ON BAIRE-1 4 FUNCTIONS

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 whic h are defined on them. For example, it is proved that a bounded functi on on a metric space K belongs to B-1/4(K) if and only if the omega(th )-oscillation, osc(omega) f, of f is bounded and in this case parallel to f parallel to(1/4) = parallel to \f\ + <(osc)over tilde>(omega) f parallel to(infinity). Also, we classify B-1/4(K) into a decreasing fa mily (S-xi(K))(1 less than or equal to xi<omega 1) of Banach spaces wh ose intersection is equal to D(K) and S-1(K) = B-1/4(K) These spaces a re characterized by spreading models of order xi equivalent to the sum ming basis of c(0), and for every function f in S-xi(K) it is valid th at osc(omega xi) f is bounded. Finally, using the notion of null-coeff icient of order xi sequence, we characterize the Baire-1 functions not belonging to S-xi(K).