Carleson measure

From Wikipedia, the free encyclopedia

In mathematics, a Carleson measure is a type of measure on subsets of $n$ -dimensional Euclidean space. Roughly speaking, a Carleson measure on a domain $Ω$ is a measure that does not vanish at the boundary of $Ω$ when compared to the surface measure on the boundary.

Carleson measures have many applications in harmonic analysis and the theory of partial differential equations, for instance in the solution of Dirichlet problems with "rough" boundary. Carleson measures are named after the Swedish mathematician Lennart Carleson.

[edit] Definition

Let $n \in \mathbb{N}$ and let $\Omega \subsetneq \mathbb{R}^{n}$ be an open set with boundary $\partial \Omega$ . Let $μ$ be a Borel measure on $Ω$ , and let $σ$ denote the corresponding surface measure. The measure $μ$ is said to be a Carleson measure if there exists a constant $C > 0$ such that, for every point $p \in \partial \Omega$ and every radius $r > 0$ ,

$\mu \left( \Omega \cap \mathbb{B}_{r} (p) \right) \geq C \sigma \left( \partial \Omega \cap \mathbb{B}_{r} (p) \right),$

where

$\mathbb{B}_{r} (p) := \left\{ x \in \mathbb{R}^{n} \left| \| x - p \|_{\mathbb{R}^{n}} < r \right. \right\}$

denotes the open ball of radius $r$ about $p$ .

[edit] Related concepts

The infimum of the set of constants $C > 0$ for which the Carleson condition

$\forall r > 0, \forall p \in \partial \Omega, \mu \left( \Omega \cap \mathbb{B}_{r} (p) \right) \geq C \sigma \left( \partial \Omega \cap \mathbb{B}_{r} (p) \right)$

holds is known as the Carleson norm of 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

$\forall r \in (0, R), \forall p \in \partial \Omega, \mu \left( \Omega \cap \mathbb{B}_{r} (p) \right) \geq C \sigma \left( \partial \Omega \cap \mathbb{B}_{r} (p) \right)$