Continuous boundary values

In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows.

• Suppose that f is a continuous complex-valued function on the complex plane that is holomorphic on the upper half-plane, and on the lower half-plane. Then it is holomorphic everywhere.

In this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the real axis. This result can be proved from Morera's theorem. Indeed, a function is holomorphic provided its integral round any contour vanishes; a contour which crosses the real axis can be broken up into contours in the upper and lower half-planes and the integral round these vanishes by hypothesis.^[5][6]

Distributional boundary values on a circle

The more general case is phrased in terms of distributions.^[7][8] This is technically simplest in the case where the common boundary is the unit circle ${\displaystyle |z|=1}$ in the complex plane. In that case holomorphic functions f, g in the regions ${\displaystyle r<|z|<1}$ and ${\displaystyle 1<|z|<R}$ have Laurent expansions

${\displaystyle f(z)=\sum _{-\infty }^{\infty }a_{n}z^{n},\,\,\,\,g(z)=\sum _{-\infty }^{\infty }b_{n}z^{n}}$

absolutely convergent in the same regions and have distributional boundary values given by the formal Fourier series

${\displaystyle f(\theta )=\sum _{-\infty }^{\infty }a_{n}e^{in\theta },\,\,\,\,g(\theta )=\sum _{-\infty }^{\infty }b_{n}e^{in\theta }.}$

Their distributional boundary values are equal if ${\displaystyle a_{n}=b_{n}}$ for all n. It is then elementary that the common Laurent series converges absolutely in the whole region ${\displaystyle r<|z|<R}$.

Distributional boundary values on an interval

In general given an open interval ${\displaystyle I=(a,b)}$ on the real axis and holomorphic functions ${\displaystyle f_{+},\,\,\ f_{-}}$ defined in ${\displaystyle (a,b)\times (0,R)}$ and ${\displaystyle (a,b)\times (-R,0)}$ satisfying

${\displaystyle |f_{\pm }(x+iy)|<C|y|^{-N}}$

for some non-negative integer N, the boundary values ${\displaystyle T_{\pm }}$ of ${\displaystyle f_{\pm }}$ can be defined as distributions on the real axis by the formulas^[9][8]

${\displaystyle \langle T_{\pm },\varphi \rangle =\lim _{\varepsilon \downarrow 0}\int f(x\pm i\varepsilon )\varphi (x)\,dx.}$

Existence can be proved by noting that, under the hypothesis, ${\displaystyle f_{\pm }(z)}$ is the ${\displaystyle (N+1)}$-th complex derivative of a holomorphic function which extends to a continuous function on the boundary. If f is defined as ${\displaystyle f_{\pm }}$ above and below the real axis and F is the distribution defined on the rectangle ${\displaystyle (a,b)\times (-R,R)}$ by the formula

${\displaystyle \langle F,\varphi \rangle =\iint f(x+iy)\varphi (x,y)\,dx\,dy,}$

then F equals ${\displaystyle f_{\pm }}$ off the real axis and the distribution ${\displaystyle F_{\overline {z}}}$ is induced by the distribution ${\textstyle {1 \over 2}(T_{+}-T_{-})}$ on the real axis.

In particular if the hypotheses of the edge-of-the-wedge theorem apply, i.e. ${\displaystyle T_{+}=T_{-}}$, then

${\displaystyle F_{\overline {z}}=0.}$

By elliptic regularity it then follows that the function F is holomorphic in ${\displaystyle (a,b)\times (-R,R)}$.

In this case elliptic regularity can be deduced directly from the fact that ${\displaystyle (\pi z)^{-1}}$ is known to provide a fundamental solution for the Cauchy–Riemann operator ${\displaystyle \partial /\partial {\overline {z}}}$.^[10]

Using the Cayley transform between the circle and the real line, this argument can be rephrased in a standard way in terms of Fourier series and Sobolev spaces on the circle. Indeed, let ${\displaystyle f}$ and ${\displaystyle g}$ be holomorphic functions defined exterior and interior to some arc on the unit circle such that locally they have radial limits in some Sobolev space, Then, letting

${\displaystyle D=z{\partial \over \partial z},}$

the equations

${\displaystyle D^{k}F=f,\,\,\,D^{k}G=g}$

can be solved locally in such a way that the radial limits of G and F tend locally to the same function in a higher Sobolev space. For k large enough, this convergence is uniform by the Sobolev embedding theorem. By the argument for continuous functions, F and G therefore patch to give a holomorphic function near the arc and hence so do f and g.