The solution sets of extremal problems in 𝐻¹

Abstract

Let u u be an essentially bounded function on the unit circle T T . Let S u {S_u} denote the subset of the unit sphere of H 1 {H^1} on which the functional F ↦ ∫ 0 2 π u ¯ ( e i t ) F ( e i t ) d t / 2 π 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 {S_u} is given in terms of an inner function b 0 {b_0} and an outer fun tion g 0 {g_0} in H 2 {H^2} for which g 0 2 g_0^2 is an exposed point in the unit ball of H 1 {H^1} . An explicit description is given for the kernel of an arbitrary Toeplitz operator on H 2 {H^2} . The exposed points in H 1 {H^1} are characterized; an example is given of a strong outer function in H 1 {H^1} which is not exposed.