The solution sets of extremal problems in $H^ 1$

Abstract:

Let $u$ be an essentially bounded function on the unit circle $T$. Let ${S_u}$ denote the subset of the unit sphere of ${H^1}$ on which the functional $F \mapsto \smallint _0^{2\pi }\bar u({e^{it}})F({e^{it}})dt/2\pi$ attains its norm. A complete description of ${S_u}$ is given in terms of an inner function ${b_0}$ and an outer fun tion ${g_0}$ in ${H^2}$ for which $g_0^2$ is an exposed point in the unit ball of ${H^1}$. An explicit description is given for the kernel of an arbitrary Toeplitz operator on ${H^2}$. The exposed points in ${H^1}$ are characterized; an example is given of a strong outer function in ${H^1}$ which is not exposed.