Math 2.0

Functions of a complex variable Function spaces and the unit disc Theorem

Fatou's theorem

Builds on Unit discHolomorphic functionPoisson kernel

Statement

Let 𝔻\mathbb{D} be the open unit disc. For θ∈ℝ\theta \in \R and α>1\alpha > 1, let Γα(θ)={z∈𝔻:∣z−eiθ∣<α(1−∣z∣)}\Gamma_\alpha(\theta) = \set{z \in \mathbb{D} : \abs{z - e^{i\theta}} < \alpha (1 - \abs{z})} be the Stolz angle at eiθe^{i\theta}. Say that a function uu on 𝔻\mathbb{D} has nontangential limit LL at eiθe^{i\theta} if u(z)→Lu(z) \to L as z→eiθz \to e^{i\theta} inside Γα(θ)\Gamma_\alpha(\theta), for every α>1\alpha > 1.

  1. Let φ\varphi be a Lebesgue integrable function on the circle (Lp space with p=1p = 1), and let uu be its Poisson integral, u(reiθ)=12π∫02π1−r21−2rcos⁡(θ−t)+r2 φ(eit) dt.u(r e^{i\theta}) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1 - r^2}{1 - 2r \cos(\theta - t) + r^2} \, \varphi(e^{it}) \, dt. Then for almost every θ\theta, uu has nontangential limit φ(eiθ)\varphi(e^{i\theta}) at eiθe^{i\theta}.
  2. Let ff be a bounded holomorphic function on 𝔻\mathbb{D}. Then for almost every θ\theta, ff has a nontangential limit at eiθe^{i\theta}. In particular the radial limit lim⁡r→1−f(reiθ)\lim_{r \to 1^-} f(r e^{i\theta}) exists for almost every θ\theta.

Remarks

Proof idea: the nontangential maximal function of uu is bounded by a constant times the Hardy–Littlewood maximal function of φ\varphi, which is finite almost everywhere. The limit holds for continuous φ\varphi, and the maximal estimate extends it to all integrable φ\varphi. For part 2, the functions f(rkeit)f(r_k e^{it}) have a weak-star limit φ\varphi in L∞L^\infty along some sequence rk→1r_k \to 1, and ff is the Poisson integral of φ\varphi, so part 1 applies.

Example: the singular inner function exp⁡(−1+z1−z)\exp\left(-\frac{1 + z}{1 - z}\right) is bounded and has radial limit of modulus 11 at every point eit≠1e^{it} \ne 1, while its radial limit at 11 is 00.

Boundedness cannot simply be dropped: there are holomorphic functions on 𝔻\mathbb{D} with a finite radial limit at almost no point. Using the inner-outer factorization, part 2 extends to every function in the Hardy space HpH^p, 0<p≤∞0 < p \le \infty.

Going deeper

For a student

Fatou's theorem is the reason Hardy spaces can be viewed as subspaces of LpL^p of the circle. It is used to define inner functions and outer functions through their boundary values. The approach must be nontangential: Littlewood showed that limits along curves tangent to the circle can fail at almost every point. The proof is a model case of the maximal function method in harmonic analysis.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectPoisson kernel, concept, no evidence levelPoisson kernelconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionUnit disc, object, no evidence levelUnit discobjectReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionFatou's theorem, statement, no evidence levelFatou's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complex numbers object
  2. Poisson kernel concept
  3. Real numbers object
  4. Set bounded above definition
  5. Unit disc object
  6. Real and imaginary parts definition
  7. Supremum definition
  8. Completeness axiom statement
  9. Existence of square roots of nonnegative reals statement
  10. Square root definition
  11. Modulus of a complex number definition
  12. Complex differentiability definition
  13. Holomorphic function definition
  14. Fatou's theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fatou%27s_theorem
About this page
TagQ2N917JQ6A
URIm2://Q2N917JQ6A
Kindstatement, theorem
WritingW0 stub
MSC30H05, 31A20
StatusThis copy of the site has no build data. Run m2 status Q2N917JQ6A.
AboutUnit disc, Poisson kernel
UsesUnit disc, Holomorphic function, Poisson kernel
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Changed2 November 2026

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