- Carleson measure
In
mathematics , a Carleson measure is a type of measure onsubset s of "n"-dimension alEuclidean 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 thesurface measure on the boundary of Ω.Carleson measures have many applications in
harmonic analysis and the theory ofpartial differential equations , for instance in the solution of Dirichlet problems with "rough" boundary. The Carleson condition is closely related to the boundedness of thePoisson operator . Carleson measures are named after the Swedishmathematician 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 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{B}_{r} (p) ight) geq C sigma left( partial Omega cap mathbb{B}_{r} (p) ight),
where
:mathbb{B}_{r} (p) := left{ x in mathbb{R}^{n} left| | x - p |_{mathbb{R}^{n < r ight. ight}
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 theHardy 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^{p} (partial D) o L^{p} (D, mu)
by
: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.
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{B}_{r} (p) ight) geq C sigma left( partial Omega cap mathbb{B}_{r} (p) ight)
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) ight) geq C sigma left( partial Omega cap mathbb{B}_{r} (p) ight)
holds, then the measure "μ" is said to satisfy the vanishing Carleson condition if "C"("R") → 0 as "R" → 0.
References
* cite journal
author = Carleson, Lennart
title = Interpolations by bounded analytic functions and the corona problem
journal = Ann. of Math. (2)
volume = 76
year = 1962
pages = 547–559
issn = 0003-486X
doi = 10.2307/1970375External links
* springer
author = Mortini, R.
id = c120050
title = Carleson measure
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