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| In [[mathematics]], a '''Carleson measure''' is a type of [[measure (mathematics)|measure]] on [[subset]]s of ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>. Roughly speaking, a Carleson measure on a domain Ω is a measure that does not vanish at the [[boundary (topology)|boundary]] of Ω when compared to the [[surface measure]] on the [[boundary (topology)|boundary]] of Ω.
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| Carleson measures have many applications in [[harmonic analysis]] and the theory of [[partial differential equations]], for instance in the solution of [[Dirichlet problem]]s with "rough" boundary. The Carleson condition is closely related to the [[bounded linear operator|boundedness]] of the [[Poisson kernel|Poisson operator]]. Carleson measures are named after the [[Sweden|Swedish]] [[mathematician]] [[Lennart Carleson]].
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| ==Definition==
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| Let ''n'' ∈ '''N''' and let Ω ⊂ '''R'''<sup>''n''</sup> be an [[open set|open]] (and hence [[measurable set|measurable]]) set with non-empty boundary ∂Ω. Let ''μ'' be a [[Borel measure]] 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,
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| :<math>\mu \left( \Omega \cap \mathbb{B}_{r} (p) \right) \leq C \sigma \left( \partial \Omega \cap \mathbb{B}_{r} (p) \right),</math> | |
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| where
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| :<math>\mathbb{B}_{r} (p) := \left\{ x \in \mathbb{R}^{n} \left| \| x - p \|_{\mathbb{R}^{n}} < r \right. \right\}</math>
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| denotes the [[open ball]] of radius ''r'' about ''p''.
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| ==Carleson's theorem on the Poisson operator==
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| Let ''D'' denote the [[unit disc]] in the complex plane '''C''', equipped with some Borel measure ''μ''. For 1 ≤ ''p'' < +∞, let ''H''<sup>''p''</sup>(∂''D'') denote the [[Hardy space]] on the boundary of ''D'' and let ''L''<sup>''p''</sup>(''D'', ''μ'') denote the [[Lp space|''L''<sup>''p''</sup> space]] on ''D'' with respect to the measure ''μ''. Define the Poisson operator
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| :<math>P : H^{p} (\partial D) \to L^{p} (D, \mu)</math>
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| by
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| :<math>P(f) (z) = \frac{1}{2 \pi} \int_{0}^{2 \pi} \mathrm{Re} \frac{e^{i t} + z}{e^{i t} - z} f(e^{i t}) \, \mathrm{d} t.</math> | |
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| Then ''P'' is a bounded linear operator [[if and only if]] the measure ''μ'' is Carleson.
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| ==Other related concepts==
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| The [[infimum]] of the set of constants ''C'' > 0 for which the Carleson condition
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| :<math>\forall r > 0, \forall p \in \partial \Omega, \mu \left( \Omega \cap \mathbb{B}_{r} (p) \right) \leq C \sigma \left( \partial \Omega \cap \mathbb{B}_{r} (p) \right)</math>
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| holds is known as the '''Carleson norm''' of the measure ''μ''.
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| If ''C''(''R'') is defined to be the infimum of the set of all constants ''C'' > 0 for which the restricted Carleson condition
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| :<math>\forall r \in (0, R), \forall p \in \partial \Omega, \mu \left( \Omega \cap \mathbb{B}_{r} (p) \right) \leq C \sigma \left( \partial \Omega \cap \mathbb{B}_{r} (p) \right)</math>
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| holds, then the measure ''μ'' is said to satisfy the '''vanishing Carleson condition''' if ''C''(''R'') → 0 as ''R'' → 0.
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| ==References==
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| * {{cite journal
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| | author = Carleson, Lennart
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| | title = Interpolations by bounded analytic functions and the corona problem
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| | journal = [[Annals of Mathematics|Ann. of Math.]]
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| | volume = 76
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| | issue = 3
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| | year = 1962
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| | pages = 547–559
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| | doi = 10.2307/1970375
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| | jstor = 1970375
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| }}
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| ==External links==
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| * {{springer
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| |author = Mortini, R.
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| |id = c120050
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| |title = Carleson measure
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| }}
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| [[Category:Measures (measure theory)]]
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| [[Category:Norms (mathematics)]]
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Wilber Berryhill is what his wife loves to call him and he totally loves this name. Distributing manufacturing has been his occupation for some time. Mississippi is exactly where his home is. To climb is something she would by no means give up.
Also visit my webpage psychic phone readings (http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews)