Non-Integrable Real Functions on the Unit Circle

In a previous paper [#!CAoRFIV!#] we showed that the correspondence between real functions and inner analytic functions established in [#!CAoRFI!#] can be extended to non-integrable real functions, so long as these functions are locally integrable almost everywhere, and so long as the non-integrable hard singularities of the corresponding inner analytic functions have finite degrees of hardness. The definition of local integrability almost everywhere on the unit circle is as follows.

Definition 2   : A real function $f(\theta)$ is locally integrable almost everywhere on the unit circle if it is integrable on every closed interval $[\theta_{\ominus},\theta_{\oplus}]$ contained within that domain, that does not contain any of the points where the function has non-integrable hard singularities, of which there is a finite number.

Although the construction used in this case, which is given in [#!CAoRFIV!#], is considerably more involved than the one for the case of integrable real functions, it is still true that given such a non-integrable real function one can define a unique inner analytic function $w(z)$ that corresponds to it, as well as a unique and complete set of complex Taylor coefficients $c_{0}=\alpha_{0}/2$ and $c_{k}=\alpha_{k}-\mbox{\boldmath$\imath$}\beta_{k}$, for $k\in\{1,2,3,\ldots,\infty\}$, and thus a corresponding unique and complete set of Fourier coefficients $\alpha_{0}$, $\alpha_{k}$ and $\beta_{k}$, for $k\in\{1,2,3,\ldots,\infty\}$, that are associated to it, despite the fact that the real function is not integrable. From the real part of this inner analytic function one can, once again, recover the real function almost everywhere by taking the $\rho\to 1_{(-)}$ limit to the unit circle. Therefore we have at hand all that we need in order to implement the proof of existence in this more general case.

In this section, using again our results from previous papers, we will establish the following theorem.

Theorem 2   : Given a real function $f(\theta)$ at the boundary of the unit disk, that satisfies the list of conditions described below, there is a solution $u(\rho,\theta)$ of the Dirichlet problem of the Laplace equation within the open unit disk, that assumes the values $f(\theta)$ almost everywhere at its boundary, the unit circle.

Proof 2.1   :

The argument is the same as the one used before for Theorem 1 in Section 2, in the case of integrable real functions, but using now the construction presented in [#!CAoRFIV!#], instead of the one presented in [#!CAoRFI!#]. Due to this, the only change with respect to that previous case is that our list of conditions on the real functions can now be upgraded to the following, still including the previous case.

1. The real function $f(\theta)$ is locally integrable almost everywhere on the unit circle, including the cases in which this function is globally integrable there.

2. The number of hard singularities of the corresponding inner analytic function $w(z)$ is finite.

3. The hard singularities of the corresponding inner analytic function $w(z)$ have finite degrees of hardness.

This completes the proof of Theorem 2.

In this way we have generalized the proof of existence of the Dirichlet problem from boundary conditions given by integrable real functions to others given by a certain class of non-integrable real functions. Note that the degree of hardness in the previous case, that of borderline hard singularities of integrable real functions, is simply zero. In other words, if all the hard singularities are borderline hard ones, then the function is simply integrable. Therefore, this theorem is a strict generalization of the previous one, and contains it.