mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Carleson measure is a type of measure on

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of ''n''- dimensional Euclidean space R^''n''. 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 A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

on the boundary of Ω. Carleson measures have many applications in

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

and the theory of partial differential equations, for instance in the solution of Dirichlet problems with "rough" boundary. The Carleson condition is closely related to the boundedness of the Poisson operator. Carleson measures are named after the Swedish mathematician Lennart Carleson.

Definition

Let ''n'' ∈ N and let Ω ⊂ R^''n'' be an open (and hence measurable) set with non-empty boundary ∂Ω. Let ''μ'' be a

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. F ...

on Ω, and let ''σ'' denote the surface measure on ∂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, :

\mu \left( \Omega \cap \mathbb_ (p) \right) \leq C \sigma \left( \partial \Omega \cap \mathbb_ (p) \right),

where :

\mathbb_ (p) := \left\

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

Carleson's theorem on the Poisson operator

Let ''D'' denote the unit disc in the complex plane C, equipped with some Borel measure ''μ''. For 1 ≤ ''p'' < +∞, let ''H''^''p''(∂''D'') denote the Hardy space on the boundary of ''D'' and let ''L''^''p''(''D'', ''μ'') denote the ''L''^''p'' space on ''D'' with respect to the measure ''μ''. Define the Poisson operator :

P : H^ (\partial D) \to L^ (D, \mu)

by :

P(f) (z) = \frac \int_^ \mathrm \frac f(e^) \, \mathrm t.

Then ''P'' is a bounded linear operator if and only if the measure ''μ'' is Carleson.

Other 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_ (p) \right) \leq C \sigma \left( \partial \Omega \cap \mathbb_ (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_ (p) \right) \leq C \sigma \left( \partial \Omega \cap \mathbb_ (p) \right)

holds, then the measure ''μ'' is said to satisfy the vanishing Carleson condition if ''C''(''R'') → 0 as ''R'' → 0.

References

External links

* {{springer , author = Mortini, R. , id = http://www.encyclopediaofmath.org/index.php?title=Carleson_measure&oldid=14223 , title = Carleson measure Measures (measure theory) Norms (mathematics)