Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-26
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The Poisson kernel on the unit disc
Definition
For and , the Poisson kernel of the unit disc is
If with , then
Remarks
The second formula is just the first one written in polar coordinates. It is the form used in the one-variable estimates and in Harnack's inequality.
Used by
- A radial Poisson limit does not control a tangential path Counterexample
- The disc Poisson integral can miss the assigned value at a jump Counterexample
- Inner, singular inner and outer functions Definition
- The Poisson integral of a finite complex boundary measure Definition
- The Poisson integral on the unit disc Definition
- A boundary atom gives an h1 function without an L1 density Example
- A singular inner function generated by a point mass Example
- An outer function with a prescribed power of a vanishing modulus Example
- Poisson extension of an indicator arc Example
- The Poisson kernel realizes the sharp Harnack bounds on concentric discs Example
- The upper half-plane Poisson boundary density Example
- Analytic Poisson integrals are exactly the measures with vanishing negative coefficients Lemma
- Bounded holomorphic disc functions have Poisson boundary data and Fatou limits under countable choice Lemma
- Finite complex circle measures are determined by Fourier coefficients and Poisson integrals Lemma
- Poisson integrals are harmonic on the unit disc Lemma
- Poisson-Jensen inequality for Hardy functions Lemma
- Properties of outer functions Lemma
- Radial p-means of a holomorphic function are nondecreasing Lemma
- Singular circle measures have Poisson integral tending nontangentially to zero almost everywhere Lemma
- The Poisson kernel is positive, has total mass one, and concentrates at a boundary point Lemma
- Fatou limits for Poisson extensions of L1 boundary data Theorem
- Fatou's boundary theorem for analytic Hardy spaces Theorem
- h1 is isometric to finite regular complex boundary measures Theorem
- Poisson density of harmonic measure on a disc Theorem
- Poisson extension is an Lp contraction and converges in finite Lp Theorem
- Poisson nontangential maximal function is controlled by circle maximal averages Theorem
- Positive harmonic boundary measures and compact normalized families Theorem
- Properties of the singular functions S_μ Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Jeremy Orloff, MIT 18.04 Topic 5: Introduction to Harmonic Functions (standard reference, not scraped)
- Sigurdur Helgason, MIT 18.112 Lecture 16: Harmonic Functions (standard reference, not scraped)