Symmetric and zygmund measures in several variables

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.