On Baire-1/4 functions

Abstract:

We give descriptions of the spaces $D(K)$ (i.e. the space of differences of bounded semicontinuous functions on $K$) and especially of $B_{1/4}(K)$ (defined by Haydon, Odell and Rosenthal) as well as for the norms which are defined on them. For example, it is proved that a bounded function on a metric space $K$ belongs to $B_{1/4}(K)$ if and only if the $\omega ^{ \mathrm {th}}$-oscillation, $\mathrm {osc}_{\omega }f$, of $f$ is bounded and in this case $\| f\|_{1/4}=\| |f|+ \widetilde {\mathrm {osc}}_{\omega } f\|_{\infty }$. Also, we classify $B_{1/4}(K)$ into a decreasing family $(S_{\xi }(K))_{1\leq \xi <\omega _1}$ of Banach spaces whose intersection is equal to $D(K)$ and $S_1 (K)=B_{1/4}(K)$. These spaces are characterized by spreading models of order $\xi$ equivalent to the summing basis of $c_0$, and for every function $f$ in $S_{\xi }(K)$ it is valid that $\mathrm {os}_{\omega ^{\xi }}f$ is bounded. Finally, using the notion of null-coefficient of order $\xi$ sequence, we characterize the Baire-1 functions not belonging to $S_{\xi }(K)$.