A boundary value problem for $H^{\infty }(D)$

Abstract:

Let $W = \bigcup \nolimits _{n = 1}^\infty {{W_n}}$ be an ${F_\sigma }$ subset of the unit circle of measure 0 and let $\{ {q_n}\} ,n \geqslant 1$, be a decreasing sequence with ${q_1} \leqslant 1$ and ${\lim _n}{q_n} = 0$. There exists an H in ${H^\infty }(D)$ of norm ${q_1}$ whose modulus has radial limit along every radius which has radial limit of modulus ${q_1}$ on ${W_1}$ and ${q_{n + 1}}$ on ${W_{n + 1}}\backslash \bigcup \nolimits _{k = 1}^n {{W_k}}$. If W is simultaneously a ${G_\delta }$ set, H may be chosen to have no zeros on C. It follows that for W countable, say $W = \{ {e^{i{w_n}}}\} ,n \geqslant 1$, there is such an H of norm 1 for which ${\lim _{r \to 1}}H(r{e^{i{w_n}}}) = 1/n$. The proof of the theorem depends on the existence of a special collection of closed sets $\{ {S_\lambda }\} ,\lambda \geqslant 1$, real, for which the function h, defined by $h(x) = {a_n} + [(\inf \{ \lambda |x \in S\} ) - n]({a_{n + 1}} - {a_n}),{a_n} = - \ln \;{q_n}$, is such that the function $H(w) = \exp ( - 1/2\pi )\smallint [(w + {e^{iu}})/({e^{iu}} - w)]h(u)$ du has the required properties. Some of the techniques used are similar to those developed in an earlier paper [1].