Carleson measure

The Carleson condition is closely related to the boundedness of the Poisson operator.

Let n ∈ N and let Ω ⊂ Rn be an open (and hence measurable) set with non-empty boundary ∂Ω.

The measure μ is said to be a Carleson measure if there exists a constant C > 0 such that, for every point p ∈ ∂Ω and every radius r > 0, where denotes the open ball of radius r about p. Let D denote the unit disc in the complex plane C, equipped with some Borel measure μ.

For 1 ≤ p < +∞, let Hp(∂D) denote the Hardy space on the boundary of D and let Lp(D, μ) denote the Lp space on D with respect to the measure μ.

If C(R) is defined to be the infimum of the set of all constants C > 0 for which the restricted Carleson condition holds, then the measure μ is said to satisfy the vanishing Carleson condition if C(R) → 0 as R → 0.