Fatou's theorem

If we have a holomorphic function $f$ defined on the open unit disk , it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius $r$. This defines a new function:

:$\backslash begin\{cases\}\; f\_r:S^1\; \backslash to\; \backslash Complex\; \backslash \backslash \; f\_\{r\}(e^\{i\backslash theta\})=f(re^\{i\backslash theta\})\; \backslash end\{cases\}$

where

:$S^1:=\backslash \{e^\{i\backslash theta\}:\backslash theta\backslash in[0,2\backslash pi]\backslash \}=\backslash \{z\backslash in\; \backslash Complex:|z|=1\backslash \},$

is the unit circle. Then it would be expected that the values of the extension of $f$ onto the circle should be the limit of these functions, and so the question reduces to determining when $f\_r$ converges, and in what sense, as $r\backslash to\; 1$, and how well defined is this limit. In particular, if the $L^p$ norms of these $f\_r$ are well behaved, we have an answer:

:Theorem. Let $f:\backslash mathbb\{D\}\backslash to\backslash Complex$ be a holomorphic function such that

::

:where $f\_r$ are defined as above. Then $f\_r$ converges to some function $f\_1\backslash in\; L^p(S^\{1\})$ pointwise almost everywhere and in $L^p$ norm. That is,

::$\backslash begin\{align\}$

\left |f_r(e^{i\theta})-f_{1}(e^{i\theta}) \right | &\to 0 && \text{for almost every } \theta\in [0,2\pi] \\

\|f_r-f_1\|_{L^p(S^1)} &\to 0

\end{align}

Now, notice that this pointwise limit is a radial limit. That is, the limit being taken is along a straight line from the center of the disk to the boundary of the circle, and the statement above hence says that

:$f(re^\{i\backslash theta\})\backslash to\; f\_1(e^\{i\backslash theta\})\; \backslash qquad\; \backslash text\{for\; almost\; every\; \}\; \backslash theta.$

The natural question is, with this boundary function defined, will we converge pointwise to this function by taking a limit in any other way? That is, suppose instead of following a straight line to the boundary, we follow an arbitrary curve $\backslash gamma:[0,1)\backslash to\; \backslash mathbb\{D\}$ converging to some point $e^\{i\backslash theta\}$ on the boundary. Will $f$ converge to $f\_\{1\}(e^\{i\backslash theta\})$? (Note that the above theorem is just the special case of $\backslash gamma(t)=te^\{i\backslash theta\}$). It turns out that the curve $\backslash gamma$ needs to be non-tangential, meaning that the curve does not approach its target on the boundary in a way that makes it tangent to the boundary of the circle. In other words, the range of $\backslash gamma$ must be contained in a wedge emanating from the limit point. We summarize as follows:

Definition. Let $\backslash gamma:[0,1)\backslash to\; \backslash mathbb\{D\}$ be a continuous path such that $\backslash lim\backslash nolimits\_\{t\backslash to\; 1\}\backslash gamma(t)=e^\{i\backslash theta\}\backslash in\; S^\{1\}$. Define

: $\backslash begin\{align\}$

\Gamma_\alpha &=\{z:\arg z\in [\pi-\alpha,\pi+\alpha]\} \\

\Gamma_\alpha(\theta) &=\mathbb{D}\cap e^{i\theta}(\Gamma_\alpha+1)

\end{align}

That is, $\backslash Gamma\_\backslash alpha(\backslash theta)$ is the wedge inside the disk with angle $2\backslash alpha$ whose axis passes between $e^\{i\backslash theta\}$ and zero. We say that $\backslash gamma$ converges non-tangentially to $e^\{i\backslash theta\}$, or that it is a non-tangential limit, if there exists such that $\backslash gamma$ is contained in $\backslash Gamma\_\backslash alpha(\backslash theta)$ and $\backslash lim\backslash nolimits\_\{t\backslash to\; 1\}\backslash gamma(t)=e^\{i\backslash theta\}$.

:Fatou's Theorem. Let $f\backslash in\; H^p(\backslash mathbb\{D\}).$ Then for almost all $\backslash theta\backslash in[0,2\backslash pi],$

::$\backslash lim\_\{t\backslash to\; 1\}f(\backslash gamma(t))=f\_1(e^\{i\backslash theta\})$

:for every non-tangential limit $\backslash gamma$ converging to $e^\{i\backslash theta\},$ where $f\_1$ is defined as above.

Here, $H^p(\backslash mathbb\{D\})$ is the Hardy space.

• The proof utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle.
• The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way.

• Hardy space

• John B. Garnett, Bounded Analytic Functions, (2006) Springer-Verlag, New York
• {{cite journal |doi=10.4310/AJM.2007.v11.n2.a2|title=The Boundary Behavior of Holomorphic Functions: Global and Local Results|year=2007|last1=Krantz|first1=Steven G.|journal=Asian Journal of Mathematics|volume=11|issue=2|pages=179–200|s2cid=56367819|doi-access=free|arxiv=math/0608650}}
• Walter Rudin. Real and Complex Analysis (1987), 3rd Ed., McGraw Hill, New York.
• Elias Stein, Singular integrals and differentiability properties of functions (1970), Princeton University Press, Princeton.

Category:Theorems in complex analysis