On bounded functions satisfying averaging conditions. I

Abstract

Let $R(T)$ be the space of real valued ${L^\infty }$ functions defined on the unit circle $C$ consisting of those functions $f$ for which $li{m_{h \to 0}}(1/h)\int _\theta ^{\theta + h} {f({e^{it}})dt = f({e^{i\theta }})}$ for every ${e^{i\theta }}$ in $C$. The extreme points of the unit ball of $R(T)$ are found and the extreme points of the unit ball of the space of all bounded harmonic functions in the unit disc which have non-tangential limit at each point of the unit circle are characterized. We show that if $g$ is a real valued function in ${L^\infty }(C)$ and if $K$ is a closed subset of $\{ {e^{i\theta }}|li{m_{h \to 0}}(1/h)\int _\theta ^{\theta + h} {g({e^{it}})dt = g({e^{i\theta }})\} }$, then there is a function in $R(T)$ whose restriction to $K$ is $g$. If $E$ is a ${G_\delta }$ subset of $C$ of measure 0 and if $F$ is a closed subset of $C$ disjoint from $E$, there is a function of norm 1 in $R(T)$ which is on $E$ and 1 on $F$. Finally, we show that if $E$ and $F$ are as in the preceding result, then there is a function of norm 1 in ${H^\infty }$ (unit disc) the modulus of which has radial limit along every radius, which has radial limit of modulus 1 at each point of $F$ and radial limit 0 at each point of $E$.