---
tag: Q2N917JQ6A
kind: statement
role: theorem
name: Fatou's theorem
msc: [30H05, 31A20]
uses: [8G5ABXRK85, 0DG0Z72CQD, TCYSWCTYYM]
about: [8G5ABXRK85, TCYSWCTYYM]
sources:
  - url: https://en.wikipedia.org/wiki/Fatou%27s_theorem
same_as:
  wikidata: Q4455033
  wikipedia: Fatou's theorem
---

## Statement

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].
For $\theta \in \R$ and $\alpha > 1$, let $\Gamma_\alpha(\theta) = \set{z \in \mathbb{D} : \abs{z - e^{i\theta}} < \alpha (1 - \abs{z})}$ be the Stolz angle at $e^{i\theta}$.
Say that a function $u$ on $\mathbb{D}$ has nontangential limit $L$ at $e^{i\theta}$ if $u(z) \to L$ as $z \to e^{i\theta}$ inside $\Gamma_\alpha(\theta)$, for every $\alpha > 1$.

1. Let $\varphi$ be a Lebesgue integrable function on the circle ([[Lp space|SASP14MA1Z]] with $p = 1$), and let $u$ be its [[Poisson integral|TCYSWCTYYM]],
$$u(r e^{i\theta}) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1 - r^2}{1 - 2r \cos(\theta - t) + r^2} \, \varphi(e^{it}) \, dt.$$
Then for [[almost every|RARTMA0ZEC]] $\theta$, $u$ has nontangential limit $\varphi(e^{i\theta})$ at $e^{i\theta}$.
2. Let $f$ be a bounded [[holomorphic function|0DG0Z72CQD]] on $\mathbb{D}$.
Then for almost every $\theta$, $f$ has a nontangential limit at $e^{i\theta}$.
In particular the radial limit $\lim_{r \to 1^-} f(r e^{i\theta})$ exists for almost every $\theta$.

## Remarks

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

Example: the singular inner function $\exp\left(-\frac{1 + z}{1 - z}\right)$ is bounded and has radial limit of modulus $1$ at every point $e^{it} \ne 1$, while its radial limit at $1$ is $0$.

Boundedness cannot simply be dropped: there are holomorphic functions on $\mathbb{D}$ with a finite radial limit at almost no point.
Using the [[inner-outer factorization|X6ZQPAQF76]], part 2 extends to every function in the [[Hardy space|PNNNFJVF6W]] $H^p$, $0 < p \le \infty$.

## Explanations

### L1 human

A bounded holomorphic function on the disc may oscillate near the boundary, but not too wildly.
Fatou's theorem says that, approaching almost any point of the circle, the function settles down to a limit.
So the function has well defined boundary values, up to a set of length zero.
The same is true for the harmonic extension of any integrable function on the circle, and the limit is the function you started with.

### L2 human

Fatou's theorem is the reason Hardy spaces can be viewed as subspaces of $L^p$ of the circle.
It is used to define [[inner functions|NN8XH5RJYB]] and [[outer functions|KGJD4A4ATY]] 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.