygmund if there exists a positive constant C such that |μ(Q+) - μ(Q-)| ≤ Cω(l(Q+))|Q+| for any pair Q+,Q- ⊂ ℝn of adjacent cubes of the same size. Similarly, μ is called an ω-symmetric measure if there exists a positive constant C such that μ(Q+)/μ(Q-) - 1| ≤ Cω(l(Q+)) for any pair Q+, Q- ℝn of adjacent cubes of the same size, ℓ(Q+) = ℓ(Q-) < 1. We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition ∫0omega;2(t)t-1dt < ∞ governs the existence of singular ω-Zygmund (ω-symmetric) measures. In the one-dimensional case, the results are well known, but complex analysis techniques are used at certain steps of the corresponding proofs.
Original languageEnglish
JournalAnnales de l'Institut Fourier
Volume52
Issue number1
DOIs
Publication statusPublished - 1 Jan 2002

Keywords

  • Doubling measures
  • Harmonic extensions
  • Quadratic condition
  • Zygmund measures

Fingerprint Dive into the research topics of 'Symmetric and zygmund measures in several variables'. Together they form a unique fingerprint.

  • Cite this